Scale: Understanding the Environment 303115732X, 9783031157325

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Cristian Suteanu

Scale

Understanding the Environment

Scale

Cristian Suteanu

Scale Understanding the Environment

Cristian Suteanu Department of Geography and Environmental Studies Department of Environmental Science Saint Mary's University Halifax, NS, Canada

ISBN 978-3-031-15732-5    ISBN 978-3-031-15733-2 (eBook) https://doi.org/10.1007/978-3-031-15733-2 © Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my wife, Mirela

Preface

Scale can be likened to a beehive. On the outside, it is a wooden box standing in a meadow. Inside, however, it is bursting with life of complexity that is difficult to grasp  – even if you remove its roof and look inside at its inner workings. More importantly, scale is unfailingly present when we relate to our environment, no matter how broadly we may define the concept of environment. These two key aspects of scale represent the main threads that run through the book. It was from their perspective that the selection of the topics was made. While the number of themes related to scale can be literally endless, having two sharp criteria at work was very helpful. In summary, the book emphasizes the understanding of scale, especially regarding the way in which scale helps us to understand the environment. These two lines of inquiry complement and support each other. Given the nature of the subject, the book is not intended to be a collection of facts and images put together after one’s journey, for visitors to see. It is meant to consist of shorter and longer trips, which the reader is invited to join: In this spirit, the author chose to walk the walk and actually take these trips anew, rather than relying on memories from the landscape. The continuous process of understanding presupposes an active involvement with questions, with problems, and readers are encouraged to confront the encountered vistas from various angles. In this context, the views of major figures in our intellectual history are sometimes mentioned: not because they would hold definitive answers, but because seeing the world from their point of view can offer the traveler a privileged experience. It can expand one’s capacity to embrace reality in more than one way and enhance one’s power of discovering new connections and new paths in the knowledge space. This is particularly true when those tall peaks bear names such as Locke, Kant, or Husserl. Unlike many concepts that can be found at the intersection of different fields, scale is right at the core of a wide variety of disciplines. The book does not intend to mention them all, but it points to this diversity throughout its inquiry on the essence of scale. For example, it refers to physical geography and to geophysics, but also to art theory, literature, and human geography. There are two main reasons for this choice. On the one hand, comprehension of a concept is better supported by a vii

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diversity of perspectives. On the other hand, by following paths of inquiry and problem solving in other fields than one’s own, one can be stimulated and inspired to find new approaches for problems in one’s area of interest. At the same time, while pursuing its main objective – to support a deeper understanding of scale and its role in our understanding of the environment – the book only briefly mentions some of the most often treaded paths. This is mainly the case for topics that are described at length in numerous books, including some intended for the undergraduate level. For instance, one may expect cartography to take up most of the book, and yet it does not. One can think of a monument sitting in the middle of a town’s central square: as it is often visited and photographed by everyone, a monograph could be entirely dedicated to it. Alternatively, a monograph may only refer to some extent to the monument, while also exploring other landmarks and artifacts, near and far, which are related to that monument in terms of symbolic meanings, means of representation, historical developments, etc.: eventually, the amount of space dedicated to the monument per se may be unexpectedly low, given its prominent presence in the town square. And yet, the study of the monument’s meaning and importance can be deepened and enhanced by this latter approach. It is a well-known fact that in some cases it is harder to explain simple things than complicated things. Complicated subjects can usually be described in terms of simpler ones. However, basic concepts can be difficult to define. Moreover, especially in one’s early stages of scholarly pursuits, after the first successful steps in a certain area, one may perceive the domain as being so clear that it becomes almost transparent. “As owls’ eyes are at noonday, so is our mental vision blind to what in its own nature is the most evident of all,” warns us Aristotle. There is not much one can do about this natural tendency. However, it might help to be aware of it. This is the reason why in this book we turn from time to time to the foundations. For example, when we talk about scale as ratio, we present the underlying framework: affine transformations. When we look at scale as rank, we open up the basics of the theory of categories. After all, from an applicative point of view, much of what is needed about scale seems to be achievable in 10 minutes of theory and another 50 minutes of practice. In terms of the above metaphor, this is almost equivalent to merely noticing that a beehive is present. Learning about its richness and about its role in our lives is a different matter. If the book will open a window into the buzzing world of the hive, and offer some notions of a language we can use to interact with the hive and with the environment, it will have accomplished its goal. Halifax, NS, Canada  Cristian Suteanu

Understanding is, after all, what science is all about. Roger Penrose, Shadows of the Mind

ix

Acknowledgments

My thanks radiate in many directions, and I am aware that I couldn’t possibly thank everyone who helped me on the path to this book. However, some forms of help stand out. My wife Mirela has a major contribution to this endeavor, from lively discussions of ideas to the immeasurable support she has been offering me at every step. Compared to the extent of her help, my thanks can only be perceived on a small, very small scale. My special thanks also go to my daughters Maria Cristina and Oana Monica for their constant encouragement and for thoughtful comments on various parts of the text. I am particularly thankful to Mamina for her relentless care and support. I am deeply grateful to my parents, who were passionate professors and role models of both scientific scholarship and family commitment. My scientific career owes much to Florin Munteanu, who was my friend and mentor from the time when he introduced me to fractal theory and dynamic systems as an undergrad student in the early 1980s all the way through my graduate studies. I was privileged to be a long-time member of his interdisciplinary research group on nonlinear science, and to work with brilliant colleagues and valued friends like Cristian Ioana and Edmond Cretu, at the Romanian Academy’s Sabba S. Stefanescu Institute of Geodynamics:  this was a remarkable research institution due to the design and leadership of Dorel Zugravescu – a passionate scholar to whom I am grateful for the relentless invaluable support and the creation of an outstanding environment for young scholars to thrive. My thanks also go to Crisan Demetrescu, an esteemed collaborator and friend for many years, and my former colleagues and friends at the Institute of Geodynamics of the Romanian Academy. It is also a pleasure to thank Mircea Rusu for his superb sessions on nonlinear physics and his mentorship. I am especially grateful to Joern Kruhl, distinguished scholar and lifelong friend and collaborator, for his continuous support and collaboration over the years. Deep thanks go to David Deutsch for his far-reaching advice and the thought-­ provoking discussions. The nonlinear science community has been very important to me. In particular, I would like to thank Shaun Lovejoy for his generous support and for fruitful discussions; I am also grateful for useful insights from Armin Bunde and Don Turcotte. At the same time, I would like to give special thanks to Bob McCalla for his genuine interest in my research and for his important support along xi

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the way. My thanks also go to Tony Charles for bibliographic help in his field of research, and to all my colleagues and friends at Saint Mary’s University who have been supportive of my work. Furthermore, I would like to thank Malcolm Bott, Eric Gaba, Gilles Messian, and Ralf Roletschek for kindly permitting the inclusion of their photographs in this book. I would also like to acknowledge permission from the National Aeronautics and Space Administration (NASA), the National Oceanic and Atmospheric Administration (NOAA), the United States Fish and Wildlife Service (USFWS), the United States Geological Survey (USGS), and the Centers for Disease Control and Prevention (CDC) to include their illustrative material in the book. Last, but not least, I would like to thank the team at Springer, in particular Zach Romano, editor, who stimulated me to write this book in the first place and has always been supportive, and Zoe Kennedy, who has been constantly helpful throughout this book project.

Contents

1

 The Meanings of Scale ����������������������������������������������������������������������������    1 1.1 Scale: Pervasive and Indispensable��������������������������������������������������    1 1.2 A Daughter of Abstraction����������������������������������������������������������������    3 1.3 The Scale’s Fields of Operation��������������������������������������������������������   10 1.4 Types of Scale ����������������������������������������������������������������������������������   16 1.4.1 Grouping the Scales��������������������������������������������������������������   16 1.4.2 Scale as Size��������������������������������������������������������������������������   18 1.4.3 Scale as Ratio������������������������������������������������������������������������   19 1.4.4 Scale as Rank������������������������������������������������������������������������   22 1.5 Understanding Scale ������������������������������������������������������������������������   23 References��������������������������������������������������������������������������������������������������   25

2

 Scale as Size in Space������������������������������������������������������������������������������   27 2.1 Setting the Stage: The System Image ����������������������������������������������   27 2.2 Scale as Size: Comparison and Change��������������������������������������������   32 2.2.1 Comparing Sizes ������������������������������������������������������������������   32 2.2.2 Size Versus Distance Change in Perspective������������������������   36 2.2.3 Size Distortion, Accuracy, and Implications������������������������   39 2.3 Orders of Magnitude ������������������������������������������������������������������������   41 2.4 Sub-Types of Scale as Size ��������������������������������������������������������������   42 2.4.1 Scale as Size with Units��������������������������������������������������������   43 2.4.2 Scale as Size Without Units��������������������������������������������������   46 2.4.3 Scale and Hierarchy��������������������������������������������������������������   49 2.5 Scale on Trial������������������������������������������������������������������������������������   51 References��������������������������������������������������������������������������������������������������   54

3

 Scale as Size in Time and in Space-Time ����������������������������������������������   57 3.1 Temporal Scale Versus Spatial Scale: What Is Different?����������������   57 3.2 Converting Size in Space to Size in Time����������������������������������������   60 3.3 Time Scales and Records of Time����������������������������������������������������   61 3.4 Scale as Size and the Granularity of Time Records��������������������������   63 3.5 Time Scales and Environmental Processes ��������������������������������������   69 xiii

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3.5.1 Scale Ranges, Accumulation, and Dissipation����������������������   69 3.5.2 Time Scales and the Renewability of Resources������������������   71 3.5.3 Scale and Equilibrium����������������������������������������������������������   73 3.6 Synchronic and Diachronic Perspectives������������������������������������������   76 3.7 The Time Scale Bias ������������������������������������������������������������������������   79 3.8 Scaling Up and Scaling Down����������������������������������������������������������   81 3.8.1 Dimension and Changes in Scale������������������������������������������   81 3.8.2 Downscaling in Climate Change Studies������������������������������   84 3.9 Scale as Size, Context, and Scale Change����������������������������������������   85 References��������������������������������������������������������������������������������������������������   90 4

 Scale as Ratio in Space����������������������������������������������������������������������������   93 4.1 Scale as Ratio in Space and Proportional Change����������������������������   93 4.2 The Basis of Scale as Ratio: Geometric Transformations����������������   94 4.3 Scaling in Two and Three Dimensions ��������������������������������������������   97 4.4 Scale and Maps ��������������������������������������������������������������������������������   99 4.4.1 Scale, Maps, and Discovery��������������������������������������������������   99 4.4.2 Specifying the Scale of Maps ����������������������������������������������  101 4.4.3 Precision and Accuracy in Scale as Ratio ����������������������������  105 4.4.4 Scale and the Environment ��������������������������������������������������  107 4.4.5 Map Projections and the Spatial Variation of Scale��������������  109 4.5 Scale in the Digital World ����������������������������������������������������������������  113 4.5.1 A New Environment ������������������������������������������������������������  113 4.5.2 Scale, Similarity, and Automated Map Generalization��������  115 4.5.3 Feature-Guided Exploration of a Spatial Database��������������  117 4.5.4 Making Use of the Spatial Informational Backbone������������  119 4.6 The Scale-Dependent View of Reality����������������������������������������������  121 References��������������������������������������������������������������������������������������������������  122

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 Scale as Ratio in Time������������������������������������������������������������������������������  125 5.1 Premises of Scale Change in Time���������������������������������������������������  125 5.2 Time Series and Time Scales������������������������������������������������������������  126 5.3 Simple Transformations in Time������������������������������������������������������  130 5.4 Event Density and Time Scale Change��������������������������������������������  132 5.5 Waves������������������������������������������������������������������������������������������������  133 5.6 Waves and Time Scale Change ��������������������������������������������������������  135 5.7 Time Domain and Frequency Domain����������������������������������������������  138 5.8 Applications of Scale in Time����������������������������������������������������������  141 5.8.1 Application Areas of Temporal Scale ����������������������������������  141 5.8.2 Perception and Temporal Scale��������������������������������������������  142 5.8.3 Time Scale and Visual Information��������������������������������������  143 5.8.4 Time Scale Change Involving a Variable Scale Factor ��������  146 5.8.5 Time Scale and Auditory Information: Sonification ������������  149 5.9 Subjective Time Scale and Learning About the Environment����������  152 5.9.1 Change and Subjective Time: The Old Masters��������������������  152

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5.9.2 Change and Subjective Time: Today’s Science��������������������  153 5.9.3 Large-Scale Time Compression��������������������������������������������  156 5.10 Time Scale in Reflection and Narration��������������������������������������������  159 References��������������������������������������������������������������������������������������������������  161 6

Scale as Rank��������������������������������������������������������������������������������������������  165 6.1 Introducing Scale as Rank����������������������������������������������������������������  165 6.2 Scale as Rank at Work����������������������������������������������������������������������  167 6.3 Scale as Rank and Categorization����������������������������������������������������  173 6.3.1 The Power of Categories������������������������������������������������������  173 6.3.2 Categorization: Basic Principles ������������������������������������������  175 6.3.3 Positive and Negative Effects of Categorization������������������  176 6.3.4 Scale as Rank Versus Changes in Categorization ����������������  178 6.3.5 Scale and Categorization: The Role of Objects��������������������  180 6.4 Main Properties of Scale as Rank ����������������������������������������������������  183 6.4.1 Directedness��������������������������������������������������������������������������  183 6.4.2 Abstractness��������������������������������������������������������������������������  184 6.4.3 Granularity: IF-Type Versus F-Type Transformations����������  184 6.4.4 Objectivity����������������������������������������������������������������������������  186 6.5 Recognizing a False Scale as Rank��������������������������������������������������  187 6.6 Histograms at the Core of Scale as Rank������������������������������������������  191 6.7 Scale as Rank and Maps ������������������������������������������������������������������  197 6.8 Uncovering Map Manipulation��������������������������������������������������������  201 6.9 Scale as Rank: Inside and Outside����������������������������������������������������  204 References��������������������������������������������������������������������������������������������������  205

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 Scale, Patterns, and Fractals������������������������������������������������������������������  207 7.1 Scale and Patterns ����������������������������������������������������������������������������  208 7.1.1 Scale and the Reality of Relationships���������������������������������  208 7.1.2 Distinguishing Patterns ��������������������������������������������������������  208 7.1.3 Defining Patterns������������������������������������������������������������������  211 7.1.4 Patterns and Scale ����������������������������������������������������������������  214 7.1.5 Patterns and Scale Change����������������������������������������������������  216 7.2 Scale, Symmetry, and Pattern Analysis��������������������������������������������  218 7.2.1 Symmetry: Old and New������������������������������������������������������  218 7.2.2 Scale Symmetry��������������������������������������������������������������������  221 7.2.3 Fractals����������������������������������������������������������������������������������  223 7.2.4 Scale as Ratio, Maps, and Fractals���������������������������������������  229 7.2.5 Scale as Ratio and the Fractal Dimension����������������������������  231 7.2.6 Scale as Ratio in Pattern Analysis����������������������������������������  234 7.2.7 Dimensions and Their Meanings������������������������������������������  242 7.2.8 Scalebound and Scale-Free Patterns ������������������������������������  244 7.2.9 Scaling and Spatially Variable Scale as Ratio����������������������  246 7.2.10 Scale and Its Role in Multiscale Pattern Analysis����������������  249 References��������������������������������������������������������������������������������������������������  250

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 Scale, Symmetry, and Nonlinearity��������������������������������������������������������  253 8.1 Insights into Artistic Currents ����������������������������������������������������������  253 8.1.1 Classicism: Scale, Symmetry, and Our Understanding of the Environment��������������������������������������������������������������������������  254 8.1.2 Romanticism: The Subtle but Pervasive Presence of Scale��  258 8.1.3 Views from the Garden: Romanticism Versus Classicism����  261 8.2 Insights into Individual Artistic Styles����������������������������������������������  262 8.3 Scale and Nonlinearity����������������������������������������������������������������������  265 8.3.1 Scale and Nonlinear Processes���������������������������������������������  265 8.3.2 Scale and Chaos��������������������������������������������������������������������  267 8.3.3 Scale and Self-Organized Criticality������������������������������������  270 References��������������������������������������������������������������������������������������������������  275

9

 The Essence of Scale��������������������������������������������������������������������������������  277 9.1 Scale Types: An Overall Perspective������������������������������������������������  277 9.2 Scale and the Guided Cut������������������������������������������������������������������  281 9.2.1 The Selective Approach to the Environment������������������������  281 9.2.2 Introducing the Guided Cut��������������������������������������������������  282 9.2.3 The Guided Cut and the Environment����������������������������������  284 9.2.4 The Guided Cut and the Logical Field����������������������������������  286 9.3 Scale Involves Mapping��������������������������������������������������������������������  288 9.3.1 Key Elements of Mapping����������������������������������������������������  288 9.3.2 Relations in Mapping and the Three Types of Scale������������  289 9.3.3 Reversibility in Mapping and in Scale Transformations������  291 9.4 Scale Involves Representation����������������������������������������������������������  292 9.4.1 Morphism and the Choice in Representation������������������������  292 9.4.2 Representation and Our Understanding of the Environment 295 9.4.3 Scale and the Modalities of Representation��������������������������  297 9.4.4 Mapping, Representation, and Scale������������������������������������  299 9.5 Scale Involves Quantitative Comparison������������������������������������������  300 9.5.1 Comparison and Memory ����������������������������������������������������  300 9.5.2 Comparison and Quantity: Looping Back to Mapping��������  301 9.6 Converging on One Essence of Scale: A General Definition������������  302 9.7 Scale as a Process������������������������������������������������������������������������������  304 9.8 Scale Is a Living Concept ����������������������������������������������������������������  307 9.9 The Open-Ended Scale ��������������������������������������������������������������������  308 References��������������������������������������������������������������������������������������������������  310

Index������������������������������������������������������������������������������������������������������������������  311

About the Author

Cristian Suteanu  is a professor in the Department of Geography and Environmental Studies and in the Department of Environmental Science at Saint Mary’s University, Canada. His research focuses on the analysis and modeling of environmental processes, and on epistemic aspects of our interaction with the environment. He teaches statistics, environmental pattern analysis, natural hazards, and environmental information management.

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Chapter 1

The Meanings of Scale

Abstract  This chapter introduces the framework through which scale is explored in the book. It starts with the process of abstraction, which lies at the foundations of scale transformation, and then it addresses the numerous ways in which scale is used and interpreted. The different forms of scale are presented in three groups: scale as size, scale as ratio, and scale as rank. These three categories are then shown to encompass the variety of scales currently in use. Each scale type is introduced, illustrated, and compared to the other scale types. The chapter thus prepares the basis for the inquiry into the concept of scale that is undertaken in the following chapters.

Keywords  Scale · Map · Mapmaking · Abstraction · Abstract space · Similarity · Scaling · Proportional reduction · Optical instruments · Mathematics · Concepts · Environment · Cartographic scale · Geographic scale · Representative fraction

During the last three decades, the various sciences have been faced with an ever-increasing number of new unsolved problems, of which many are linked to questions of scales. — Laurent Nottale (2011)

1.1 Scale: Pervasive and Indispensable If you can live in some environment without making use of any idea of space or time, you can also live without the concept of scale. Maybe. Scale is an unusual concept. “Provided that no one asks me, I know. If I want to explain it to an inquirer, I do not know,” we can almost hear Saint Augustine sighing. He was not referring to scale, of course, but to time. We feel that we understand rather well concepts such as time or space or life, but defining them – that is another matter. Scale is not such a concept: in fact, the opposite is true: “Scale is a slippery concept, one that is sometimes easy to define but often difficult to grasp” (Lock and © Springer Nature Switzerland AG 2022 C. Suteanu, Scale, https://doi.org/10.1007/978-3-031-15733-2_1

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Molyneaux 2006). Indeed, scale does not lack definitions. One of the problems with its understanding is precisely the fact that it is defined in many different ways. But surely there is nothing wrong with having a concept defined in more than one way? That might be true, unless the various definitions point in distinct directions. This indeed seems to be the case here: sometimes it is not easy to see how – or even if – the different views on scale are compatible with each other. Are we using scale to determine the length of a tectonic fault based on a digital map? Are we studying a small-scale ecosystem like a pond, or a large-scale ecosystem such as a tropical rainforest biome? Or are we exploring the large-scale structure of the universe? Are we specifying, using scale, the level of analysis chosen for a complex system? Or are we applying a medical scale, to monitor a patient’s impairment after brain injury? One can rightly ask if we can even talk about scale as being a single concept, or if we are facing nothing but a collection of meanings, loosely related or even unrelated to each other. How can we begin to identify and absorb the meanings of scale, when we feel as if we were standing in a room full of mirrors, wondering about an actual object, while being surrounded by innumerable images? The book aims at answering these questions: taming the jungle of images by detecting structure in the multitude of notions; showing that, when carefully examined, numerous instances in which scale is applied reveal their common roots; and last but not least, presenting practical examples from a variety of fields to illustrate the power offered by scale when we ask questions and address problems in the environment. Analyzing a concept that is broad and widely applied in many fields involves specific dilemmas. On one hand, there is a natural tendency for the author to look at the subject from various angles while still dwelling in one particular domain. Such a positioning of the author tends to be reflected in the book’s approach, even when it concerns topics outside that specific specialty. On the other hand, if the book’s intention is to reflect the concept in its depth, revealing the value of its generality, a disciplinary confinement is not helpful. One must then face the challenge of looking at the concept without letting the boundaries of one’s field to be decisive for one’s perspective. This may significantly change the standpoint that shapes the whole endeavor. However, the change is worth the effort, because a barrier-free approach to the core of the concept can support its better understanding. For this reason, this book was written in an attempt to leave behind the author’s “daily hat,” mostly related to system analysis and modeling in Earth and environmental science, to step out of a particular field, and to adopt a more extensive view. Based on this choice, the presentation shifted its focus: from disciplinary issues to essential traits of the concept, its ramifications, and their implications. In this context, the application space is wide, while our target is the understanding of scale and its role in our comprehension of the environment. Scale is a powerful concept. Its meaningful presence in our daily lives begins with our intellectual development as children (Tretter et al. 2006; Taylor and Jones 2009), including our skills of recognizing and classifying objects (Preuss et  al. 2014). Currently, scale lives in many areas of our endeavor: from medicine to engineering and economics, from geography to geology and geophysics, from biology

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to physics and astronomy, from the environmental sciences to the social sciences. Scale looks indispensable to our understanding of reality. As far as we can tell, it was always there, in our distant past, and it has gradually evolved, helping us arrive where we are now. It is a fundamental concept in its own right. Scale plays a key role in every field of knowledge that involves space or time, or both: therefore, as soon as we try to make sense of anything in our environment, we realize how vitally important it is to us. Moreover, scale does not only apply to space and time: it also captures other properties in the natural and the social sciences, playing a key role in a multitude of scientific approaches and methods. But what is so special about what scale can do? Why is it so broadly applicable? Why is it so widely valued? How does scale actually help us comprehend reality? This book is engaging with such questions by approaching them from various angles. To immediately issue one brief answer, we can start with the last question, thereby also touching upon the others: What does scale do in our interaction with reality? We should make it clear that when we normally mention words such as “the world” or “reality,” we do not actually mean “everything that is,” but rather a tiny portion of that. Considering everything humanity has ever been able to understand, in every possible way, throughout its learning history, there is no way for us to even have the slightest idea of how little we know of what the “whole of reality” actually is. The latter “veiled” realm is called by the physicist Bernard d’Espagnat “independent reality,” in contrast to “empirical reality” (d’Espagnat 1989), which consists of the areas that are “closer” to us in space and in time – those on which we have been spending by far the largest amount of interest and effort and which we keep understanding increasingly better. It is to such portions of reality that we assign here the term “environment.” This is the context in which scale currently reveals its power. To take some examples, scale enables us to assess and compare sizes and distances, to distinguish features of sounds and relate them to their sources, and to reconstruct complicated processes starting from scant sensory input: scale is linked to key features of abstract thinking (Peer et al. 2019). With scale, our senses can operate, and abstraction can work its magic. With scale, we can explore our environment. Using scale, we can apply transformations to space and time. Thereby, we can understand change and, in turn, produce change. If it were only for the fact that it helps us handle space and time, scale could be considered nothing short of a “miracle concept.”

1.2 A Daughter of Abstraction How did we get here? How did we end up with an instrument of such complexity, such versatility, and such power? Did we first discover it by chance in the world around us? By observing an image of the forest in a drop of water hanging from the tip of a leaf? By noticing rocks that looked the same, but which turned out being very different in size? Did we then take a daring leap from one world to another,

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from material objects to abstract entities? Or did we start with a plainly abstract, profound idea and brought it all the way down to the world around us – splitting it, bending it, turning it into an increasingly distinct intellectual tool? One way or another, abstraction lies at the core of the concept of scale – and of its seemingly misty clouds of meanings. Once we realize this, abstraction becomes a valuable workspace. It can enable us to pierce through these peculiar clouds. To perceive order, simplicity, and value, all in one major root concept, and to identify several branches, each with its specific powers. We have always been aware that size is meaningful. Appreciating the size of an animal, whether prey or predator, was a matter of survival. This is also true about animals themselves. Not only do they routinely estimate size: some species even put the size effect to work by trying to look bigger, as part of their intimidating or “deimatic” behavior, or in male-male competition (Kelley and Kelley 2014). Meanwhile, correctly assessing the size of a crack in rocks to find the right moves to jump over it, or judging the height of an obstacle and deeming it passable, has been less easy than appreciating the size of a deer. Rocks, like many other features in our environment, are tricky: they have the strange property of looking the same – or almost the same – when seen from different distances. You cannot tell if the cracks and blocks in an image are big or small unless you discover (or add yourself) a completely different object in the picture (Fig. 1.1). In anticipation of the story about scale and similarity, which is told in Chap. 7, we can add that this scale-related indistinctness can be turned to our advantage: for example, when it is embedded in our analysis of

Fig. 1.1  Is this image as wide as a football stadium, or as narrow as a fingernail? We can only find the answer if known features, which are completely different from the actual rock, are also present in the picture – in this case, notice the grass and the leaves. (Courtesy of USGS)

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strongly irregular features, it can lead to interesting and useful environment-­ sounding tools. Scale considered as size – the size of an object – appears to be a rather crude way of viewing our environment. What can be simpler? And yet, size appears to hold a key role for the way in which we represent the environment in the mind, for the way in which we navigate, etc. It is involved in processes that are anything but simple (Ayzenberg and Lourenco 2020). Scale as size begins to operate right there, in the environment. This is where the defining comparison takes place: a certain part of the environment is confronted with another part of the environment. The latter is special, and it is chosen by us to be so: it is the declared “unit of measurement.” The outcome of this confrontation ends up in an abstract space. This outcome includes two components. One is entirely abstract from the beginning – a number, an entity with an intricate history, laden with strangeness (Wolfram 2021). The other originates in the environment: the selected measurement unit, which is marking its territory in the abstract space where it was planted. For instance, whatever the number that will be found by the measurement, it will refer to “meters.” As a consequence, following the encounter between a crack in the rock and a measuring device, and the processes involved in the crossing into abstract space, a new object surfaces in that space – for instance, “20 meters.” As we will see in the following chapters, measurement is only an early stage in the emergence of “scale as size.” Suffice it to say at this point that scale as size includes a vigorous abstraction stage but one that is strongly rooted in the environment: environment-bound elements dominate the concept of scale as size. Recent research has led to valuable insights into the deceptively plain concept of scale as size and its role in understanding the environment, as further shown in Chap. 2. Scale is different when it is associated with a representation such as a map or a statue, i.e., a transition from one part of the environment to another. In this case, a scale transformation takes place – a transformation that is accessible to daily experience. One only needs light and a plain device – for instance, a lens. Usually, such a transformation is applied when the studied component of the environment – or “object”  – is either too small or too far away for the human eye to see it at the desired level of detail. Euclid (300 BC/1945) brought forward this issue, stating that “things seen under a larger angle appear larger, those under a smaller angle appear smaller, and those under equal angles appear equal.” Later, in the second century, Ptolemy more clearly explained the relationship between the object’s size, its distance from the viewer, and the angle over which it is observed: the same increase in the viewing angle can be a consequence either of a change in position or a change in the size of the object (Fig. 1.2). In other words, Ptolemy showed that increasing the size of something that is small and bringing closer something that lies far away are two actions that rely on the same principle. Little did he know that both effects were soon going to be materialized in optical devices called microscopes and telescopes, respectively. Optical transformations of scale may have already been performed long before Euclid. However, although such transformations were known since antiquity in various parts of the world, it was only in the thirteenth century that they were applied

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Fig. 1.2  Did the circle come closer, or did it get larger? Ptolemy showed that based on the angle increase from a to b, it is impossible to tell the difference

in devices for daily use – in particular reading (Grant 1974). Several more centuries passed until the creation of instruments for scientific observation took off. This happened almost at the same time both for the study of the small and for the detection of the remote – microscopes and telescopes became increasingly common after the beginning of the seventeenth century (Bardell 2004). Although the two types of devices were usually developed by different instrument makers, the fact that the same general principle was underlying them all was already well known. Scale-transforming processes made possible by such inventions had major consequences. In 1610, using his telescope, Galileo made observations on planetary bodies in the Solar System, which provided decisive support to the heliocentric model. The scale change brought parts of our environment closer, and, by doing so, it opened a new window towards our surroundings. In this way, the groundbreaking worldview was not supported only by hard-to-digest models: reality was staring us in the face, and it was enough to look into the scale-changing instrument to see it (the fact that some among the powerful of those days stubbornly refused to look is a completely different matter). This was one of the spectacular early occasions when a spatial scale transformation performed upon the environment offered vital support to abstract thinking. The subsequent paradigm shift was so deep and pervasive that it is impossible to imagine how our history would have evolved, had it not happened then. The discovery that “bigger” and “closer” reveal situations that are similar to each other must have felt new and powerful. Not because opportunities to observe them were rare. Humans have always been dynamic beings, getting closer, retreating further, and observing now from here and then from there, always on the move. The explored environment has been so diverse in its turn. This diversity must have played a role in the development of our capability of acquiring and pondering multiple views. But all this was not enough. The idea that one and the same underlying procedure can both make something bigger and bring something closer can only arise if one gets to approach the observed surroundings in a different way: by

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turning objects into abstract entities. As long as a tree is only seen as a tree, one can never think that a trunk that is thicker and a trunk that is closer can be perceived as one and the same thing. One must completely give up a large number of “details” – in fact, almost all of them – in order to think about the tree trunk as a cylinder and of its cross section as a circle. It is then – and only then – that the strangely analog character of the two situations, bigger and closer, can reveal itself (Fig. 1.2). The explorers of scale domains – those who used scale change to travel to the unthinkably remote or to the unreachably small – made, quite early, an additional discovery. There were striking similarities between newly encountered spaces and the already familiar ones. Galileo must have lost his breath when he realized that he was watching something that looked like a small planetary system around a planet that belonged to a planetary system itself. In the world of the small, van Leeuwenhoek, who in the seventeenth century revealed an extraordinarily rich and diverse world under the microscope, could see that minuscule grains of sand have shapes that are similar to those of chunks of rock. The implications of the fast-developing science of scale change, which relied on optical instruments, were impressive. But what is this telling us about the abstract scale transformation? Was it inspired by material processes? The relatively late development of optical instruments stands in contrast to the early emergence of a subtle and abstract form of scale transformation: mapmaking. The isotropic scaling concept (described in Chap. 4) that lies behind the production of numerous maps is a highly abstract concept itself. It involves the consistent application of operations that are not trivial: for instance, the shortening of all sizes using the same reduction rate, which leads to a representation that enjoys important properties. It is indeed remarkable that the resulting map looks similar, in a way, to the mapped reality. Moreover, more significant abstraction processes are at work here – so radical, in fact, that they look almost impossible to perform. Hence the conspiration enthusiasts’ explanation that humans must have been transported high above the land in extraterrestrial flying objects in order to come up with the idea of producing a map of the land. This way of thinking started probably from the assumption that a map includes all items reduced in size. However, this leaves out a key aspect involved in the design of a map. A careful selection operation must take place: very few elements are kept, and everything else is ignored. Moreover, those retained elements are represented with the help of symbols. The selection/abstraction operation is in fact taken to the extreme: this is not a reduction from ten lines to one, or from one thousand dots to ten. The different aspects of the material environment  – for instance, everything one finds in a village, in a forest, in and around a river, etc. – are converted either into a concise sign or into nothing. This is not mere information reduction. It is information implosion (Fig. 1.3). The occurrence of this information implosion is a quite baffling phenomenon. Let us immerse ourselves in the spaces that are involved in this transformation. We stand at the edge of the road, the map in our hands. We look around us; we listen; and we breathe in the scents of the summer. Engaged in both realms, we compare them. We can see the road running endlessly in both directions; on the map, it is shown as a winding line. On both sides of the road, there are tall, colorful trees – on

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Fig. 1.3  Example of information implosion involved in abstraction: from a myriad of structures and shapes and colors to one dot

the map, nothing. All along the road, a green environment beaming with life, the air full of buzzing insects and flower scents – on the map, nothing. And if such abstraction looks too stern, we can repeat the experiment elsewhere – on the seashore, for example. This time, our map includes a large patch of blue, in an apparent attempt to do justice to the ocean stretching endlessly before us. The wavy line of the shore is quite easily recognizable on the map. This is encouraging, so we keep comparing. In front of us, interweaved shades of blue – on the map, an empty surface (nothing). At our feet, foamy waves breaking with roaring sound – on the map, nothing. Cool wind and ocean spray reaching our face; farther in the water, colored flashes of fish schools; behind us, on the beach, seaweed and shells and the penetrating aroma of the ocean. On the map: nothing. Most elements that we find worth remembering from our immersion in the environment are absent from the map. We could also create a different kind of map, one specifically dedicated to certain elements in the environment, like the trees, and this would save them from disappearance into nothingness. Even then though,

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innumerable other elements would still be lost. Abstraction is merciless. What must be left out is left out – no exceptions. How did humans become capable of making such a selection? Those who experienced the wind, and the ocean, and the bird songs, and the trees, and the bustling city life, on one hand, and those who represented the environment on maps, on the other hand, seem to truly belong to different species altogether. Some of the described operations look as if they relied on a long experience of applying scale change. One can only wonder: How did abstract scale handling develop independently, so early, so long before scale change could be explicitly witnessed – or produced – in our environment? For instance, the proportional reduction implied the manipulation of concepts such as similarity. As mentioned above, similarity was conveniently offered to us by certain optical devices. And yet, those devices were developed and applied long after highly abstract mapmaking skills had left their proof of existence in many places on the planet. In later times, we can find another type of scale, which looks even more abstract than those mentioned above. One of its manifestations consisted of the following: if what we needed to measure did not match the requirements for measurement, a “scale” was projected into abstract space, and arrows were shot from one space to the other. Abstract arrows, of course. The scale consisted of abstract “boxes” piled on top of each other, broad intervals that were clearly discernible from each other. The arrow shooting rules were conceived in such a way that for each possible starting point in the material world, an arrow would land in the proper box. In this way, wind strength could be assessed in terms of its effects on the sails of a ship and assigned a number between 0 and 12. There was no need to worry about finer wind speed differences. The overall outcome was remarkable: a process that was taking place in the environment  – whether vague, unapproachable with a measurement device (as in the case of physical pain), or complex and strongly variable (as in the case of the wind and its impact on the sails) – could be tamed, pigeonholed, assigned a well-defined interval, an interval in abstract space. When scale is applied in this way, it is the position of the destination interval, its “address,” that is the definitive outcome. This result is not linked to measurement units, and it is not caught in a web of signs like those of a map either. It is a mere address, like “box number 4.” Thereby, although the process ends this time in abstract space, it can provide decisive information about what happens around us: for instance, “the wind matches a 12 on the Beaufort scale: no canvas sail is expected to withstand it.” As we could see from these brief examples, there are different instances in which scale plays a meaningful role in our interaction with the environment. In each case, abstraction dominates some of the core stages involved in scale handling. Moreover, the points in which abstraction intervenes and the role it plays can help us to distinguish more clearly the scale’s operating modes. It is therefore beneficial to put abstraction to work and let it help us to get a better insight into the meanings of scale.

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1.3 The Scale’s Fields of Operation Perhaps no transformation is as radical as the process of abstraction. Even its simplest forms are puzzling. You start with some aspect of reality, and you end up with something that is inherently of a different nature altogether. You see a brick, and you leave with nothing but a brick in your mind. You see a nut, and now you have it in your mind. In some way, you transferred them both to another realm. How do we know that the transfer led to anything “like” the brick or the nut? How can we be sure that the transfer to another space, an abstract space, didn’t end up as a meaningless jumble, a worthless soup unrelated to those “things”? We know this because abstract space is reachable. We can revisit it, and we can work within it. For instance, we can operate on the brick and the nut in abstract space, and we can combine them there into a process: when we return to the environment and we get to see the brick and the nut again, we can apply the abstractly created process and crush the nut with the brick – and it works. We can generate or discover certain relations in the abstract space and verify them in the environment. In fact, had that not been the case, we would not be around at all. Our success at abstraction has been making our survival possible. It has made our endless becoming possible as well. More importantly, something happens when we go beyond putting together two objects in our mind and being amazed that the outcome can be applied in the material environment. The mystery deepens in extraordinary ways when we dwell in another universe, which is purely abstract from the very beginning – the universe of mathematics – and find that its complex, coherent network is related to innumerable instances of the material universe. Many brilliant minds have wondered about this apparently endless applicability of mathematics to science; famous names include Eugene Wigner (1960), John Wheeler (1966), and Roger Penrose (2006). Some interesting ideas have been proposed as possible paths to an explanation, such as the “similarity between knowing-within-mathematics and knowing-in-general” (Posina et al. 2017) and the hypothesis that “the physical world really is completely mathematical” (Tegmark 2014). The fact that we do not express our astonishment regarding the connection between the material and the abstract realms more often might be confusing. It could make us believe that it is because we are making progress in addressing this mystery. This is not the case. Most scholars admit the reality of the unexplained connection and focus on other questions. Of course, there is no guarantee that in abstract space we couldn’t still end up with some useless soup, which has little or nothing in common with actual things (as we know, for instance, from the discourses of some politicians). Generally, however, the process of abstraction is so reliable and so safe that we can live together, work together, and basically agree upon most aspects of what we call reality. A “space” associated with processes performed by an individual/and attributed to what we call the mind, a space that is consistently related to – but different from – the material world surrounding us, was envisaged by many scholars throughout history. Contemporary examples include Popper’s (1972) “world 3,” the ideosphere of Hofstadter (1985), or the structured information space of Boisot (1995). Given

1.3  The Scale’s Fields of Operation

11

the scope of the book, such spaces will not be explored here, nor will the simply named “abstract space” be investigated in detail or endowed with specific properties. We will mainly use it to distinguish it from the space-time in which we experiment with material elements. Abstraction, however, involves much more than the transfer of elements from one space (called the environment) to another (called abstract space). Vivid discussions about what abstraction is, how it works, and with what outcomes have been raging, with endless ramifications, for centuries (Laurence and Margolis 2012), and are nowadays far from reaching convergence – let alone consensus (Falguera and Martínez-Vidal 2020). Here we will only briefly touch upon some key aspects of abstraction, which are important to our exploration of scale. Arguably one of the most interesting points of divergence regarding abstraction is the fruit of nominalism, from its medieval outlines to its contemporary shapes. We can summarize this problem of abstraction with the help of the following simplified picture. Of the two main varieties of nominalism, one denies the existence of “universals,” and the other denies the existence of abstract objects. While these two positions do not coincide, at the root level, they have much in common. In other words, nominalists claim that only particular objects actually exist: the things we see around us. Only these objects are real. This horse, that hammer. The so-called concepts, like “horse” and “hammer,” are secondary to actual things: they are mere names attached to sets of objects. It goes without saying that the latter apparently innocent observation triggered intellectual explosions when it entered the powder house of analytic philosophy. According to the alternative direction of thought, abstract objects do indeed exist, and they are important components of reality. Concrete things – the particulars – are instances of the general, abstract concepts, in relation to which they play a secondary role. Fortunately, we do not need to dwell in the forest of richly ramified arguments fed by this controversy. There is a solution that can better serve the purpose of this book. To this end, we leave behind the controversial elements that specify what is believed to come first, or what is considered to be secondary to what. There are solid reasons why we can do that. First, both concrete and abstract objects are, in fact, objects of our experience. Second, any considerations concerning their precedence or their secondary status have no bearing on our approach to scale in this book. As long as they change neither our methodological paths, nor our insights on their outcomes, we can confidently move beyond such differences. The important decision here, however, is to admit the evident existence of both concrete and abstract elements – this is an essential step in our exploration of scale, which we can often find in this book. As Wolfram (2021) points out, as long as every single item in the environment is unique to us, we never feel the need for a concept: it is the encounter with multiple objects of the same kind that triggers what he calls “symbolization.” Of course, Wolfram’s account was preceded by a millennia-long, often turbulent history of intellectual developments; he seems to extract a simplified outline of key aspects involved in this process. For instance, when we see objects that have much in common, a major transformation occurs in the mind. Something new can appear. Tree – rather than “this tree.” River. Apple. Rock. A concept is created, and a symbol – a

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word or a graphical sign – is attached to it. With a name assigned to it, the concept becomes easier to handle. It is therefore used more often. As a further consequence, in turn other concepts become easier to be built, as they can rely on the previously existing concepts. A network of concepts and relations between concepts can thereby grow through a snowballing effect that starts from basic abstraction operations. The starting point looks easy – and yet, this change implies a process that is so unique that it can only stand next to very few other achievements of the mind. It is a transition from the concrete to the abstract. Why then does it look so easy? Let us start with an example. Consider an animal – we will pick one that is less familiar for many of us: a wombat (Fig. 1.4, on the left). Let’s assume that we know this particular animal very well: for us, she is “Womby.” In order to come up with the concept of wombats, we must first realize that, in some way, wombats are all “the same.” Surely we know that they are not. You can’t find another Womby on the whole planet. If we are perceptive enough, we can notice what is specific to each individual. However, in order to see all of these animals as being “alike,” we may find that we do not actually need to see each individual animal in more detail. On the contrary, it seems that it may help if we would see them less well. If we only had a blurred image of each individual, and we did not distinguish their details, we might feel one step closer to the concept of the wombat (Fig. 1.4, on the right). In other words, it looks as if it takes less information to form concepts in the environment than to describe its elements as unique as they are. Such perceptual premises of abstraction appear to be informationally efficient. As we will see in the following chapters, the operation we have just performed – as inadequate as it may be in its reflection of abstraction – includes a transformation of scale, which has direct implications for the amount of information we handle. This conclusion supports our earlier impression that abstraction should not be difficult. Nevertheless, as the reader might have already noticed, there is a problem

Fig. 1.4  An animal identifiable as an individual that we call “Womby” (a common wombat or Vombatus ursinus), vs. a set of less sharp images that can make us think of all such animals as “wombats”

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13

with this approach. The smaller images in Fig. 1.4 may be pictures of wombats, all of them, but we cannot be sure about that: as far as one can tell from the less detailed images, there may be pictures of dogs or rabbits among them. The trick that promised to bring us closer to generalization, to enable the concept of “wombat” to pop up, turns out to be the source of the problem: by decreasing the amount of perceived detail, we also become progressively less certain about the set of individuals we wish to capture in one single pool and address by one single name. Abstraction by mere simplification might not work that well after all. In fact, no matter how we consider them, we must admit that the complex tasks of abstraction and concept building must rely on preexisting concepts, which must in turn have been produced in advance. The ample processes of comparison, selection, and interlinking of features must involve abstract space. Identifying, selecting, and representing relevant relations between concepts are among the key components of abstraction. Relations between concepts are essential because concepts can only operate in a network. They cannot be suspended in void space. If they are isolated, they are not even rescuable – they are simply dead. All these operations require quite different skills and a different level of training, compared to the creation of images based on the fine-tuning of their level of detail. Several examples of effort to comprehend abstraction might help us, like flashes of lightning in the stormy intellectual space, discerning possible obstacles and/or ways of moving further. Visiting the tracks of thought of others, especially of illustrious minds, can often be a fertile learning experience. In fact, Aristotle might not have been pleased with the line of thinking described above. It was clear to him that we cannot succeed by simply ignoring the details acquired from individuals. We must rather identify their “essence” that “something” that those individuals have in common, their structure. This is a much more difficult task than just changing the focus of our observation lens. Because this goal is so vague and so little is offered in the way of specific steps to follow, the task of essence identification based on these clues remains difficult. Insights into how to perform such a task are offered, for instance, by John Locke (1690/1975): he recognizes the importance of casting out details in order to reach a general representation. However, he also discovers more about the mechanism at work: abstraction does not start from a blurred picture of many individual instances. Neither does it require an exhaustive understanding of numerous instances of the same entity, to be fed into the burgeoning concept (which would be overwhelming to the point of potentially rendering the task impossible). The proposed mechanism begins with in-depth knowledge of only a few individuals. The process then builds on that understanding, while one gets acquainted with increasingly more individuals. Approaching this question from a different angle, Kant’s Critique of Pure Reason (1787/2007) provides a more precise image of the challenge. It confirms that our method should not rigidly rely on common properties and similarities, but it shifts emphasis from property selection to the establishment of relationships, which should support the identification of the actual function of the concept. It is not surprising that the acts of reaching abstract concepts and operating with them are notoriously difficult. It is not straightforward to follow, step by step, even

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one’s own path of thought, when it is involved in abstraction. It is surely not easier to guide someone else’s steps towards – and in – abstract space: teaching “abstraction” has never been a clear-cut, algorithmic endeavor. There are, nevertheless, several things that we have learned in this regard. First, we must ensure that there is ample exposure to well-selected instances of the considered notion: this is a valuable premise for the crystallization of an abstract entity. This is one of the reasons why we find it so beneficial to be offered concrete examples when we are taught something new (similarly, a large variety of training cases are offered to artificial intelligence applications, in a process that resembles learning). Second, persistent and carefully guided practice should be based on objectives of gradually increasing difficulty. Third, there is a point that turns out to be specifically useful in the case of scale, given its essential connection to the environment: offering learners meaningful opportunities to move in space while interacting with objects in the environment. Abstraction processes can become highly intricate, and it is increasingly problematic to follow their development and transformation at a higher and higher level of complexity. For instance, turning a landscape  – including villages, roads, and rivers – into a collection of symbols drawn on a small, flat surface implies a high level of abstraction. We have to conceive of the enormous metamorphosis of sizes and distances, which is needed if we want to fit a whole “land” on a palm-sized plane, which is as real as the land itself – a sheet of paper, the wall of a rock, or the sand surface on a beach. We must accomplish this “teleportation” act by handling a transfer between distinct realms. To this end, we use an intermediate stage, outside of space and time: in abstract space. We move from the “large” realm into the abstract space and from the abstract space into the “small” realm of the environment. Both phases, which are often interweaved and not always perceived as being distinct, involve specific operations that imply abstraction. Abstraction can be almost miraculous. Remarkably, information transfer between realms works in both directions, if we include an intermediate element of high complexity: ourselves. For instance, as shown above, you are starting out with a village and all that it comprises: houses (with their painted walls, windows, roofs, and everything that is inside), fences, roads, fountains, people, animals, etc., and you are ending up with … a dot. The village is the dot. Nothing more. And then you connect the dot with another dot, using a line. Admittedly, a line is much more similar to a road than a dot is to a true village or city. But then the miracle also goes in the opposite direction: by looking at the position of the lines and the dots, you can identify a dot, and you can make the village open up in your mind – in some way. This does not recreate the whole city and every crack in its walls, although you may very well remember a specific crack in a specific wall in a specific house that is important to you, and that crack may shine for you in abstract space as soon as you recognize the meaning of the dot. One may argue that the dot on the map is not even meant to represent an actual city. It is designed to specify the location of a city in relation to other features on the map, which is true  – to some extent. However, in our discourse, almost without exception, we do not refer to dots as only pointing to locations on the map. We

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15

would almost always say, by pointing to a particular dot on the map: “this is Athens” and not “this is the location of Athens” (unless a particular context would compel us to do so). For most of us, the dot on the map “is” the city, and not its mere position. Abstraction is deeply embedded in our functioning. Terms such as transformation, metamorphosis, etc. applied above may be misleading. We have not actually transformed anything. The village, the road, etc. are still there. What we have done was to open up what we called an abstract space and to populate it with elements meant to correspond to those in the environment  – Chap. 9 will take a deeper look at this process. What is important at this point is the fact that not only do we benefit from an abstract space that is rich in elements, their properties, and relationships: we can also act upon these. This only becomes possible if we know how to operate in this very different kind of space, in which nothing is completely disconnected from the environment, but neither is it quite the same as in the environment. If that is the case, we can perform in abstract space acts that are impossible to accomplish in the actual environment. We can incrementally construct plans. We can try out various scenarios. We can follow the implications of possible changes, if they were to be performed in the environment. So, unavoidably the question arises, whether or not we can be confident that processes in the environment would match those in abstract space. The answer is a crisp “no.” Since abstraction implies drastic information reduction, in abstract space we get to handle an infinitely small but presumably relevant fraction of the features of the material environment. As long as the selected portion of the features absorbed into abstract space is decisive for the actual processes of interest, we may reach results that are relevant. In general, however, the possibilities of landing on erroneous conclusions are wide open, and moving back and forth between these realms may be necessary in order to minimize errors, as long as a “back and forth” shift is possible. At this stage, we can consider the above operations to work acceptably well at this level of detail. This will allow us to handle scale types, in useful ways, in this and the next five chapters. The problem that lies at the core of scale is, however, much more extensive and interesting and will be more rigorously examined in Chap. 9. Abstraction associated with maps and scale has the peculiar property of not leading to a merely static result, no matter how interesting or useful that might be. If you look at a photograph of a running person, or at an image of a leaping animal, you can distinguish in the static picture the dynamics of the captured action. With maps, one can argue that their dynamic nature is integrated in them: symbols on the map are not just lying there like insects pinned to a corkboard. The road indicates travel; it highlights the potentiality of movement. The various elements on the map are there for a reason; their relative positions have a meaning. They silently tell us about influences various elements can exert upon each other. They are signs of actual or potential interaction, of obstacles to interaction, etc. – they speak about what can happen in the actual environment. Maps are both subtle and intricate types of expression of dynamic features. The map, an apparently static representation by its very nature, is subject to a form of tension, by extending a network of interactions between the elements that

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were chosen to be represented. Maps are capable of meaningful change: they can influence our values; they can communicate, as well as shape identity (Kent and Vujakovic 2017). A map is nothing like a nice picture on the wall. Unless, of course, one wishes to use the map as a nice picture on the wall. It is there, and you can look at it at any time, but you are missing the life that was enclosed in the map. Such elements of abstraction tend to reveal themselves gradually, while one is getting acquainted with the life of maps.

1.4 Types of Scale 1.4.1 Grouping the Scales Scale can be a frustrating concept. It has multiple meanings even in a single scientific domain, and there is significant variation in meanings across domains. —Zhang et al. (2017)

There is a definite danger associated with the many-sided nature of scale, or, in fact, “scales.” As Geoffrey West (2017) points out, “scale can mean a lot of different things to a lot of different people.” Such a situation can make the life of a concept really difficult. One feels compelled to check if these diverse notions have common roots and, if they do, to look for their points of intersection, clearing away the variety of layers of meaning that had accumulated through daily use. The vital role played by abstraction in the functioning of the different kinds of scale offers itself as a starting point in this endeavor. There is no science without abstraction. There is no life for us without abstraction either. Given the pervasiveness of abstraction, we could use it as a filter to separate different types of approach to scale, and at the same time to bundle together multiple instances of scale and distinguish common approaches. While all of the discussed instances of scale invariably assign a fundamental role to the process of abstraction, they do so to varying degrees and following distinct routes. Abstraction can take on specific forms. By distinguishing these forms, we are able not only to get a better grasp over the possible grouping of versions of scale: we also develop better insights into their role in our understanding of the environment. A diversity of meanings of scale can be identified, even in the natural sciences alone. The most common one is map scale: a unitless number representing the ratio between a well-defined distance on a map and its corresponding distance on the ground. The map scale, or the “cartographic” scale, is a powerful instrument that helps us to effectively “translate” distances, in both directions, from one realm to another, the material environment and some material form of the map. When the map leaves behind its material body and moves into a digital environment, the scale’s meaning persists, but in a different form. While distances in digital space are not established by measuring them on a screen, the graphical scale shown on the screen – which can change, depending on how much one zooms in or out – can play the same role as its precursor on a paper map. Here, however, a novel significant

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element emerges, in the process of the transformation performed between the real environment and its digital representation: resolution, which is related to scale. In contrast, scale is also used in the sense of “spatial extent.” Confusingly enough, this scale is also called “geographic scale.” Since it is often important to take into consideration the spatial extent over which certain processes occur, an “operational scale” is also recognized. Finally, a certain “scale” is applied to refer, for instance, to the intensity of tornadoes, the air quality, or the pain experienced by an individual, by assigning to them a numerical value, chosen from a set of accessible values. Most – if not all – of the forms of scale mentioned above can be encountered not only in space but also in time and/or space-time. The resulting spectrum of meanings of scale is quite wide. Under these circumstances, the first question that unavoidably emerges is whether or not a significant meaning of scale, and not just of various scales, can be identified if we look deeper into the numerous notions of scale, which seem to be so different from each other. A second question is whether the recognition of a common root of all these “scales” would considerably support our understanding of scale in general and of its various particular forms. The mere articulation of these questions has already disclosed the fact that this book’s answer to both questions is “yes.” A careful analysis of the different ways in which the label “scale” is applied and used in various fields reveals that all the abovementioned forms can be seen as clustered in three groups. We will call these groups “scale types.” Among the forms of scale listed above, some refer unambiguously to a certain size, which can be a distance in space or a time interval (sometimes it also refers to other measurable quantities). This scale category includes the one referring to “spatial extent” or the “geographic scale,” as well as the “operational scale.” Both of these can be found in time as well. We will group all these forms of scale in one scale type, called scale as size. Map scale is distinctly different. Map scales, in all their forms, belong to the same category, in spite of the differences between hardware maps and their digital “equivalents.” What characterizes them is a ratio between sizes in the two domains involved in a map (the environment and its symbolic representation). Scale enjoys specific properties in the case of digital maps and remote sensing images; its relation to resolution distinguishes it from its paper map ancestor. On the other hand, a similar meaning is associated with scale of different kinds of models beyond maps, including three-dimensional models. Thereby, this type of scale is relevant to various forms of objects, from artifacts of ancient times to models in science. Moreover, scale-based transformations are also performed in time, for a variety of purposes. All these forms of scale will be assigned the scale type called scale as ratio. Last but not least, scale is used in order to allocate one number, one position from a discrete set of possible positions, to an otherwise often misty and complicated situation. This use of scale is also markedly different from the ones in the above categories. It can take a wide diversity of forms, and it can be applied in many different ways. All of these forms will be grouped together in the third type of scale, which we call scale as rank.

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We will start with a brief overview of the three scale types. Each of the three groups will then be addressed in detail in the next five chapters.

1.4.2 Scale as Size Although it is rarely – if at all – called as such, scale as size represents the most widely used group of meanings of scale. Among other categories of scale that are presented in literature, two of them overlap, in specific ways, with scale as size. One is the “observational scale,” which refers to the spatial extent of the studied area, while the other is the “operational scale” or “process scale,” which is the extent (in space or in time) over which certain processes operate (Zhang et al. 2017). These two scales imply different interpretations regarding their application. However, they both correspond to the type “scale as size” in terms of the scale process they involve: (i) they start in the physical environment, where they refer to a spatial extent, (ii) they compare that extent to a reference system (a measurement system, or a conventional system concerning the meaning of “small,” “large,” etc.), and (iii) they provide the result of the comparison as output. To apply scale as size, we refer to a part of the environment by focusing only on magnitude. This allows us to refer to the studied phenomena by using phrases such as “small-scale processes,” “large-scale features,” etc. but also “on a scale of kilometers, …,” “on a scale of millimeters,” etc. Implications of the resulting perspectives are then transferred to abstract space. The quoted statement examples are meant to carve out a part of the environment and to attach one single property to it: the size – in these examples, in space – to which we refer when we mention the scale. Since by using scale as size we mostly stay embedded in the environment, with only magnitude being sent over into the abstract realm, this type of scale involves the lowest degree of abstraction among the three types analyzed here. It is simpler and easier to apply to our surroundings than the other two types, from the microscopically tiny to the large features in the universe. A commonly applied alternative is to see scale simply as representing size (Lovejoy 2019), which is frequently the case in scientific literature. In such circumstances, scale benefits from the precision that characterizes the size associated with it. The concept allows us, for instance, to scan certain scale ranges and to refer to targeted processes while studying phenomena of interest. For example, if we work on research concerning the occurrence of landslides, we can focus on the submillimeter to millimeter scale: pores in the soil – their size and shape, their interconnectivity, the presence of water, etc. Then, at the millimeter to meter scale, we can also study the presence of vegetation in the soil – their type, their spatial density, the shapes and sizes of roots with their multiple ramifications, and so on. We can also analyze the topography of the area, which can be done on multiple scales: the outcomes on a scale of meters will reveal a surface shape variability that can be very different from the overall slope that is distinguished on a scale of tens and hundreds of meters or more. On the other hand, certain geometric properties of topography can be remarkably independent of

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scale, and this can be the case over a wide range of scales. The distinction of “scale-­ free” vs. “scalebound” features in our environment (presented and exemplified in Chap. 7) can have valuable implications for our understanding of the environment. The term “size” is commonly associated with a spatial context. Indeed, it is mostly in space that scale as size has accumulated its richest and most successful history. However, if we think about it as “magnitude,” it becomes clear that the concept can also be applied to other variables. Among them, time has been playing the leading role, as shown in Chap. 3. With the fast development of theory and methods dedicated to complex environmental systems, the role of time scale has been increasingly recognized and fruitfully applied. A common way of putting this scale type to work is by expressing scale ranges in terms of “orders of magnitude,” which is particularly useful when we study system properties that are unchanged on a wide range of scales, whether in space, or in time, as discussed with examples in Chaps. 7 and 8. Scale as size is also used in a different form that does not specify measurement units, nor does it involve spatial or temporal meanings. An example referring to social interactions is presented in Fig. 1.5.

1.4.3 Scale as Ratio “If I eat one of these cakes,” she thought, “it’s sure to make some change in my size” […] So she swallowed one of the cakes, and was delighted to find that she began shrinking directly. —Lewis Carroll, Alice’s Adventures in Wonderland

The challenges of mapmakers cannot be overcome as easily as they were by Alice. As an essential instrument of geographers and scholars from many other fields, the map is characterized by a scale or a set of scales, which are designed to unambiguously and quantitatively specify the relation between spatial sizes on the map and those in the represented environment – hence the name “scale as ratio.” For Alice still to look like Alice, her transformation – whether shrinking or growing – had to occur in a very precise manner (based on the scaling transformation, which is discussed in Chap. 4). The key parameter in this process was the ratio ruling her size change: the scale. Something similar applies to maps. However, as intricate as it may be, a change in the number of dimensions is not the only problem. The surface of our planet can never be made to match a flat surface. Something has to give: to crack, to stretch, etc. In simple situations, when we can ignore such distortions, scale can be represented by a single number – a single ratio – which is applied to the whole map. Fortunately, this is a common situation. One unit on the map stands for a number of units on the ground. In a more general case, this issue can become more intricate and interesting: maps may include a range of scale values, which depend on the actual map area, as shown in Chap. 4. The ratio between a distance on the map and its corresponding distance on the ground relies on the assumption that both are measured in the same units. For instance, we can write the scale or “representative fraction” as 1:50,000. This property of map scale allows us to use any units we want, whether centimeters, inches,

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Fig. 1.5  Symbolic representations of social networks. (a) A small-scale network of social interaction: five nodes and six links. Network understanding includes listening to each member’s discourse, watching their facial expression, etc. (b) A large-scale social network: tens of thousands of nodes and millions of edges. Network understanding relies on a range of pattern analysis methods

or something as unique as the size of a half-used pencil, and the result does not change. Neither does the scale precision suffer for this reason. Scale as ratio refers strictly to a relationship. It only relates the environment to its representation by specifying the characteristic of the transformation from one to the other. We can notice that while scale as size is (or can be) associated with physical units, scale as ratio does not involve units at all. In spite of its lack of units, scale as ratio can be – and usually is – precise, from a mathematical point of view. The reality of the problem of precision is different, of course (it is discussed in Chap. 4). There is a striking difference between these two scale types. Because scale as ratio has the numerator equal to one and the denominator equal to the distance on the ground measured in the same units as those chosen on the map, the larger the distance in the environment, the smaller the value of the ratio. A smaller ratio means

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a smaller value of scale as ratio. In other words, maps that cover larger areas are characterized by a smaller scale. On the contrary, large-scale maps have a smaller denominator of the representative fraction, and smaller areas are therefore covered. Because of the ways in which scale as size and scale as ratio are defined, the labels “large” and “small” have opposite meanings. When a large part of the surface of the planet is captured on a map, that is a small scale for scale as ratio but a large scale for scale as size. One must carefully avoid confusions arising from this difference in meanings, especially when “geographic scale” (Lam et al. 2004) has the meanings of scale as size, while “cartographic scale” refers to scale as ratio. Another way of specifying scale as ratio is offered by the scale bar, which has the significant advantage that changes in map size do not affect their scale (Fig. 1.6). When we leave paper maps behind and step into digital territory, scale does not become obsolete: on the contrary, it takes on new forms and new functions. It can acquire a dynamic character, offering seemingly endless flexibility, as long as we know how to operate with it. In this context, scale as ratio is the key to rich information sources, ranging from simple digital maps in popular apps to scientific images based on modeling and remote sensing. Scale as ratio remains meaningful even in situations when, if we change it, the studied pattern itself does not change: in this case, scale and scale change transform themselves into observation instruments for uncovering special attributes of the pattern. They support an in-depth characterization of certain elements of our environment, which helps us to better understand them. The discovery of such properties of the natural world have led to the creation of powerful analysis methods applied both in space and in time (as shown in more detail in Chaps. 7 and 8), in fields as different as geography, geology, environmental science, biology, medicine, or finance. Scale as ratio is best known for its association with maps. However, it also plays a significant role in the case of three-dimensional representations, especially in relation to modeling. Its impact on applications in time is relevant in many different fields. The most obvious applications of scale as ratio in time are associated with

Fig. 1.6  Slopes in Valles Marineris, close to Mars’ equator; the dots indicate the position of possible signs of liquid water. Since topographic shapes can be similar on a wide range of scales, the scale information provided in graphic form is highly significant. (Courtesy of NASA/JPL-Caltech/ University of Arizona)

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sound. Compressing sound over time or stretching it out leads to a shift in frequencies. This can help us to move the recorded signals into the desired domains: for instance, we can bring them in the audible range, or modify the time scale in ways that enable us to better perceive patterns that are relevant for the studied processes, and which otherwise would go unnoticed. Examples include animal communication, sounds produced during volcanic eruptions, or vibrations associated with earthquakes.

1.4.4 Scale as Rank Scale as rank is the third and last of the proposed groups of notions. It involves the highest degree of abstraction among the three categories: neither space nor time has to be involved here. A brief comparison with the other groups might better illustrate its apparently strange character. Scale as size is used to lift entities from the environment and bring them to the realm of abstract thinking, where comparisons can operate upon them. Scale as ratio is meant to perform a reliable transition between a part of the environment and its abstract form, and from there to the material representation (a model, such as a map). In contrast, scale as rank is to be applied to such entities that often cannot be put side by side with a measuring stick and compared. Such entities can be too vague, too deeply hidden, too variable, or too complicated for those other types of scale to be applicable. Examples include the effect of the wind on the sea and on the boats’ sails (Table  1.1) or the intensity of headaches (Loder and Burch 2012).

Table 1.1  Criteria applied for the Beaufort wind force scale (RMetS 2018) Wind force 0 1 2 3 4 5 6 7 8 9 10 11 12

Description Calm Light air Light breeze Gentle breeze Moderate breeze Fresh breeze Strong breeze Near gale Gale Strong gale Storm Violent storm Hurricane force

Specifications Sea like a mirror Sea rippled Small wavelets on sea Large wavelets on sea Small waves, fairly frequent white horses Moderate waves, many white horses Large waves, extensive foam crests Foam blown in streaks across the sea Wave crests begin to break into spindrift Wave crests topple over, spray affects visibility Sea surface largely white Sea covered in white foam, visibility seriously affected Air filled with foam and spray, very poor visibility

While wind speed intervals are also associated with each of the categories, for this example only the verbal description is provided, since it represents the original criteria for category distinction

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Under these circumstances, scale as rank is used to project a measuring framework in a space in which we wish to operate. It may be imagined as a set of light beams sent high into the night sky, allowing us to compare and measure things that would otherwise be floating around up there, ungraspable. It sets boundaries between which the studied entities are recognized, and then it divides up the space between these boundaries in a series of discrete categories. Thereby, an otherwise hard-to-distinguish feature can be associated with a well-defined box: according to this scale, it offers an alternative to the element of interest being measured. Unlike the other two types of scale, scale as rank implies a creative effort of setting up a window in abstract space and providing rungs to mark specific levels inside this window. In some cases, not only the category limits but even the overall boundary values are difficult to circumscribe in an objective way. The projected framework may be unavoidably vague. And yet, this approach offers us the capability of evaluating phenomena in our surroundings, leaving the mist behind and stepping into a structured space. This space benefits from limits with specified, albeit imprecise, positions. To illustrate this principle with an extreme example, scale as rank makes it possible for us to answer questions such as “on a scale from 1 to 10, with 1 being so and so, and 10 being so and so, how do you feel in terms of …?,” and suddenly our hazy, formless impressions can be associated with a number. Pinned to a scale. This is surely not as accurate as a physical variable that is measured with an instrument, but it offers a reference frame, a space in which to think and work, where before there wasn’t any. In this way, a large number of measuring frameworks has been created, which can be applied like off-the-shelf instruments: as long as we consistently use the same scale, we can rank the objects of our investigation and usefully compare them to those measured by other studies. Scale as rank offers us efficient means for quickly evaluating new elements in our surroundings. In fact, once the framework with its categories is set up, the operation of associating a new element to its place in the framework can be fast and easy. Valuable as it is, this instrument also has its limits and its drawbacks. Chapter 6 explores it in more depth, illustrating it with practical applications.

1.5 Understanding Scale The marriage between the physical sciences and the social sciences is far from trivial. […] one of the most important conceptual challenges to that union [is] the concept of scale. — Gibson et al. (2000)

There are times when we tend to focus more on our tools – whether instruments, methods, or concepts. This is not to say that a major shift in interests occurs at the same time in all fields. Nor does this imply that an enhanced emphasis on our tools would take away from the energy that is deployed in the processes of discovery, which are spreading at an accelerated rate in so many directions. On the contrary, it is often the wildfire of discovery, which is fueled and supported by the creation and application of tools, which pushes methods and concepts into new realms, possibly

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driving them to the limit of their applicability. In this context, we realize from time to time with utmost clarity the need to turn our attention to a more in-depth understanding of our means of discovery and understanding. This happens even if we have been using those means for a long time, and even if they have been proving their reliability in unquestionable ways. Sometimes it is the new territory to which they have been transferred; sometimes it is the construction of other, new elements that rely on older, familiar means; and sometimes there may be other reasons altogether that make us look back at time-proven, stable, reliable tools, with new eyes. Scale is essential to geographic representation, but its importance reaches far beyond such endeavors. It plays a major role in the design of research frameworks in a wide range of scholarly areas. These comprise data acquisition, data integration, and data analysis (including spatial analysis), visualization in numerous areas of scholarly research, identification of relevant aspects of social processes, creation of instruments for insightful exploration in the humanities, etc. While growing from a common place, the abstracting arms in the presented scale types embrace reality in different ways and achieve different outcomes. The book will first focus on each of the presented types, highlighting their main characteristics and putting them to work in application examples. Chapters 2 and 3 investigate scale as size (in space and then in time and in space-time). This investigation begins with the key issue of size comparison and includes the topic of perspective  – in everyday life, and in art; aspects of accuracy and distortion are addressed, and scale subtypes and their properties are then presented. Chapter 2 also includes a discussion regarding the scale debate in human geography. Chapter 3 presents challenges that are specific to scale in time, including effects of sampling and granularity, and shows scale at work in relation to the natural environment, including the impact of scale and scale change on the outcome of our investigations. Afterwards it focuses on approaches to synchronic and diachronic views and their implications, and on the meanings and the effects of scaling up and scaling down operations. Subsequently, Chaps. 4 and 5 analyze scale as ratio (in space and in time, respectively), starting with the theoretical basis of transformations and continuing with practical applications. Topics associated with maps, as hard copies and in digital form, dominate Chap. 4. Key topics include precision and accuracy in scale as ratio, map projections, automated map generalization, and feature-guided exploration of spatial databases. Chapter 5 includes a brief introduction to waves and transformations in time, as well as a range of applications of time scale change, from scientific applications to considerations regarding time scale in narration and literature. Chapter 6 is dedicated to scale as rank. Its foundations in terms of the theory of categories are presented first, after which the scale’s main properties are presented, and their implications, including the recognition of pseudo-scale as rank, are analyzed. Properties of histograms, their sensitive dependence on the interval size, and their implications for map representations are presented along with concrete examples. It is shown that the characteristics of scale as rank can be used in ways that distort reality, both in graphs and on maps. Chapter 7 introduces key notions of symmetry, focusing on scale symmetry, patterns, and pattern analysis. Fractals and the fractal dimension are presented and defined, and an effective multiscale pattern analysis

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method is presented in detail, along with practical application examples. Chapter 8 makes use of the concepts presented in the preceding chapter and expands the application area both in the field of art (cultural currents) and in the field of science (nonlinear processes). The last chapter, 9, uses the insights gained about each of the scale types, presenting a broad picture of scale. The elements that are at work in all three scale types are discussed in sequence, including the guided cut, logical field, mapping, morphism, and representation, and the emerging portrait of scale in its most general sense is presented. The applied framework is also summarized in graphical form. Concluding thoughts end this chapter and the book.

References Ayzenberg V, Lourenco SF (2020) The relations among navigation, object analysis, and magnitude perception in children: Evidence for a network of Euclidean geometry. Cognitive Development 56:100951. doi:https://doi.org/10.1016/j.cogdev.2020.100951 Bardell D (2004) The invention of the microscope. Bios 75(2):78-84 Boisot M (1995) Information space: A framework for learning in organizations institutions and cultures. Routledge, London d’Espagnat B (1989) Reality and the physicist. Cambridge University Press, Cambridge Euclid (300 BC/1945) Optics (trans: Burton HE). Journal of the Optical Society of America 35(5):357–372 Falguera JL, Martínez-Vidal C (2020) Abstract objects. For and against. Springer, New York Gibson CC, Ostrom E, Ahn TK (2000) The concept of scale and the human dimensions of global change: A survey. Ecological Economics 32:217–239 Grant E (ed) (1974), A source book in medieval science. Harvard University Press, Cambridge, MA Hofstadter DR (1985) Metamagical Themas: Questions for the essence of mind and pattern. Basic Books, New York Kant I (1787/2007) Critique of pure reason (B). Penguin Books, London Kent AJ, Vujakovic P (2017) Maps and identity. In: Kent AJ, Vujakovic P (eds) The Routledge Handbook of Mapping and Cartography. Routledge, Abingdon, p 413–426 Kelley LA, Kelley JL (2014) Animal visual illusion and confusion: The importance of a perceptual perspective. Behavioral Ecology 25(3):450–463. doi:https://doi.org/10.1093/beheco/art118 Lam NS-N, Catts D, Quattrochi D, Brown D, McMaster RB (2004) Scale. In: McMaster RB, Usery EL (eds) A research agenda for geographic information science. CRC Press, Boca Raton, FL Laurence S, Margolis E (2012) Abstraction and the origin of general ideas. Philosophers' Imprint 12(19):1-22 Lock G, Molyneaux B (eds) (2006) Confronting Scale in Archeology. Springer, New York Locke J (1690/1975) An essay concerning human understanding. Oxford University Press, Oxford Loder E, Burch R (2012) Measuring pain intensity in headache trials: Which scale to use? Cephalalgia 32(3):179–182 Lovejoy S (2019) Weather, Macroweather, and the Climate. Oxford University Press, Oxford Nottale L (2011) Scale relativity and fractal space-time. A new approach to unifying relativity and quantum mechanics. Imperial College Press, London Peer M, Ron Y, Monsa R, Arzy S (2019) Processing of different spatial scales in the human brain. eLife 8:e47492. doi:https://doi.org/10.7554/eLife.47492 Penrose R (2006) What is reality? New Scientist 192:32-39 Popper K (1972) Objective knowledge: An evolutionary approach. Oxford University Press, Oxford Posina VR, Ghista DN, Roy S (2017) Functorial semantics for the advancement of the science of cognition. Mind & Matter 15(2):161-184

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Preuss SJ, Trivedi CA, Vom Berg-Maurer CM, Ryu S, Bollmann JH (2014) Classification of object size in retinotectal microcircuits. Current Biology 24:2376–2385 RMetS  – Royal Meteorological Society (2018) The Beaufort Scale. https://www.rmets.org/ resource/beaufort-­scale. Accessed 24 July 2021 Taylor A, Jones MG (2009) Proportional reasoning ability and concepts of scale: Surface area to volume relationships in science. International Journal of Science Education 31(9):1231-1247 Tegmark M (2014) Our mathematical universe: My quest for the ultimate nature of reality. Vintage Books, New York Tretter T, Jones G, Andre T, Negishi A, Minogue J (2006) Conceptual boundaries and distances: Students’ and experts’ concepts of the scale of scientific phenomena. Journal of Research in Science Teaching 43(3):282–319 West GB (2017) Scale: The universal laws of growth, innovation, sustainability, and the pace of life in organisms, cities, economies, and companies. Penguin Press, London Wheeler JA (1966) Curved empty space-time as the building material of the physical world: An assessment. Studies in Logic and the Foundations of Mathematics 44:361-74 Wigner E (1960) The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics 13(1):1-14 Wolfram S (2021) How universal is the idea of numbers. Conference presentation at Numerous Numerosity, Society for Multidisciplinary and Fundamental Research, 25 May 2021. https:// semf.org.es/numerosity/ Zhang J, Atkinson PM, Goodchild MF (2017) Scale in spatial information and analysis. CRC Press, Boca Raton, FL

Chapter 2

Scale as Size in Space

Abstract  This chapter is dedicated to one of the two aspects of scale as size presented in the book; it thus focuses on space (scale as size in time is addressed in Chap. 3). It introduces elements of the ground framework applied in the book, starting with the system image and its role in research. Significant characteristic aspects of relations between scale as size and spatial extension are discussed, including comparison, accuracy, distortion, orders of magnitude, and hierarchy. Scale change as a function of distance and position, in images and on maps, is related to the problem of perspective. Two main sub-types of scale as size are presented and exemplified. The debate on scale in human geography and in areas of the social sciences is reviewed and discussed. Keywords  Scale · System · Map · Spatial extension · Foreshortening · Perspective · Stommel diagram · Networks · Urban environment · Fisheries · Hierarchy theory · Ecology · Human geography · Globalization · Marginalization

2.1 Setting the Stage: The System Image One can safely argue that every tool can only be useful if it is properly applied. Scale as size is no exception. We must know where we want to apply i, and why. We can then also find out how. Such questions and their answers become particularly important every time our goal is to address a real problem in the real world: when the instruments we deploy must be applicable in practice and work effectively. It is in this spirit that aspects that are specific to scale as size are approached in this chapter. Cutting out pieces of the environment and analyzing them separately has its drawbacks, and this approach has been justifiably often criticized. However, while it implies a limited point of view, it also represents an operationally applicable approach in our inquiries into reality. Without setting clear limits to the explored reality, we may end up with an image that is hard to read, even if we believe that “everything” is there (it never is). Cutting the system out of its environment is © Springer Nature Switzerland AG 2022 C. Suteanu, Scale, https://doi.org/10.1007/978-3-031-15733-2_2

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unavoidable, at least in a scientific context, but the cut must be made with care while being aware of possible consequences. We know that due to the virtual separation of our system from its surroundings, many of the relationships tying it to the environment are ruthlessly removed, and this can have major implications for our study. We should therefore always keep track of the boundaries we draw and of the criteria we applied to draw them. For our communication on the topic of scale to be functioning smoothly, we need to lay down some common ground, by agreeing on a couple of terms. At this point, there are two of them: system and system image. For the purpose of this book, we will use a general and widely applicable way of defining a system: a set of items considered as a whole. While this may not be the most common definition in many areas, it is expected to be helpful in the applications discussed here. On one hand, it makes it clear that – contrary to the perception that is often still conveyed in school – there are no “systems” out there. It is sometimes surprising for students to learn that unlike mountains, rivers, foxes, and trees, systems do not exist in the physical world: a system is whatever we choose to define as such. The well-known systems around us are so widely accepted because they proved to be very useful. A system is not a thing to discover, but an intellectual tool to apply to the surrounding world. On the other hand, depicting a system in this way helps us to move beyond the disciplinary restrictions imposed by other definitions. We have the freedom to select the elements that represent, together, the system we are going to study. We are the ones who choose where to draw the boundary between the system and everything else. This freedom comes, however, with a price: we must take responsibility for our choice. Certain systems thus defined can turn out to be very useful, others not so much. There is no exact rule to pick the “right” systems. The choice unavoidably depends on the questions asked. Since the answers depend on these choices, it is important to carefully decide upon the boundaries and to properly document them: when communicated, the outcomes of our exploration must be provided along with the specification of the actual analyzed system. On the other hand, the delimitation of the system can also be reviewed in the process of exploration: our systems can be redefined. Every time systems are (re)defined, the scale(s) in space and in time must be thoroughly (re)considered; as we will show in Chap. 7, a multiscale approach can often be useful. Research outcomes may, in fact, change if the system is delimited in a different way. An example may illustrate this idea. Consider the earthquake epicenter locations shown in Fig. 2.1: one can see distinct clusters in space, which are associated with geological features such as volcanic vents and fracture networks. If we wish to characterize one of these clusters, the exact location of boundary lines for the area taken in consideration can make a significant difference. Specifying the boundaries is usually the first step in pattern analysis. A system is usually not characterized by its elements alone. Relationships between the system’s elements and between those and the outer environment can be highly relevant. However, these relationships can vary over time, without such variability undermining the existence of the defined system. Emphasizing elements

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Fig. 2.1  Big Island of Hawaii, earthquakes recorded between June 29 and July 28, 2021. (Courtesy of USGS)

(rather than relations) has practical advantages in a large diversity of situations in which scale is applied as a tool. One can notice that scale as size is – implicitly – already involved here. It is present from the very beginning, when we establish the identity of the system we want to investigate. It plays a role in the system boundary-decision phase. The fact that scale is embedded in a concept considered to be as fundamental as the system is quite remarkable. Decisions concerning the identity of the system, its components, its boundaries, the extent to which relations between the system thus defined and the environment are taken in consideration, etc., have implications for the applied approaches, the methodological tools to be used, the interpretation of the results, and, eventually, the outcomes of the investigation. Let us consider, in several steps, an environmental issue as an example: chlorofluorocarbons and their impact. Step 1: Chlorofluorocarbons (CFCs) were discovered towards the end of the twentieth century, but it was in the 1920s that their synthesis was improved, and afterwards their use developed fast, leading to a variety of applications – in fire extinguishers, as refrigerants, aerosol can propellants, solvents, electric equipment cleaning products, etc. CFCs were stable: they did not react with other gases in the atmosphere, and they stayed at ground level rather than rising in the atmosphere. At this stage, the system consisted of the CFC molecules and the surrounding air, located at ground level. The spatial scale was defined by the tested chemical reactions and by the physical properties of the gas, all confined to the size of the laboratory.

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Step 2: In the 1970s, an unusual phenomenon of atmospheric ozone depletion was noticed, and eventually CFCs were identified as the main cause. In fact, CFCs turned out not to be as “ground-bound” as they seemed. In a landmark paper, Molina and Rowland (1974) showed that they gradually reach the stratosphere, even if it takes more than a decade for them to do so. There, CFCs cease to be the harmless, chemically inert gases they were at ground level: broken down by ultraviolet radiation, they trigger a chain reaction with major ozone depletion effect. At this step, the size of the system extended all the way to the stratosphere. The “stable” situation in the system defined in the first step was unstable in the newly defined system. The uncovered chain reaction showed that transformations occur at an accelerating rate. Step 3: In 1985, the so-called ozone hole was discovered above the Antarctic, and this triggered a new level of alarm among scientists. Moreover, while the main sources of CFCs were located on the northern hemisphere, drastic changes were seen on the southern hemisphere. The system had to be increased in size again, to include the atmosphere at the planetary scale. Following sustained advocacy efforts of scientists and various organizations, the Montreal Protocol was then signed in 1987, and this led, in several steps, to major transformations regarding the worldwide production and consumption of CFCs and the protection of the ozone layer. A range of lessons can be drawn from this significant experience: • Different scales can reveal distinct system characteristics. The scale at which we study a system is a decision that must be made carefully, based on current scientific knowledge. Scale is a critical component of system delimitation. • Interactions between system components are typically nonlinear. They can span a range of scales, and they may vary as a function of scale. This enhances the importance of system definition and boundary tracing: different choices can lead to considerably different or even radically different outcomes. • The relevance of the defined system can change over time. It is worth reviewing system definition along with our progress in scientific understanding. • When research involves complex natural systems, as it is often the case, it is useful to consider more than one single scale of analysis: a multiscale approach can be highly effective (this will be addressed in more detail in Chap. 7). Certain approaches in physics assign more weight to processes than to objects, to “becoming” than to “being” (Prigogine 1980): they see a world made of events, not of things (Rovelli 2019). Such approaches have been highly successful providing insights into the workings of complex systems. Looking at the environment through the lens of change can be rewarding indeed; in some cases, doing so can be even a requirement. This does not mean, however, ignoring its material components in order to concentrate solely on flows, on processes. As we will see in the next chapters, it is not only justified, but even essential to identify the elements involved in the studied phenomena, including those elements that are reliably expected to be stable throughout our study. Therefore, designing systems around those elements – such as people in a social network, species in an ecosystem, etc. – is in most cases a productive starting point in our exploration of the environment.

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Our exploration is not performed, however, directly on the system. This is true at least for an investigation that strives towards a better understanding of our environment (rather than one at the level of emotional experience, which, albeit important, is not addressed here). In a variety of ways, we obtain a system image (Fig. 2.2). The system image is made of everything we know about the system, based on observation, the application of instruments (whether material instruments or methodological ones), information sources that we have assimilated, as well as a range of transformations that we have operated ourselves upon the available image (various forms of knowledge integration, new connections between disparate items, etc.). The system image is a dynamic feature, subject to change. It develops due to accumulating facts from our own studies and/or from external sources, as well as from new insights and our deepening understanding of the elements and processes involved. We have access to parts of the environment, which we have identified and called “systems,” through their system images. All the pictures, all the data, all the relationships we are aware of, everything we understand, is part of the system image. And it is the system image that we keep analyzing. Not the lake, not the community, but their system image. To conclude, we apply an abstract entity, scale, to an abstract configuration, the system image. Everything happens there. In abstract space. Not in the field, not where people interact. The place where people interact, the lake where we study an

Fig. 2.2  A simplified image of principles regarding the production and use of the system image. (Photo courtesy of USGS)

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ecosystem, they are all important points where our understanding starts, but not where our understanding ends. It is in this context that we can begin to apply our tools based on scale. The fact that in science we apply methods (which have an abstract nature), to acquire, handle, and understand information about the world around us (this information has an abstract nature too), leads certain scholars to intriguing views, such as doubting the existence of a mind-independent reality. We will leave any ensuing debate aside and walk along the paths of our increasing understanding of the environment. The real one. The one in which wonders occur, such as minds asking questions and reaching answers that trigger more questions, in their turn, like fireworks.

2.2 Scale as Size: Comparison and Change 2.2.1 Comparing Sizes The simplest meaning of scale is size. It is not surprising, therefore, that “scale as size” in space begins, in fact, with identifying and handling size. As mentioned earlier, assessing distances and sizes as accurately as possible has been important for a very long time. After having played a prominent role throughout our history, size or spatial extension even reached the point when it became the defining trait of matter itself. With Descartes, the world turned out to be fully understandable to the human mind, since matter could be captured by the concept of extension, and extension was accessible to the penetrating power of geometry. Thereby, scale as size opened a wide and bright mathematical door to the study of reality, and this had deep and persistent implications for the subsequent becoming of humanity. How relevant is size for us today? Perhaps more than we tend to realize. It concerns the most different aspects of life. We pay more attention to it when faced with explicit challenges or questions – whether our kitchen door is wide enough to allow the new refrigerator to go through, if this dog is tiny or huge, if that parking space is large enough; all these can make a difference. Anyone would probably expect size to be a critically important feature that humans can identify in their environment. This expectation was confirmed by studies organized in a wide variety of circumstances. One of the relevant findings refers to the relation between the size of our objects of interest (“targets”) and the way in which we pay attention to them: according to investigations by Hopf et al. (2006), larger targets trigger attention-­ related neural activity in higher level areas of the visual cortex, compared to smaller targets. In other words, even the neural processing area changes with size. Larger targets are distinguished from smaller ones very early, during the first several hundreds of milliseconds, and a higher level of attention is assigned to the larger ones. This occurs long before other processing phases take place (e.g., recognizing the object, evaluating its significance in the given spatial context, etc.). Comparing sizes of different objects is a skill of particular importance. In a study involving subjects of all ages from elementary school to graduate students (Tretter et  al. 2006), all

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students were more accurate when comparing the size of objects (evaluating their relative size) than when establishing their absolute size; moreover, their assessment accuracy was higher on “human scale” ranges than in size domains of the very small or the very large. Developed early on in children, size perception is deeply involved in object classification as well as in spatial orientation and movement, and this connection is even reflected in the spatial position of cortical areas that are responsible for such tasks (Ayzenberg and Lourenco 2020). Scale as size related to distance is critical to spatial orientation, as well as to location retrieval based on spatial cues (Commins et al. 2020). The relation between the size of an object and a person’s distance to the object involves specific problems to solve. In the case of obstacles encountered by persons who walk, for instance, object size estimation can be an even more urgent and critical task than the evaluation of the distance to the object: knowledge about object size offers key premises for the person’s safe navigation in space. Moreover, size estimation is particularly valuable in a context featuring complex arrangements of many objects of different sizes (Diaz et al. 2018), which is highly relevant since the natural environment is often characterized by such a context. In spite of the apparent simplicity of the task, assessing the size of objects that lie a certain distance away from us can be challenging: the perceived size depends both on the size of the object and on the distance that separates it from us. If we can only perceive an object without being acquainted with it and without knowing the distance (looking, e.g., at a cylinder floating in the air), we cannot find out the size of the object, because we cannot differentiate the effect of the size from the effect of the distance to the object. If we knew one of them, we could – based on experience – approximate the other. This can happen if we are either familiar with the object, thereby guessing how far it is located, or if we know approximately what the distance is, so that we can assess the size. The task of estimating the size s based on the known distance r is simplified by the fact that for small angles α, the tangent of α can be approximated with α. In other words, we can consider that the size s is proportional to the distance and the viewing angle. As we can see in Fig.  2.3, this approximation holds even for angles that are not close to zero: depending on the purpose of the estimation, it can be reasonably accurate for larger angles as well, and it is useful in a wide range of real-life situations. The eye-brain system can discover relatively quickly how to determine sizes based on their dependence on angle and distance, but it can do so much more than that. Its sophisticated ways of making rapid assessments of complex configurations can also be the reason for some of those occasions in which it commits errors. For instance, Robinson (1998) showed that the same object typically looks longer if we see it in the vertical position than in the horizontal position (Fig. 2.4, left). A common practical implication is that we tend to overestimate the height of trees. The sources of incorrect visual estimation of sizes are usually rich and complex, and not limited to one factor only, such as the spatial orientation of the object. Even for this example, one may find instances when the vertical orientation of the object does not necessarily lead to the perception of a larger size. Figure 2.4, right, shows the same tree trunk, in a different configuration. This time, the illusion of a taller

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Fig. 2.3  Size of a tree trunk seen at a distance under the viewing angle α. The graph compares size change based on the tangent of the viewing angle with the linear approximation of the angle tg(α) ≈ α

Fig. 2.4  Left: Which tree trunk is the tallest? Robinson (1998) showed that vertical objects typically look taller. Right: Errors in size assessment can have other sources too. Here, the vertical trunk does not look longer than the other three. The relative position of the items must also play a role

vertical pole is weakened, if not suppressed altogether. It is a good idea to avoid assuming that a vertical structure is necessarily less tall than it seems. The dilemma of size vs. distance is usually promptly solved when we recognize either the object, or some of its elements, or certain items that lie close to the object, i.e., equally far away from us. In the case of a tree, we might see its leaves, or grass or bushes underneath, or the height at which the tree begins to branch out, and if we are close enough, we might even distinguish a pattern on its bark, etc. When all such

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elements are missing, it might not be possible to tell the size of the tree, especially because the tree is an example of a self-similar feature: various parts of the object, on different scales, are similar to each other (Fig. 2.5): they all correspond to the same “pattern” (a notion that will be closely addressed in Chap. 7). For the same reason, it is difficult to tell the size and/or the distance to mountain peaks that are far away: topographic surfaces also enjoy this property, as do many features in the natural environment. In order to make sense of the environment in such circumstances, we need to adopt a different approach, as explained in Chap. 7. Patterns that are similar on different scales do not have to raise difficulties in interpretation: on the contrary, when approached with an appropriate methodology, the multiple, consistent occurrence of similar shapes can offer, with its geometric redundancy, enhanced reliability to our pattern evaluation endeavor. In contrast, elements that are unique can be more easily misevaluated, especially as a function of the spatial context in which they appear.

Fig. 2.5  Various parts of the tree branches look similar (though not identical) to each other

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Context is involved in many ways in the process of understanding the environment. For instance, the difference in size between an object and the items that surround it may lead to a misevaluation of the object size (the Ebbinghaus illusion, Roberts et al. 2005); a size misevaluation also occurs when one associates the target object with other elements in the same image, which are known to be larger or smaller (the “assimilation effect,” Sarcone and Waeber 2002). In general, while the context often has a significant impact on the apparent size of the analyzed object, its role is not reducible to simple rules regarding sizes, shapes, and relative positions, considered separately. More often than not, it is the synergic effect of such features that influences the way in which objects are seen. Incorrect size estimation can have major implications for the way we examine and understand the world in and around us (Ross and Plug 1998). Such problematic situations, including the evaluation and interpretation of the magnitude of objects, the comparison of sizes of different items, depth assessment etc., extend from the microscopic scale (e.g., in histology) to medical imaging at the human scale, all the way to planetary-scale exploration (Alexander et al. 2021; Anderson 2011; Liu and Todd 2004).

2.2.2 Size Versus Distance Change in Perspective While estimating the objects’ size is an important step in our learning endeavor, recognizing change in the perceived size of an object is another step. It was certainly easier to notice the apparent size change of an object with distance, than to correctly grasp the actual relationships between sizes involved when such changes occur. The problem of relationships of size in a triangle had been elegantly solved mathematically in the sixth century BC (by Thales, unless this was accomplished sometime before him). Making use of this knowledge, however, was not enough to move from the detection of spatial configurations to an understanding of the relationships that are involved, along with their multiple implications. This task was so difficult that it took generation after generation to watch, to reflect, and to experiment, until the light was turned on in the intellectual space of size-distance alliance. The breakthrough happened when the laws of change in size were uncovered, and the linear perspective was born in all its mathematical clarity and beauty. The name often associated with this achievement is the one of Filippo Brunelleschi, architect, sculptor, and engineer from Florence. Nevertheless, his fifteenth century breakthrough was preceded by numerous other more-or-less connected and more-­ or-­less successful steps. The simplest part had been to notice that the size of an object seen across the line of sight, whether vertical or horizontal shrinks with increasing distance. Moreover, it was also noticed that an object’s size along the line of sight looks shorter than across the line of sight (in what is called “foreshortening”). Consider the example in Fig. 2.6. All the parallel lines produced by connecting the corresponding parts of the different shelves seem to converge into a point located far ahead – the “vanishing point.” The distance between the shelves on the left and those on the right can be seen indeed decreasing when we look at shelves farther and farther away from us.

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Fig. 2.6  The vanishing point in this photograph is indicated by the lines connecting analog elements on each shelf row. Note also the shortening of the horizontal beams, as well as the changing distance between successive shelves. (Photograph by Ralf Roletschek)

On the other hand, the distances between successive shelves, on either side, are shortened in their turn. Transferring this insight to in paintings or drawings means more than just proving one’s understanding of relations between distance, size, and position. The moments in history when humans have become capable of mastering linear perspective, including its underlying theoretical framework, can be viewed as milestones on the journey of understanding. There is a consistent tendency to associate the birth of linear perspective with the Italian Renaissance. This was, indeed, the time when the representation of spatial perspective reached both unprecedented levels of perfection and wide adoption in painting. We know, however, that graphical representations incorporating foreshortening, if not perspective as such, were already made in ancient Greece as early as the sixth century BC, while perspective was applied, in a different way, in ancient China and India (Cucker 2013). More importantly, the painter Agatharchus from Samos (fifth century BC) not only used what might be called “perspective” principles in his paintings: he even laid down the theoretical basis for this representation in a dedicated treatise (Arafat 2012): in this way, his work became widely known, and even Democritus and Anaxagoras built on it. His work was lost, however, and while a remarkable rendering of spatial features was accomplished in certain instances (as in Pompeii, 200 years ago), both the lack of a proper technique with its theoretical basis and a different cultural setting led to the subsequent loss of our understanding of the interplay of size, distance, and position.

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The fact that images of the environment had to be distorted in order to look like reality – in fact, to be perceived in two-dimensional pictures as if they were seen in three dimensions by our own eyes – might have seemed strange in the beginning. And before the interdependence of size and distance could be deciphered (or re-­ deciphered), attempts at opening up the three dimensions of the environment on the flat surface of the canvas often led to strange results. When Brunelleschi made his discovery, he combined his mathematical background and his experience as an architect with – surprisingly enough – experiment. After grasping features of spatial transformations by producing drawings of ancient monuments (which might have indeed acted as problem openers and question raisers), he devised a unique study setup. He made a small hole in the middle of his painting, which he held close to his eyes, facing away from him. In his extended arm, he held a mirror. By looking through the hole in the painting, he could see his painting in the mirror. When he moved the mirror away, he could see the painted building in front of him: by moving the mirror in and out, he could thereby compare his work with the three-dimensional environment out there, making corrections, comparing again, learning. Eventually, according to Vasari (1568/1991), he even invited people on the street to have the same experience, in what became a sensational event. Brunelleschi’s technique was quickly adopted by other painters, but it was mainly when his ideas were soon afterwards crystallized in a theoretical framework, “coded” in a step-by-step methodology by Leon Battista Alberti, that the new way of understanding space perception really took off (Edgerton 2009). The main idea behind the linear perspective was to extend imaginary lines from all parts of the represented environment towards the chosen vanishing point (Fig. 2.6). One takes note of where those lines would penetrate through the plane of the painting, virtually located between the vanishing point and the represented part of the environment. The ghostly intersection points between the imaginary lines and the imaginary plane are then materialized on canvas. The result of this gradual modification of scale, obtained by consistently following a mathematical rule, is an image that can be surprisingly close to what a three-dimensional environment would look like for a viewer in that position. A flat surface is transformed into a new kind of space. This way of capturing the relation between size and distance led to a new and spectacular representation of reality (Fig.  2.7), assimilated by many artists. However, it accomplished more than that. It contributed to an expanded worldview. Such transformations based on (linear) perspective are usually applied when the viewer is supposed to look at the surroundings from a position approximately situated at eye level. What happens when we do not look horizontally at the features of the environment, but vertically, right from above? Can we apply a similar reasoning in that case too? The question is relevant in relation to maps, which are designed to show the represented part of the environment as seen from above. We can imagine a simple spatial arrangement, including vertical features, so that the question asked above can be addressed with enhanced clarity (Fig. 2.8a). For simplicity, we choose the position of the viewer above one of the features. Thereby, at least one of the items can be simply represented by its projection on the horizontal plane.

2.2  Scale as Size: Comparison and Change

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Fig. 2.7  An illustration of perspective with one vanishing point: Hans Vredeman de Vries, sixteenth century. (Courtesy of the National Library of the Netherlands)

The lines connecting the cylinders to the viewer’s eye highlight the two distinct situations we are facing: one item is seen as a circle, the other one (along with all the remaining cylinders) is still perceived as a cylinder – with the necessary change in size imposed by spatial perspective. The result may seem complicated, but this would surely not be a more challenging problem than applying linear perspective in the horizontal direction (as in Fig. 2.7): all the necessary theory is there. And yet, we do not represent vertical views in this way – certainly not on maps. On a map (a very large scale one), the six features would be represented as in Fig. 2.8b (if not simpler, by using a conventional sign). We act as if we moved the viewer’s position for each item, locating it right on the same vertical with each feature, one by one. In other words, we explicitly choose a suppressed perspective. What we end up showing is an impossible view to get in reality, regardless of our position. There is thus another level of abstraction involved in the creation of the map, long before we decide to select only a small number of features from the actual environment.

2.2.3 Size Distortion, Accuracy, and Implications The relation between the objects’ size and the distance that separates us from them is thus not as straightforward as it may seem. As it often happens, something that appears to be simple when considered one item at a time can quickly gain in intricacy when interacting items are admitted into the picture. As we can see in Fig. 2.7,

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Fig. 2.8 Suppressed perspective: (a) Three-­ dimensional features seen from above: even if one of the columns is seen directly from above, as a circle, all the others are perceived as having a height that is shortened depending on the distance. (b) The corresponding map, where each element is seen directly from above, as if the perspective had been suppressed

the perceived sizes shrink when we look farther and farther from the bottom of the image. Sizes differ, however, also along the horizontal direction. In the end, every area in the image corresponds to a different change in scale, and those differences are all interdependent. Compared to the operations involved by proportional change in size, which is applied in the case of many maps and two-dimensional and three-­ dimensional models, this representation has a higher degree of complexity. These intricacies are not mere artifacts meant to embellish a picture or to surprise the viewer with an unexpected scene. They are designed to create a view to be recognized as what we would actually see in our three-dimensional environment. The way in which every piece of the resulting image is distorted in a specific manner is, in the end, providing us with knowledge about the size of the objects, their distance to us, and their spatial relation to each other. The calculated distortion has the goal of recreating the kind of distortion our own view relies on when faced with the environment and thereby to offer us an accurate image of reality. A groundbreaking event like this one – the capturing of relations between the objects’ size, their location and position, and their perceived spatial extent – was not expected to sit quietly in a dusty corner of history and be forgotten. Indeed, it shook the artistic view, the scientific view, and, eventually, the worldview of those times,

2.3  Orders of Magnitude

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reaching more and more minds across Europe. Acting like sparks added to the already accumulating “fuel” of optical devices – the telescopes – the new insight contributed to the spreading intellectual fire that burned increasingly brightly, presenting the world in a different light. While not always explicitly named as such, scale was there, at the origin of the fire, both in the fuel of optical devices and in the sparks of perspective, and kept being embedded in the newly expanding understanding of the world. The extent to which more and more artists, on a wide geographical area, challenged viewers with mindboggling new images, stimulating them to question their familiar vision, must have been astounding. One can only expect that their work had a significant contribution to the widespread transformation that occurred in the landscape of thinking. And a change in the ways of thinking – especially when it affects a broad, interwoven community  – can produce major leaps in understanding. A meaningful example is offered by Galileo (Edgerton 2009). He was not only knowledgeable regarding linear perspective. He had assimilated it in depth. He had explored the relation between aspects of the environment and their representation. Having studied early in his life the art and science of applying linear perspective in drawings, he thoroughly practiced the depiction of shadows that appear due to protuberances and channels on a curved surface. It is this training, Edgerton (2009) finds, that made Galileo able to immediately recognize what he actually discovered in the image of the moon, using the telescope that he had built right after it was invented. Galileo did not see just shades of gray, commonly attributed at the time to the moon’s “translucent internal composition.” He recognized mountains, valleys, a topography as irregular as here on Earth. The moon was not a perfect, smooth sphere, as taught by Aristotle, a view that had persisted for two millennia. Benefitting from his understanding of perspective, Galileo turned a mere observation into a major discovery, with significant implications for our history.

2.3 Orders of Magnitude To do order-of-magnitude physics is to distill, to make insightful, simplifying assumptions […]. It is hard to overestimate the empowerment that order-of-magnitude estimation brings. —Eugene Chiang (2015)

Scale as size often relies on the use of orders of magnitude, which are often applied in science. Magnitude, frequently considered equivalent to “size” or “quantity,” is particularly useful when one refers to values that are very different from each other and to variables that cover a wide range of values. Two numbers are separated by one order of magnitude when one is approximately ten times larger than the other. In a general sense, orders of magnitude refer to values that are k times larger from one order of magnitude to the next. However, since multiplication and division by ten are so advantageous and so widely used, normally one order of magnitude refers to the range between a given value and another value that is about ten times larger.

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Orders of magnitude offer a concise and useful way of referring to variables expressed in international system units. For instance, a distance of one or several kilometers is three orders of magnitude larger than a distance of one or several meters and is six orders of magnitude larger than a size of millimeters (this certainly works better than for other units such as the mile, which is equal to 5280 feet, which comprises in its turn 12 inches). The frequency of words such as “about” and “approximately” in the preceding paragraph stands in stark contrast to the usual precision requirements of scientific language. How are then orders of magnitude useful in science? They are expected to filter out everything but the important aspects of the comparison between very different quantities. For instance, if a person is 1.7 m tall, and a basophil (a type of white blood cell) in the blood is 15 μm in diameter, their ratio is 1.7 × 106 μm/15 μ m = 1.133 × 105. When values are that different, the actual number 1.133 is pretty useless, and we would rather retain the fact that the basophil is five orders of magnitude smaller than the person. The notion of “orders of magnitude” was exactly what we needed. The concept’s capability of selecting relevant information proves its effectiveness in many other ways. For example, when we must compare quantities given with different levels of precision, or when our information about a quantity is vague, orders of magnitude can help. The notion is also helpful, for instance, when we find that certain quantities are “of the same order of magnitude.” In spite of its apparent lack of precision, this form of scale as size is a useful intellectual instrument. Since the logarithm of a number n is the exponent to which we must raise another value representing the base, to get that number n, orders of magnitude can be conveniently represented using logarithms. The use of logarithms, in its turn, can decrease the precision with which the represented values can be discerned, by “squashing” the differences: Fig.  2.9 illustrates this effect using a set of random numbers (Y1) and their logarithm (Y2). However, depending on the circumstances when it is applied, this loss-of-precision effect does not have a negative impact on the outcome of the operation. In Fig. 2.10 the successive intervals, each of them ten time larger than one preceding it, are relatively narrow. No exact limits would be necessary or useful here. In this case, as in many others, we are mainly interested in scale size ranges, in intervals of distance values, which can be suitably represented in a logarithmic scale.

2.4 Sub-Types of Scale as Size Scale as size is conceived and applied in more than one way: in this regard, it is not different from the other two categories of scale addressed in this book. We will present the main sub-types according to their degree of abstractness.

2.4  Sub-Types of Scale as Size

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Fig. 2.9  A set of random numbers (Y1) and their logarithm (Y2). X indicates their succession. Note the amount of scatter in each of these two cases. Y2-values are shifted vertically in view of the comparison with Y1

Fig. 2.10  Scales of size in space. The boundaries have approximate values

2.4.1 Scale as Size with Units Scale may refer to a certain size by specifying an actual value, along with the units used. In this case, its precision is the same with the precision associated with that variable. It can be also be applied in statements such as “on a scale of tens of kilometers….” The specified range can also be wider, involving more than one unit, e.g., “on a scale of millimeters to kilometers….” Often applied in science, it may be surprisingly useful, considering its relative vagueness compared to many other

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scientific concepts and instruments. In this latter form, scale is not precise, neither is it expected to be. Biomes offer a good example of ecosystems on a wide range of scales in space, from global climatic zones to continental and sub-continental scales, to spatial scales of soil and water distribution, to even smaller scales like those of plant growth (Mucina 2019). On the other hand, habitat heterogeneity and biodiversity studies show that species richness is related to area (Kallimanis et al. 2008); as it is often the case, we keep specifying spatial scale in terms of a unidimensional quantity: scale as size is simply given as a length, even when we refer to two- or three-­ dimensional aspects of the environment. A particularly useful aspect of differentiating scale intervals consists of the distinction of scale-specific dominant processes. For example, the formation of gullies (erosional landforms – small valleys or channels with steep sides, often produced by fast-flowing water) can imply a variety of factors. While the numerous physical processes implied are interlinked on many scales, one can recognize factors that are dominant on certain scale intervals (Wang et al. 2021): for instance, land use, slope value, and slope length on smaller scales, and rainfall and soil type on larger scales. An example of interdependent processes acting on a wide range of space and time scales is offered by the climate system. It is a complex system involving interlinked feedback loops (which act over a wide interval of scales), instabilities, and forcings, which are characterized by scale-range dependent variability (Ghil and Lucarini 2020). Turbulence in the atmosphere extends over many orders of magnitude. Cloud structures can be observed from the scale of the planet to smaller and smaller scales, with structures being observable down to a scale of millimeters (Lovejoy 2019). Interactions across scales occur over many orders of magnitude: microphysical in-cloud processes have an impact on large-scale precipitation variability (Hagos et  al. 2018). Scale as size is not just relevant, but essential to our understanding. This sub-type of scale usually refers to scale intervals rather than individual values. Examples include the spectrum of electromagnetic radiation by wavelength, from long radio waves to gamma rays, or the Udden-Wentworth scale for grain size in soil or sediment (distinguishing grain size domains corresponding to clay, silt, sand, and gravel, each of these including a series of sub-categories). In most of these cases, the exact domain boundary values do not have particular relevance. Physical processes occurring in the environment cover a range of scales in space and in time. It can be useful for our understanding of the phenomena of interest to represent relevant processes in a “space” defined by spatial and temporal scales. For instance, Stommel diagrams (named for Henry Stommel, who first used them in oceanography) usually have space on the X-axis and time on the Y-axis and use a logarithmic scale. The logarithmic representation is favorable to the grasping of scale ranges spanning many orders of magnitude, which can support the visualization of features and processes that are otherwise difficult to compare. At the same time, the difference-crushing effect of the logarithmic scale can distort the perception of phenomena when such diagrams are applied. An example is shown in Fig. 2.11, where relevant scales in space and in time are represented for

2.4  Sub-Types of Scale as Size

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seconds century 109 108 year 107 106 105 day 104 hour 103 102 minute 101 second 100 10-1 metres

Humans

Fruit flies

10-4 10-3 10-2 10-1 100 mm m

101

102

103 104 105 km

106

Fig. 2.11  Spatial scale vs. temporal scale for two forms of life. An effect of logarithmic axes in a Stommel diagram: fruit flies and humans do not look so drastically different from each other, especially in terms of their life span

Drosophila melanogaster or fruit fly (Perrimon et al. 2016) and for humans. On one hand, we can notice in this case again that precision is not the foremost requirement for the diagram, given the logarithmic nature of the graph: this level of imprecision matches the intended message, since it does not make sense to specify “exact” life span and travel distance values. On the other hand, we can see that as a side effect of the otherwise useful logarithmic representation, we obtain a potentially misleading image: the scale ranges characterizing fruit flies and humans do not look radically different from each other. This is an illustration of the fact that the advantage of bringing together processes that occur over very different scale intervals comes with a price. Figure 2.12 presents an application of the Stommel diagram in relation to coastal processes. It offers a comprehensive, yet easy to interpret image reflecting space and time scales associated with the relevant physical processes. One can thus make a meaningful comparison of the factors of interest, simultaneously in space and in time, with a remarkable clarity and simplicity. In this example, the diagram also highlights a scale-range-defined area that is relevant to a certain group of specialists (in this case, coastal managers). Although spatial and temporal aspects of the studied systems dominate instances when scale as size is applied, other quantities can also be involved. The variables implied may even be nondimensional. For example, we may be interested in the relationship between the size of an area and the number of elements of interest that are involved in that area. An example where both axes in a graph are not only non-­ spatial and non-temporal, but nondimensional, is shown in Fig. 2.13: the modeling of earthquake patterns using networks (M. Suteanu 2014) includes the study of the size distribution of nodes, where the size of a node is represented by the number of

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Fig. 2.12  Stommel diagram regarding coastal processes. (Courtesy of USGS)

edges connecting it to other nodes. Such relationships between values of scale are further discussed in Chap. 7.

2.4.2 Scale as Size Without Units Most of our language is inexact. Yet what we mean is communicated. —St. Augustine (fourth century/1998)

With a higher degree of abstractness than the preceding one, this subcategory of scale still refers to actual magnitude in space or in time, but without making explicit use of units. In contrast, a frame of reference is usually offered, to clarify what is meant by “small scale” and “large scale.” The decreased precision it involves is compensated by the reader’s expected background and, more generally, by context. One of the most common ways of using this scale is meant to distinguish “local” from “global” scales. Usually when local and global scales are presented in opposition to each other, only a vague relation with spatial quantities is implied. Even the levels of administrative units that are implied can remain unspecified: for instance, it is not always clear if by “local scale” one means a town or a whole province or region. And yet, the message meant to be conveyed is usually unambiguously recognized.

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Fig. 2.13  Logarithmic graph showing the relationship between two instances of scale that do not represent spatial or temporal variables: the number Nn of nodes having a number Ne of edges, in a network used in the modeling of earthquake patterns. (With data from Mirela Suteanu)

One can encounter such an approach in relation to local scale vs. global scale in a wide diversity of applications, such as projects focusing on social justice, on supporting local communities through links on an international level, on community capacity building and global knowledge exchange, on policy development to address common challenges at the local level through worldwide collaboration, etc. Interestingly, local and global scales are frequently mentioned by authors who propose that they should be abandoned and replaced by a more flexible framework, or who advocate for major shifts in their relationship to each other (Shuman 2021). Other scholars recommend that we abandon reference to these scales altogether, and “the linear metaphor of scales, such as the stretching from the micro level to the macro level […] should be replaced with the metaphor of connections” because trajectories in the social space “are neither macro nor micro” (Urry 2003). Significantly, however, even the authors who recommend that we should forget about the concepts of local and global scales make ample use of these same concepts while writing against them. It appears that the mere mentioning of “connections” or “flows” is insufficient: it is leaving these newer proposed concepts floating in the air, making them unusable, unless they get anchored in reality, with the help of scales. The fact is that the use of local and global scales has not been abandoned. We are currently witnessing a considerable growth in interest regarding the significance and the characteristics of these two levels of scale, as well as the evolving relationships between them. According to the Google Books Ngram Viewer (2021),

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over the last 50 years, the relative frequency of the phrase “local to global” in published books has increased more than 20 times and is still growing. During the same interval, the phrase “global scale” has grown twice as fast as “local scale.” An incursion into the controversial issue of scale in human geography is made in Sect. 2.5. In some cases, scale as size is indicated indirectly, by making use of subsystems or building blocks of the analyzed system. For example, urban characteristics identified for smaller units, at the household level, can differ from those found on larger scales, at the street segment or at the neighborhood level (Umar et al. 2021). In this case, building blocks are more relevant than their corresponding spatial scales. When people’s perception of their urban environment is studied, it turns out that this distinction can be more subtle. Eizenberg et al. (2020) identified three classes of scale: intimate (“a few buildings close to my home”), intermediate (“a few streets around my home”) and neighborhood. All three are only loosely connected to actual spatial scales, but these are the scales that truly count for the people who live in that environment. On the other hand, a larger scale – the size of a city – is relevant from many points of view; for example, a strong correlation was identified between a city’s increase in spatial scale and the intensification of socioeconomic segregation (Monkkonen et al. 2018). Scale as size, with or without units, proves to have a major significance on many levels. Scale as size of this sub-type is also successfully applied when the targeted aspects do not involve spatial and temporal aspects alone, but also several other parameters, which must all be considered together. An example is offered by research on fisheries. Here, the terms “small scale” and “large scale” offer a compact, synthetic view of a multidimensional feature. Some of the aspects included in their assessment are purely quantitative, such as the annual catch, the mass of fish destroyed per year as by-catch, the number of fishers employed, capital cost per job on fishing vessels, etc. (Carvalho et al. 2011). However, quantitative variables are far from telling the whole story. As long as we are aware of this reality, we can still include quantitative parameters in our evaluation. These parameters can be used to construct a space in which the representative points of fisheries are seen from a different perspective than by looking at numerical values only. Two- or three-­ dimensional graphs can be applied for this purpose. An example is shown in Fig. 2.14, which relies on data from Carvalho et al. (2011). There is a striking difference in terms of the number of fishers employed (0.5 million in large-scale fisheries and 12 million in small-scale fisheries) and the quantity of fish destroyed as bycatch (11 million tons, on average, for large-scale fisheries and virtually none for small-scale ones). Such a graphical representation can also be used when one wishes to follow changes in a comparative perspective, e.g., in certain communities, for specific fisheries, etc., by showing the trajectories followed over time by their representative points in this type of space. It is indeed methodologically helpful to operate in a space defined by quantitative variables. Nevertheless, it is essential to be aware that we cannot exclusively rely on such variables, if we want to reach an in-depth understanding of fisheries and their links to communities and to ecosystems. This is particularly the case in a context driven both by economic and climatic change. We must also consider,

2.4  Sub-Types of Scale as Size

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Fig. 2.14  Small-scale and large-scale fisheries represented in three-dimensional space, in terms of the number of fishers employed, annual catch for human consumption (in tons), and bycatch destroyed per year (in tons). (Based on data from Carvalho et al. (2011))

against the backdrop of scale, other aspects that are not readily quantifiable, including social relations and ecological relations, seen in their dynamic development, as pointed out by Charles (2021) and Smith and Basurto (2019).

2.4.3 Scale and Hierarchy The tree is hierarchy. But where do you see one part dominating another? —Antoine de Saint Exupéry (1948)

Addressed in increasing order of abstractness, the sub-types “scale as size with units” and “scale as size without units” are followed by “scale in virtual space.” The latter is not explicitly related to spatial or temporal quantities. It often operates in abstract space alone. One of its specific features is the fact that it comes in a variety of “flavors.” For instance, it can be framed as “level of analysis” or as “scale of analysis,” as it is often the case in history, sociology, and political science. The exact meaning of this scale can be heavily dependent on context and confined to a single field. Cognitive hierarchy theory offers a good example of such a domain-specific use, in relation to social and economic processes (Gracia-Lázaro et al. 2017). The various incarnations of “level of analysis” can take different shapes, more or less clearly related to the broader understanding of scale, especially of scale as size without units (e.g., “small scale” to “large scale”), and they are usually focused on a well-delimited application area. In contrast, a more general approach is offered by hierarchy theory.

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Hierarchy theory occupies a notable position in several scholarly fields. A key reason for this is its perceived capability of addressing the problem of scale (Gibson et al. 2000). Perhaps no field has been benefiting so much from – and contributing so much to – hierarchy theory as ecology, even if its scope has been overflowing disciplinary boundaries (Ahl and Allen 1996). Hierarchy theory absorbed key insights and methodological values from its deep roots in general systems theory, which flourished after the second world war. For the reader of the twenty-first century, general systems theory books – like the domain-­ defining one by Bertalanffy (1968) – might be surprising. One tends to flip back every now and then to the first pages of the book, to check when the book was published – so surprisingly advanced are many of the ideas put forth. Surprisingly, the meaningful developments in nonlinear science, which started in the 1970s in mathematics, physics, biology, and other fields, and which have been quickly expanding both on the theoretical and the applicative level (Suteanu 2005), did not have the same impact on hierarchy theory. Common key features of complex nonlinear systems (such as the lack of a characteristic scale, the sensitive dependence on initial conditions, etc.) do not play in hierarchy theory the role one would expect, considering the major advances from which it could have benefitted. Therefore, one may speculate that its power of applicability is more limited than it might have been, had it kept assimilating the relevant findings developed in the study of complex nonlinear systems, which are currently applied in a wide diversity of disciplines. Hierarchy in a system is defined by the existence of a structure consisting of layers (levels), which have asymmetric relations among them (Wu 2013). For instance, if in layer A there is a sub-layer A1 (along with sub-layers A2, A3, etc.), A includes A1, but A1 does not include A. In this case, when hierarchy is based on inclusion, it is called nested hierarchy: a system includes a set of subsystems, each of which includes subsystems in its turn, and so forth, all the way down to the level of so called “elementary” subsystems. Examples of nested hierarchy can refer to biological systems, various human organizations, classification schemes, etc. Similarly, in another example of hierarchical asymmetry, A has a dominating position versus A1, and not the other way around. Hierarchy in this case, which does not involve inclusion, is called non-nested. We can think of examples such as command hierarchy, postal codes, trophic levels (the organisms’ position in a food chain), etc. (Wu 2013). Among the two, it is nested hierarchy that has a direct relation to scale as size. To better distinguish the relation of hierarchy types to spatial scale, another way of characterizing hierarchy can help (Suteanu et al. 1993): a rigid hierarchy is one in which all subsystems at the same level are associated with the same spatial scale. Therefore, the hierarchic level of a subsystem can be identified based on its size. In contrast, if the levels of subsystems cannot be distinguished by scale (or size) alone, they form a distributed hierarchy. In this case, one and the same hierarchic level can span a significant range of sizes. The two concepts are illustrated in Fig. 2.15. Most natural systems for which a hierarchical structure can be detected are of the distributed hierarchy type. A rigid hierarchy is mostly related to human-made systems, such as organizations, classifications, or conventional subdivisions of various quantities.

2.5  Scale on Trial

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Fig. 2.15  Illustration of the concepts of rigid hierarchy (on the left) and distributed hierarchy (on the right). In the first case, we can distinguish the hierarchic level of a subsystem based on its size. In the second case, if we relied on the size of the subsystems, we would reach incorrect conclusions: for instance, an element at a certain hierarchic level can have a similar size with an element at another hierarchic level

2.5 Scale on Trial Does scale actually exist out there, in the environment? Or is it merely an invented instrument that we apply to various aspects of the surrounding world? The answer offered in this book should clearly emerge from the presented perspectives and arguments. However, to promptly address this issue here, we have an affirmative answer to the second question – but with two amendments. First, we eliminate the word “merely”: yes, scale is an invented instrument, but an exceptionally deep, pervasive, and valuable one. Second, scale is not a device like a grid, randomly thrown over parts of reality, with the hope of catching fish, or butterflies. One of its miracle powers is the fact that it matches so well the most diverse features in the environment, that it actually works in such a variety of circumstances. The roots of these unusual powers might be hidden in a deeper space than we can explore here. However, one can assume that this instrument has been undergoing a long and complex process of emergence and refinement, which is embedded in our intellectual history. To use the terms of scale categorization, in most cases when the reality of scale was questioned, this happened in relation to “scale as size,” usually “without units.” This is the reason why the ontological problem of scale must be discussed in this chapter. Perhaps no field has focused so intensely on the ontological status of scale, on its construction, and on its implications, as human geography. A wave of explicit doubts about its existence as a “real” feature in the world began mainly towards the end of the twentieth century and continued in various forms afterwards. Clear-cut statements were made, whether brief and stern such as “there is no such thing as a scale,” or more comprehensive, like “we may be best served by approaching scale

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not as an ontological structure which ‘exists,’ but as an epistemological one – a way of knowing or apprehending” – the first statement belongs to Nigel Thrift (1995), the second to Katherine Jones (1998), both papers being discussed by Marston et al. (2005). The ontological condition of scale was supported, among others, by authors who were aware of its possible negative consequences, but still saw scale not just as a useful tool, but as an ontologically valid category. For instance, Neil Smith (2008) assigned a special role to three scales – the urban scale, the global scale, and the scale of the nation-state, all considered in an economic and political context. A paper that sent long-lasting waves throughout the world of human geographers was “Human geography without scale” by Sallie Marston, John Paul Jones III, and Keith Woodward (Marston et  al. 2005). The authors state openly: “despite the insights that both empirical and theoretical research on scale have generated, there is today no consensus on what is meant by the term or how it should be operationalized. In this paper we critique the dominant – hierarchical – conception of scale, arguing it presents a number of problems that cannot be overcome simply by adding on to or integrating with network theorizing. We thereby propose to eliminate scale as a concept in human geography [our emphasis].” With respect to the disciplinary space of the discussion, the paper refers mainly to “urban, political, economic, feminist and cultural geography, as well as political ecology.” The authors make a strong argument about scale not representing an ontological category, but rather an instrument for understanding the world. This, however, is by far not their main message. There is something much more important for them than the ontological status of scale. The authors suggest that considering scale to be real (which unavoidably means for them adopting a hierarchy-centered approach) must be associated with a wealth of implications. One can feel that those implications point, more or less explicitly, to changing power relations, globalization, and factors affecting social justice, threatening and affecting cultural diversity, fueling processes of social exclusion and marginalization, and aggravating social struggle. Scale, tightly associated by the authors with hierarchy and a hierarchy-dominated vision, should thus be eliminated and replaced with what the authors call a “flat ontology,” one that does not make use of scale, neither vertically nor horizontally, but rather consisting of self-organizing systems and their complex network of dynamic relations. However, one can notice that even when we consider networks, dynamically interacting and changing systems, etc., i.e., when “small” and “large” are used in a non-­ spatial context, when we think of a large impact or a small influence, we ultimately often rely on spatial thinking. Spatial concepts often inconspicuously penetrate “non-spatial” thinking. Subsequent developments on this important issue led to solutions according to which scale “is just one spatiality among many,” or should be treated along with other concepts such as mobility, phase space, and topology (Jones et al. 2017). As we can see, such solutions mostly represent attempts at diluting the concept of scale, without removing it from our discourse, and more importantly, from our way of thinking. They do not address the core problems raised by Marston et al. (2005). One could say that on the contrary, by bringing back scale in the guise of mobility, phase space, and topology seems to offer the illusion of avoiding the burden of

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scale, while still leaving it there, in its key position. Other scholars feel the need of scale as a sharp and effective instrument of observation. They still perceive an undeniable tension between the (local) detail, which is easily visible and understandable, and the (wider) system, which is less transparent and much harder to conceive (Nir 2017). This debate on the role of scale in human geography, launched along with the well-articulated vision of Marston et al. (2005), may not explicitly enjoy today the same level of attention as it did in the past. However, for many, the questions raised by those authors have not found an acceptable answer yet. We will only add a few points, making use of the framework dedicated here to scale. On one hand, it is useful to make explicit choices regarding the role we assign to concepts. We will follow Popper (1992), who pointed out that while concepts are very important, there is a danger to reach dry and unproductive outcomes when the conversation shifts its emphasis to the meaning of words: our attention should mainly focus on problems, not on definitions. Yes, clearly defining concepts is essential. Marston et al. (2005) justifiably criticize the fact that “there is substantial confusion surrounding the meaning of scale as size – what is also called a horizontal measure of ‘scope’ or ‘extensiveness’ – and scale as level – a vertically imagined, ‘nested hierarchical ordering of space’.” One finds their critical stance particularly valid, especially since this problem can be overcome by a carefully designed classification of the types of scale, based on consistent criteria (which is one of the main objectives of this book). Another problem detected by the authors is that “hierarchical scale is bound to […] a God’s Eye view.” Creating a permanent bond between the concepts of scale as size and hierarchy, instead of clearly distinguishing them from each other, does not help: the consequence of such a bond is that one of the two must be dragged along, whenever the other one is applied. In fact, the authors suggest that the involvement of hierarchy at every step where scale is applied is what actually happens, whether we want it or not. This is, in other words, not the making of scholarship, but of discourse practice. It is the latter point that sheds doubt, in our view, on the benefits of eliminating scale. We cannot help feeling that the proposed changes would not have real opportunities to be adopted and to produce a major transformation in the social, political, and economic discourse, in the dominating patterns of practice. Social and political processes are unlikely to change due to the elimination of scale from scholarly discourse. Even if pushed, through certain concepts and methodologies, towards other ways of understanding their surroundings, humans are expected to always return to fundamental factors that define their existence in the world, like space and time; scale is one of our main links to both space and time. Not by giving up scale, and by purging our discourse accordingly, will our thinking become more encompassing, more realistic, and more just. Most importantly, one does not become more compassionate by avoiding the concept of scale. If, on a scholarly level, we want to provide a helpful alternative to the discussed images of scale and hierarchy, of “local” and “global,” of “down here” and “up there,” we would propose another approach. It might not be painless or unproblematic to be inspired by methodological approaches from other fields, especially those

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that lie on the other side of a deep disciplinary chasm (Suteanu 2021). However, given the stakes, it might be worth trying. The multiscale approach applied, for instance, in pattern analysis (presented in Chap. 7), has been highly successful in the natural sciences, mainly due to its fundamental principles. It involves a large number of scales, of different points of view, none of which plays a privileged role, none of which stands out. The multiscale approach can keep the concept of scale alive while making sure that it does not absorb the baggage of damaging implications produced by parasite meanings. A successful application of such an approach outside the circle of scholarship, in a way that would more widely penetrate worldviews and world practice, might help. It could work, unless the problems of the local and the global, of domination, marginalization, and exclusion, lie in a completely different space and require a different approach. But if the latter is the case, this only means that we cannot afford to hope that solutions will emerge out of thin air and that we have to keep innovating. Fortunately, this is what humans do, when they are aware of problems, and when they care.

References Ahl V, Allen TFH (1996) Hierarchy Theory: A vision, vocabulary, and epistemology. Columbia University Press, New York Arafat KW (2012) Agatharchus. In: Hornblower S, Spawforth A, Eidinow E (eds) Oxford classical dictionary, fourth edition. Oxford University Press, Oxford Alexander RG, Yazdanie F, Waite S, Chaudhry ZA, Kolla S, Macknik SL and Martinez-Conde S (2021) Visual illusions in radiology: Untrue perceptions in medical images and their implications for diagnostic accuracy. Frontiers in Neuroscience 15:629469. https://doi.org/10.3389/ fnins.2021.629469 Anderson BL (2011) Visual perception of materials and surfaces. Current Biology 21(24):R978–R983 Augustine, Saint (fourth century/1998) Confessions. Oxford University Press, New York Ayzenberg V, Lourenco SF (2020) The relations among navigation, object analysis, and magnitude perception in children: Evidence for a network of Euclidean geometry. Cognitive Development 56:100951. https://doi.org/10.1016/j.cogdev.2020.100951 Bertalanffy L von (1968) General system theory: Foundations, development, applications. George Braziller, New York Carvalho N, Edwards-Jones G, Isidro E (2011) Defining scale in fisheries: Small versus large-scale fishing operations in the Azores. Fisheries Research 109:360–369 Charles A (ed) (2021). Communities, conservation and livelihoods. IUCN, Gland, Switzerland, Community Conservation Research Network, Halifax, Canada. Chiang E (2015) Order-of-magnitude physics: Hand-waving as performance art. https://w.astro. berkeley.edu/dta_statement_chiang.pdf, accessed August 9, 2021 Commins S, Duffin J, Chaves K, Leahy D, Corcoran K, Caffrey M, Keenan L, Finan D, Thornberry C (2020) NavWell: A simplified virtual-reality platform for spatial navigation and memory experiments. Behavior Research Methods 52:1189–1207. https://doi.org/10.3758/ s13428-­019-­01310-­5 Cucker F (2013). Manifold mirrors: The crossing paths of the arts and mathematics. Cambridge University Press, Cambridge

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Ross HW, Plug C (1998) The history of size constancy and size illusions. In: Walsh V, Kulikowski J (eds) Perceptual constancy: Why things look as they do. Cambridge University Press, Cambridge, p 499–528. Rovelli C (2019) The order of time. Penguin, London Saint-Exupéry de A (1948) Citadelle. Gallimard, Paris Sarcone GA, Waeber MJ (2002) Dazzling optical illusions. Sterling, New York Shuman M (2021) Going local: Creating self-reliant communities in a global age. Routledge, New York Smith H, Basurto X (2019) Defining small-scale fisheries and examining the role of science in shaping perceptions of who and what counts: A systematic review. Frontiers in Marine Science 6:236. https://doi.org/10.3389/fmars.2019.00236 Smith N (2008) Uneven development: Nature, capital, and the production of space. University of Georgia Press, Athens Suteanu C (2005) Complexity, science and the public: The geography of a new interpretation. Theory, Culture & Society 22(5):113–140 Suteanu C (2021) Disciplinary chasm: Questions on identification and mending. Belgeo 4:1–12. http://journals.openedition.org/belgeo/52842 Suteanu C, Ioana C, Munteanu F, Zugravescu D (1993) Fractal aspects in solids fragmentation. Experiments and model with implications for geodynamics. Revue Roumaine de Géophysique 37:61–79 Suteanu M (2014) Scale free properties in a network-based integrated approach to earthquake pattern analysis. Nonlinear Processes in Geophysics 21:427–438. https://doi.org/10.5194/ npg-­21-­427-­2014 Thrift N (1995) A hyperactive world. In: Johnston R J, Taylor P, Watts M (eds) Geographies of global change: Remapping the world in the late twentieth century. Blackwell, Oxford, p 18–35 Tretter T, Jones G, Andre T, Negishi A, Minogue J (2006) Conceptual boundaries and distances: Students’ and experts’ concepts of the scale of scientific phenomena. Journal of Research in Science Teaching 43(3):282–319 Umar F, Johnson SD, Cheshire JA (2021) Assessing the spatial concentration of urban crime: An insight from Nigeria. Journal of Quantitative Criminology 37:605–624 Urry J (2003) Global Complexity. Polity, Cambridge Vasari G (1568/1991) The lives of the artists. Oxford University Press, Oxford Wang H, Luo J, Qin W, Zhang B, Liu H, Deng Q, Qin F, Xuan F (2021) Effect of spatial scale on gully distribution in northeastern China. Modeling Earth Systems and Environment. 7:1611–1621 Wu J (2013) Hierarchy Theory: An Overview. In: Rozzi R, Pickett S, Palmer C, Armesto J, Callicott J (eds) Linking ecology and ethics for a changing world. Ecology and Ethics, vol 1. Springer, Dordrecht, p 281–301. https://doi.org/10.1007/978-­94-­007-­7470-­4_24

Chapter 3

Scale as Size in Time and in Space-Time

Abstract  This chapter explores multiple aspects of scale as size when a temporal dimension is involved. It presents the process of time discretization and the implications of the sampling rate and illustrates them with examples. Scale as size in time is shown at work in relation to natural hazards and natural resources. Implications of time scale for the interpretation of equilibrium are discussed. The distinction between synchronic and diachronic perspectives is shown to have significant implications for the study of the environment. Operations of scaling up and scaling down are presented, including their dependence on the number of dimensions. The concepts of context of production and context of interpretation in relation to scale in space-time are described and shown to support our understanding of the environment. Keywords  Scale · Time · Discretization · Sampling rate · Natural hazards · Natural resources · Renewable resources · Environment · Equilibrium · Synchronic perspective · Diachronic perspective · Downscaling · Space-time · Time series · Sustainability

3.1 Temporal Scale Versus Spatial Scale: What Is Different? It is quite common to hear about our incapability to perceive time. “We do not have a sensor for time” is the usual statement. A quite curious statement after all, given that we do not have a sensor for space either. We do not perceive space as such; what we do perceive is size in space. We also perceive changes in size: those are changes affecting something we already know – something we recognize: they are the so-­ called distortions that we use to reconstruct in our mind various distances and relative positions, as discussed in Chap. 2. Similarly, we also only perceive size in time. If we identify two clearly distinguishable events, we compare the size in time between them with the size in time attached to other instances  – to other pieces of “duration” (whether perceived in relation to other events, or internally produced by various timing mechanisms): the © Springer Nature Switzerland AG 2022 C. Suteanu, Scale, https://doi.org/10.1007/978-3-031-15733-2_3

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brain’s capacity of evaluating the length of time intervals involves distinct circuits as a function of time scale (Buhusi and Cordes 2011). There is, however, a major difference between appreciating size in space and in time. In space we have a myriad of elements that are known to us. We immediately recognize them and figure out their positions based on their multiple, intercorrelated size and shape distortions. In time, we cannot do that. What distinguishes perception in time from perception in space is not the absence of many familiar elements but their availability. If we want to navigate the concept of size in time using what is so reliable in space, we need access to the elements employed in our comparison. Therefore, the elements that are not “internally produced” must be stored; they must be frozen and arranged, ready for our timeless comparison. Alas, when we look at the memory-stored set of elements, we find a much poorer landscape than we have in the case of spatial assessment. In other words, while in space we usually have everything we need before us, in time we make use of very different tools, including memory. Implications of this difference are reflected in the distinct perspectives we can apply to our surroundings, which are addressed in Chap. 2. In a way, scale changes seem to be simpler in time than in space: there is no foreshortening and no perspective transformation. Distortions still exist in the case of time; time perception is affected by so-called telescoping bias: distances in time to past events can be perceived as shorter or as longer than they are (Mazur 2020). However, this does not help very much. While numerous, apparently complicated distortions occur in space, they consistently obey objective and clearly identifiable rules. The sizes we perceive in space are rarely the actual sizes of objects. Nevertheless, based on the way in which they are changed, we can quickly confidently reconstruct their picture, including their distances relative to us and their relative positions. In contrast, temporal distortions can be highly variable. Beyond distance in time, there are other aspects of lived events that often play a major role for the perceived size of durations. In fact, Leibniz pointed out as early as 1704 in his New Essays on Human Understanding (eventually published after his death, in 1765) that powerful events can have drastic effects on our perception of time succession and its meaning (Leibniz 1765/1996). What seems simpler and one-­ dimensional, like a set of beads sliding along a thread, becomes more intricate than the three-dimensional image in space: we do not have access to the individual beads, nor can we consistently decipher the complex, multidimensional, intangible environment that is impacting our perception. We will not give way to the temptation to enter here the thorny but fascinating debate on the nature of time, where the very ideas about its relevance or even existence are questioned (Barbour 1999) or reinforced (Smolin 2013). We must add, however, that there are deep chasms that cross the field between the scientific understanding of time and other views on time, e.g., psychological, philosophical, etc. Perhaps the best illustration of this clash of views is offered by the legendary debate between Albert Einstein and Henri Bergson in the spring of 1922. Expectedly, the two scholars talked about two very different kinds of time, which did not even seem to have the needed common ground to make room for comparisons. On one hand, there was time as applied in physics, explained by Einstein in the light of the

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all-­famous, quite recently established theory of relativity. On the other hand, there was time as a constituent of experienced life, as acclaimed by Bergson, one of the most prominent philosophers of the day, who had addressed the topic in an earlier book (Bergson 1889/2013). Those who actually assumed that a unique “conclusion” would emerge from these discussions must have been disappointed. However, this extraordinary meeting between extraordinary people, in its wider context and with long-lasting implications, produced a rare picture of the human spirit in motion (Canales 2016). The event has a particular significance for the concept of scale as size in time. One of the aspects of “physical time” that Bergson found to be alien to our perception of the world refers to what he called its “numerical multiplicity.” Any and all time segments seem to be always equally present. They are readily comparable – in a quantitative way. They are “external” and discontinuous. This is the multiplicity that Bergson had primarily associated with space. He opposes it to the “internal multiplicity” of experienced time characterized by continuity, qualitative rather than quantitative differences, etc. It is the former image of time, the scientific one, that is of interest here: early on, Bergson had identified the implicit operation of spatialization that we apply to time. As long as we see time as a spatial dimension, both the characteristics assigned to space and the methodologies developed to address spatial features could be transferred to the elusive temporal realm. The mentioned conceptual conflict about time appears like a collision in slow motion, which has not yet ended. While this may seem to be an issue of secondary importance these days, one can argue that, on the contrary, the depth of the topic and the stark contrast between the implied worldviews – especially between those of the scientists and of the public – contribute to an intellectual climate of fragmentation, which is shown by David Bohm (1980) to undermine what should be a coherent, collective effort to understand and better address our environment. The challenges implied by the evaluation of distances in time are compounded by the existence of innumerable types of variability that can be observed around us, at various scales. Moreover, temporal features are not as easy to visualize as spatial ones (Lovejoy 2019). For many reasons, including those mentioned above, the way in which processes occur over time might be more difficult to capture and to understand than their aspects in space. It is therefore not surprising to see that temporal traces of the systems we study are often represented and viewed as if they were spatial features, in line with Bergson’s views. Since in most cases scale as size refers to one-dimensional space – specifying, for instance, processes on a scale of “kilometers”  – even when we refer to two-­ dimensional areas, or even to three-dimensional volumes, the parallel between scale as size in space and scale as size in time could be easily identified and put to work. This step has been a useful one; it can bear fruit as long as we keep in mind the main aspects that distinguish temporal features from spatial ones. Many of our graphical representations of spatialized time (which is virtually the only way of graphically representing time) considerably blur, or even suppress the differences between temporal and spatial aspects of the presented processes. For instance, time and space in Stommel diagrams (such as those in Figs. 2.11 and 2.12) do not seem to be different

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from each other. One can obviously argue at this point that indeed there should not be any difference, considering the “geometrical” joining of space and time. As early as 1754, d’Alembert declared time to be a “geometric parameter,” and in 1794, Lagrange called the field of dynamics “a four-dimensional geometry” (Prigogine 1980). The actual fusion was then performed by Minkowski, followed by Einstein’s theory of relativity, which sees the world living in a space-time continuum that joined the temporal and the spatial dimensions in an inseparable way. Notwithstanding the worldview opened up by the theory of relativity, the space and time scales on which we consider the environment in most fields do not require a view of time as a dimension in space-time. The spatialization of time is very attractive. We can treat time  – and especially see time, because the resulting view can have deep and ­multilayered effects – as if it were a spatial dimension with all the benefits that it can bring. We may gain insights by considering a time interval to be equivalent to a piece of the road, seen from above, which we can analyze on any scale, by looking at any segments, in any order. However, by doing so, we also lose time-specific gifts: succession and, in some cases, rhythm, or speed. The diagrams that show time scales vs. spatial scales are now blind to the sequence that is embedded in information on time. Information on temporal succession can be valuable in situations such as those concerning strongly irregular fluctuations of certain variables (precipitation, river discharge, wind speed), or various aspects of irreversible processes, in which the order of event occurrence is important to the investigation (Suteanu and Ioana 2007). These will be analyzed more closely in Chap. 7. Some of the other key time scale-specific aspects of scale as size will be discussed below.

3.2 Converting Size in Space to Size in Time The highlighted differences between spatial and temporal scales are so significant that the act of fusing them together into a unique system seems quite unnatural and strange. On the other hand, if we apply a different perspective to the way in which these concepts are put at work, we can see that they can be brought to a form that makes them more easily comparable. A basic distinction between spatial and temporal “sizes” starts with the procedures of measuring them. If we can unify those, we may develop a useful common ground, attaining a perspective from which they might not look so drastically different from each other. We start by asking how we can measure the size – for instance, the length – of an object, by performing the measurement process in time. One possibility is to move past the object, from one end to the other, and to measure the time it takes to do that. If we know the relative speed between ourselves and the object, we can find the length in spatial units. In order to consistently make use of the measured time interval without referring at each instant to a particular speed, we can choose a solid reference for the latter: the speed of light in the vacuum, c. By doing so, we can find any length as the time needed by light to travel that distance.

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If we apply this approach to sizes in space, we can refer to all sizes, in space and in time, in the same way – in time units. We would thus refer to the height of a person of 6 nanoseconds (ns), to a room of 12 by 15 ns, and a distance between two cities of 300 microseconds (μs). Would that lead to a unique scale as size, equally applicable in space and in time? There are at least two answers to this question. The short first answer is “no.” Not only for the reasons highlighted in the previous section but also because of the deep differences between the two categories of size, as revealed, for instance, by the synchronic and diachronic approaches to objects in our environment (Sect. 3.6). The second answer is that we are not even inclined to produce such a unifying conversion: as we show in Sect. 3.3, for concrete reasons, we are rather interested in making time treatable as space, and not the other way around. However, if we wanted to handle the same type of units for sizes, both in space and in time, and we preferred those units to be length-like and not duration-­ like, we could use the same reasoning that was applied above: by converting time intervals into distance units, corresponding to the distance traveled by light over those time intervals. This is, in fact, exactly what we do when we refer to large distances by measuring them in light-years. Although handling the same spatial or temporal units, respectively, both for sizes in space and in time, can at times be useful, this unit conversion does not help bridge the specific features of these two categories. The value of melding them together in space-time for the breakthrough made by the theory of relativity cannot be overemphasized. However, for most problems we address in relation to our environment, the mentioned specific features of sizes in time and in space are important: sometimes they may be seen as difficulties to be overcome, but they can also be uncovered as precious windows into the intricacies of reality.

3.3 Time Scales and Records of Time I know a man who once, when falling asleep, heard the clock strike four, and counted the strokes as “one, one, one, one.” It then seemed to him that there was something absurd about this, and he shouted out: “That clock must be going mad; it has struck one o’clock four times!” —René Descartes (1641/1996)

Descartes’s story offers a good example of the implications of scale as size in time. On a scale range of up to one second, one can never interpret “four o’clock” by listening to the clock striking. No matter when we listen to it, all we can hope to find out is the fact that something has happened, since the clock has struck  – once. Something similar happens if due to some form of memory impairment one could only capture a limited time interval, assimilating new facts but continuously forgetting the older ones. If one only had this “temporal buffer” interval available for interpretation, it would be impossible to make sense of a story that would extend over a longer time period. While an actual memory impairment associated with a very short time interval can make this phenomenon strikingly obvious, the limited temporal buffer issue can be considered to be quite general. In many ways, this is

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indeed how we are normally operating. With his characteristic precision and clarity, Eco (1991) shows how, in the process of semiotic interpretation, thought is moving on and on towards new targets but keeps cutting its bridges towards the past while doing so. This limitation in the grasping of temporal events reveals a first major distinction between our views of features in space and features in time. True, our capacity of perceiving space in the environment is limited as well, but in principle we can always change our point of view, our orientation, and even our spatial location altogether, thereby exposing wider areas and discovering new aspects of our surroundings. When it comes to time, such an exploration is not possible  – unless we spatialize it. Even then, our exploration possibilities are subject to firm constraints. Not only is it impossible to move back in time: even moving forward has to happen in a way we cannot control. Spatializing time, however, is something we can do, and we have been successful at it. This operation offers us many of the opportunities we enjoy in space, albeit in an unavoidably impoverished way. In fact, when we record processes occurring over a certain time interval, we then obtain access to any part of the recording; we can refocus on a short time splinter, replay it, and analyze it indefinitely; we can freeze a fraction of it – an image; and we can move forward and backward. We can handle recorded time as if it were a strip of space cut out of the dynamic environment – but one brought into the lab to be studied. Almost akin to a rock sample. The outcome is thereby severely impoverished. Our understanding can only take flight starting from that limited object. Rather than being free to explore the feature of interest any way we want (as we would with a spatial feature – like a new island, or a spot deep in the ocean), we are limited to the content of the recording. This means that we look through a narrow and non-expandable window, to see an image that will never change. Ultimately, the intrinsic limitations and the characteristic properties of recordings of change in time make us look in two opposite directions to see distinct implications. Both are true. Both are worth our attention, as they can help us in our theoretical and practical investigations. The first one is limitative. The second one is opening unending perspectives. These two contradictory outlooks, which are reminiscent of Nicolaus Cusanus’ coincidence of opposites, meet in one place: the actual time recording to be analyzed. Lifting fingerprints off processes that are flowing with time gives us something that is not unlike a picture: usually sets of zeroes and ones, preserved in capsules meant to be as immune to the stream of time as possible. Such records remind us of the silence of writings denounced by Plato’s Socrates: “writing shares a strange feature with painting. The offspring of painting stand there as if they are alive, but if anyone asks them anything, they remain most solemnly silent” (Plato fourth century BC/1997). As signs of events from the past, they are not expected to change: they can only deliver one and the same message. The second outlook does not deny the statements of the first one. The message carried by the “picture of time” is indeed fixed and ageless. However, it is rich. So rich, in fact, that it can even be considered inexhaustible. The ordered series of bits

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that make up the picture, which represent the time series, can be assessed in many different ways. In fact, the number of possible ways of characterizing a time series is infinite. This can be easily seen if we think of various ways in which the values of the time series can be used to obtain even one single number to be associated with the time series and deemed to describe it. Some of these evaluation operations are more useful than others, but new algorithms able to capture interesting and relevant features of time series keep being created. Therefore, one could say that what we pictured as a rigid and limited message includes, at the same time, an infinite potential for us to engage in an exploratory dialog with the recorded data, and thus with the processes we investigate.

3.4 Scale as Size and the Granularity of Time Records We know that we do not live in a continuous world. When we look deep down, far enough, we realize that a discrete character of the environment can be observed everywhere. On the other hand, we feel that we live in a continuous world. It would indeed be safe to say that it is quite normal not think about the world at quantum scales. For most of us – all except the esteemed colleagues who focus on a different range of scales, facing the fascination and the weirdness of the quantum world – reality is continuous. In most cases, we would not run the risk of being led astray by working under this assumption. Quantum physics itself offers us the peace of mind of operating in a “continuous” macroscopic environment. (This is not to say that there are no larger scale processes that can only be explained by looking at the quantum level – the black body radiation spectrum is a famous example, but even more mundane phenomena like seeing the colors of the rainbow in an oil patch, or even the reflection in a mirror, belong to this category. However, we will not even come close to the quantum level in our current endeavor.) While we see our environment as being continuous both in space and in time, this is not how we get to look at it when we capture parts of it in our investigations. All we acquire from the environment takes the form of discrete data, which can be seen as consisting of bits. It is true that in the past an illusion of continuity was provided by so-called analogue devices. However, ultimately what that apparent continuity often meant was that we were dealing with an actual lack of continuity that we were unable to control. For example, the light entering the lens of our camera ended up on photographic material with a granular structure, even if we had access to “fine-­ grained” material; similar issues were encountered with magnetic material. Pendulum inclinometers, which were able to continuously measure very small changes in the inclination of a surface, traced a graph using light reflected by a mirror, which reached, again, photosensitive material. On the other hand, we should make the distinction between “continuous” as the opposite of “discrete” and “continuous” meant as a way of grasping “all” the intermediate states between distinct stages in a process or elements in an image. For example, seismometers designed to record mechanical vibrations could eventually trace their graphs with ink, on paper.

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However, their measurements were dependent on the mechanical properties of the instrument: only certain components of the spectrum of vibrations could be recorded, which was another source of discrete and selective information picking. Now we know: we are capturing discrete samples from the environment, both in space and in time. Moreover, the way the samples are selected can have a major impact on the resulting data. Here we encounter another significant difference between the spatial and the temporal scale as size. Granularity also influences the resulting image of a spatial feature, and it should be taken in consideration when we analyze the image; however, in the case of temporal features, sampling plays an even more decisive role. When we place our instruments in the environment to capture relevant information, we always select – sometimes only implicitly – space and time rules according to which we decide upon the number and the position of measurement points. These decisions are usually more important when we record a property that is changing in time, compared to instances when we take what is an essentially static picture. It is much easier to assess measurements when they consist of equally spaced samples (in space and in time) than when distances between samples vary. This condition is easier to fulfill in spatial images. Uniform sampling is normally desirable – and possible – for temporal sampling too, but in some cases, it is not applicable (examples of variable time-step information are offered by ice cores or lake deposits). However, data obtained with nonuniform sampling can also be fruitfully used if approached with the proper methodologies (Marvasti 2000). When the variable of interest  – the signal  – is digitally recorded, its assumed continuous shape is sliced, sampled, and sets of discrete values are obtained instead. An example is shown in Fig. 3.1. Rather than the continuous curve, what is captured only consists of the temporal positions of the samples and their values on the curve, i.e., the coordinates of the circles marked by the dotted lines. The outcome of the sampling process can crucially depend on the temporal density of the samples (the sampling rate). If the sampling rate is increased in the case illustrated in Fig. 3.1, we get a more detailed and more accurate image of the recorded variation. On the other hand, if the sampling rate is too low, the collected information cannot help us to properly reconstruct the actual temporal variation. Figure  3.2 shows a sampling example in which the selected sampling rate leads to a very different image of the variation (the dash-dot line connecting the sampling points). The simplest rule that helps us avoid such distortion is given by the Nyquist criterion, according to which the reconstruction of the signal can be free of distortion if the sampling frequency is at least twice the highest frequency of the signal; the problem of signal reconstruction, even when affected by noise, was elucidated by Claude Shannon (Marks II 1991). By using sampling, we can thus obtain glimpses of the changing system, captured from time to time. These images are separated by void space  – which can never be explored. We can actually know more about black holes (which exert an influence on their surroundings) than about the void spaces between samples. For us, such black spaces simply do not exist. A quite dissatisfying situation, especially since we know that those seemingly empty intervals are not empty. Our system has

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Fig. 3.1  Sampling of a presumably continuous signal (in arbitrary units)

Fig. 3.2  Sampling of the signal in Fig. 3.1 with a five times lower sampling rate, which does not fulfill the conditions of the Nyquist criterion. In this case, the sampling does not lead to a distortion-­ free capturing of the recorded variable

continued to act, the world has continued to change, and real time has continued to pass. However, the reality is that while we cannot witness phenomena occurring during our “reading blackout” intervals, this does not mean that we cannot find out

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about what occurred in the meantime. Depending on the system, in some cases we can apply a range of methods capable of helping us to bridge the void between samples. A comparison with a multiple exposure image (Fig. 3.3) can put this peculiarity in perspective. We are entitled to assume that had those images been taken at shorter time intervals, more intermediate positions would have become visible – and we would be right. This, however, is not what happens in the case of wind, for example. When we capture wind speed at increasingly dense time intervals, we start to see completely new elements of the picture each time. For the launch pad image, it is as if anything could change between two recorded images, and we would not notice. The launch pad could even be painted in another color and repainted back again, or the space vehicle could altogether be replaced with another one: we do not know what happens between two samples of wind speed records. Increasingly high sampling rates keep leading to different pictures. Therefore, when we assess wind variability, the results can significantly vary with the sampling rate; this certainly

Fig. 3.3  A historical multiple exposure photograph: launch of Gemini X spacecraft, July 18, 1966. (Courtesy of NASA)

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depends on the evaluation method, but some of the most interesting aspects of wind, such as wind speed variability, its time scale dependence, and its change over time, typically belong to this category. Using an anemometer, we acquire data at equally spaced time intervals. What should those intervals be? To illustrate the sampling issue and the relation between sampling rate and scale, we will use a dataset that was already sampled at the relatively high density of 0.13 seconds per sample (all the data used to produce Figs. 3.4 and 3.5 are courtesy of Google Wind Data Collection Project 2011). With these data we can look at the time series as if we had a sparser sampling, which is always easy to do (unlike the opposite operation, i.e., moving from a lower to a higher sampling density). Figure 3.4 shows wind speed values in m/s for samples taken every 13 seconds. Even before we apply any analysis method, we realize that wind speed is strongly variable in time. The “void space” issue can be noticed here if we increase the sampling rate. Figure 3.5 shows what happens if we take our snapshots ten times more frequently: every 1.3 seconds. Having this safety net allowed us to plunge without the fear of getting lost into the initially void space between two samples of the series in Fig. 3.4. One single sample interval from Fig.  3.4 now reveals ten new samples. Without inserting ten more sampling operations (which had all been performed, but ignored initially), we would not have seen the solid line pattern in Fig. 3.5. Due to the initial denser sampling, we can perform such an expansion, which would normally not be available. We can even dive deeper into the void between values and uncover ten new samples between every two successive solid line/circle samples. At this point we have reached the sampling rate of 0.13  seconds, the highest rate that was

Fig. 3.4  Wind speed record sampled every 13 seconds

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Fig. 3.5  One single sample in the wind speed record shown in Fig. 3.4 can be seen in a different light, when an increased sampling rate of 1.3 seconds is applied (solid line). If the sampling rate becomes even denser – 0.13 seconds – we can see new features emerging between every two successive points in time (dotted line). The wind portrait changes as a function of time scale

available for this dataset. The same phenomenon would have kept occurring for increasingly short time intervals between samples (Lovejoy 2019), if we had sampled the time so densely. In practice, in most cases data at such time density are not readily available: usually, these values are prepackaged through averaging over time intervals of 10 or 15 minutes (Gross et al. 2020). Depending on the goals of the research, after this merging operation, the data can still be valuable; however, from the point of view of their time scale-dependent variability, the data fusion process has major pattern-transforming effects. Therefore, when wind pattern characteristics provided for different locations are compared, we must proceed cautiously; for certain pattern characteristics, the applied sampling rates and the box sizes used for data coarsening in the compared locations must be carefully considered. As we can see from these examples, temporal sampling is unavoidably connected to scale as size in time. The size – or the length – of a recorded time interval is given by the number of samples and the size of one sampling interval. It might be worth underlining that none of the samples speaks about time: samples are as timeless as any numerical value can be. Time is not captured by the samples themselves – this is reminiscent of the metaphor about time slipping through our fingers: time is only caught by the interval between samples. This is why digitally recorded data can be useless if we do not know the sampling rate at which they were acquired. Our access to various scales in time can be severely constrained by the sampling rate. Therefore, as mentioned above in the case of wind data, if we use data acquired in other projects, their suitability for our investigations must first be assessed.

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Sampling rate and possibly subsequent data averaging (sometimes improperly named data “integration”) operations will be key points on our checklist. This sampling effect in the time domain is one of the factors that make temporal sampling different from spatial sampling. In fact, sampling did not pose notorious challenges in the case of records of spatial information before the advent of high-resolution imaging, image compression and reconstruction, etc. In our current approach to the environment, both space and time require the collection of information from discrete points. What we do, in space and in time, when we explore our surroundings is to send out sets of minuscule needles to collect information, regardless of how dense we can make the needle framework. And yet, there is a remarkable difference between space and time sampling. As we could see above, when we reveal the variation of wind speed between two samples (based on even denser sampling we had actually performed), new shapes almost magically appear. This does not normally happen with sampling in space when we take digital photographs: if we use a camera with a higher resolution, it is highly unlikely that we will suddenly discover a wolf popping up between the two sheep we could distinguish in our photograph before.

3.5 Time Scales and Environmental Processes 3.5.1 Scale Ranges, Accumulation, and Dissipation Time scales are particularly relevant when scale range by scale range we discover different pictures of the environment (Liu et al. 2007). Not only can the picture of variability change as a function of time scale: distinct time scale ranges are often associated with specific processes. Even for one and the same phenomenon, like the variability of wind speed over time, one can discern time scale-dependent implications. For instance, over scales of seconds and less, each part of the turbine is shaken by the wind in a different way (in fact, the extent to which vibration patterns change with the distance from the rotor, and from one spot to another, is more pronounced over specific time scale intervals). Further on, variability over time scales of minutes and hours is reflected in fluctuations of the turbine power output. Over longer time scales, wind speed variability is important in terms of power availability, which can also be considered over increasingly large spatial scales, due to the integration of multiple turbines in a wind farm, multiple windfarms, and possibly other energy sources in wider energy networks. The various aspects of variability in the environment are expressed over a wide range of scales. The time scales associated with physical processes involved in natural hazards, for instance, can be strikingly different from each other. Figure  3.6 presents examples of hazards related to climate change. We can notice major differences in terms of both the lower and the upper bounds of time scales. Some types of hazards can span a much wider time scale range than others. Two notes should be added here. First, mass movement representation was limited in this figure to events

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Fig. 3.6  Time scales involved in climate-related natural hazards (approximate values). Some of these (floods, mass movement) include different subcategories of hazards, some of which are dominated by distinct physical processes. Mass movement that involves longer time scales would most likely be less related to climate. There is no actual reason for the represented limit of drought, the rectangle of which can be considered (in a pessimistic scenario) open-ended

of up to 1 year in length, even though such slow processes can last much longer, whether or not they are dominated by climate change effects. Second, drought events were shown here up to a limit of 10  years: obviously, they could become much longer. When analyzing such representations, it is important to keep in mind that the graph uses a logarithmic scale: therefore, time scale intervals that look similar may still be quite different from each other. For comparisons, it is thus important to look beyond the intervals drawn in the graph and pay attention to the time units specified on the X-axis. If the environment is subject to ample change, as is the case with climate change, the frequency of occurrence and the intensity of these hazardous phenomena can and are expected to also change. However, the processes of change, on one hand, and the actual occurrence of natural hazard events, on the other hand, are associated with very different time scale ranges. Due to the nonlinear character of the processes involved in climate change, we may see such transformations advancing in leaps: time intervals of gradual shifting of patterns may be followed by abrupt variations and transitions to quite different functioning regimes of natural systems. Time scale (in all its forms) is therefore an important element to watch closely while monitoring climate change. A relevant trait characterizing the nonlinear nature of such processes consists in some cases of the discrepancy between the time scales over which energy is

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accumulated, gradually, over time, and the time scales over which energy is dissipated. The behavior of a jammed door can be a suggestive example. While one pushes against a door that is stubbornly stuck against a bump in the floor, potential energy is accumulated in the system (e.g., due to the elastic properties of the wood that we are bending while pushing). When the force exerted on the door makes it overcome the friction with the floor, the door bursts open. A gradual process of energy accumulation is also involved in the growth of trees, with the sun – our main source of energy – playing the key role. The stored energy can then be abruptly released when trees catch fire. Gradually occurring processes, due to the relative movement in geosystems at different spatial scales, also lead to the slow accumulation of energy, which is dissipated in earthquake events during sudden slips on faults. The word “sudden” must be interpreted, again, in comparison to the time scales related to energy accumulation: in reality, the slip on a fault does not take place instantly and not even on the whole fault surface at the same time. The rupture propagates in space, and this takes time – the process can last seconds, tens of seconds, or even minutes. Both examples are illustrated in Fig. 3.7. In this case too, the logarithmic time scale is very useful – it makes it possible to have both the accumulation and the dissipation stages on the same graph, in spite of the enormous difference in the time scales involved. However, here again, the logarithm effect can be misleading, since extremely different time scales seem to be comparable: had we used a linear scale instead, the dissipation scale intervals would have collapsed into a tiny dot, compared to the accumulation time intervals. Determining and representing relations between spatial and temporal scales can be relevant for certain processes, as proven by the frequent use of Stommel diagrams. In many cases, shorter spatial scales are associated with shorter temporal scales, and large scales in space tend to be related to longer-lasting phenomena, although exceptions to this general trend also exist. The problem of the relationships between space and time scales can be particularly difficult to establish when both physical processes and social processes are involved, as in studies in the field of water management (Wallace 2003).

3.5.2 Time Scales and the Renewability of Resources Complementary processes that occur over strikingly different time scales can be important to natural resources. In this case, we are focusing on the relation between consumption rates and recovery or replenishment rates. In fact, in terms of time scales, this relation can be decisive for the “renewable” or “nonrenewable” status of resources. A relevant example concerns groundwater. In principle, groundwater could be a renewable resource, since water sources are continuously refilled and purified through natural processes, with no need for human intervention. It is thereby endlessly made available again, which characterizes a truly renewable resource. However, consumption rates can be faster, even orders of magnitude faster than aquifer replenishment rates, leading to a (sometimes rapid) drop of the water

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Fig. 3.7  Time scales involved in accumulation and dissipation stages in two types of natural hazards: (a) forest fires, (b) earthquakes

table, eventually moving the water resource into the category of nonrenewable resources, often with ample, even disastrous consequences. While this picture may be helpful, it can be misleading. First, focusing only on water quantity depletes the problem of its meaning. It is water quality, along with its change through human use and its transformation through long-lasting processes, each with their associated time scales, which actually close the circle and frame the

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alarming nature of the problem. Second, it is clear that if you take out something faster than it is replaced, the source will be depleted. But the water story is developed in circumstances of extreme variability in space and in time. Water resources are unevenly distributed in space. Therefore, the degree to which water problems are identified as major challenges as opposed to life-threatening crises varies among countries and even regions within the same country. Water availability is also highly variable in time. Temporal fluctuations  – regarding precipitations, snow melting, and river runoff – can be great, spreading over orders of magnitude. In turn, this variability depends on geographical position. The temporal variability in human water use (especially on a time scale of a day or of a year) adds to the variability problems related to water resources. Another often used example to illustrate the time scale implications for natural resources concerns fish stock depletion. In this case too, a simple input-output balance along with the associated time scales can be suggestive, but it ends up proving to be utterly incomplete. The complex interrelated processes that are involved, the temporal variability of their occurrence, and the difficulties implied by the estimation of the fluctuating status of the fish stocks all contribute to a highly intricate dynamic picture (Zhou et al. 2017). As these examples suggest, the relation between time scale and the “renewable resource” status is richer than we initially outlined. It involves multiple scales and scale ranges both for space and time. It implies challenges for grasping useful patterns. Moreover, this relation between time scale and renewable resources is often affected by ample pattern transformations that can occur gradually, over long time scales, but also abruptly, on comparatively short time scales. Therefore, it is often wiser not to refer to “the” relation between time scale and renewability, but rather to realistically acknowledge the multitude of nonlinear phenomena that are at work in interrelated ways, to admit the reasons for uncertainties and to emphasize the role played by scale range estimation in the endeavor of establishing the renewability status of resources.

3.5.3 Scale and Equilibrium The word equilibrium stands for a large number of concepts, which differ in many ways from one field to another. Distinct meanings are associated with it even within a specific field  – whether this be physics, biology, chemistry, economics, social theory, etc. Some of those definitions are more precise and more rigorous than others. Most of them focus on some form of “balance” (which, strictly speaking, borders on a circular definition): a way in which different forces, tendencies, influences, etc. tend to cancel each other out, leading to a state of the system that is not expected to experience drastic change. As annoying as such vagueness can be, for instance, for the physicist, one is compelled to admit a broad semantic aura of the term equilibrium, in order to consider its meanings and its relations to scale in a variety of circumstances relevant to our understanding of the environment. We will thus not

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afford to dwell in the crystal clear order assigned to such concepts, for example, in thermodynamics by Carathéodory (1909). Equilibrium and the closely related concept of balance enjoy powerful positive connotations. Equilibrium is often seen as the unquestionably desirable state of virtually any system around us. Equilibrium tends to make us feel at peace. It allows us to breathe, without anxiously watching for new dangers assailing us – certainly a sought-after situation that has been relevant for a very long time throughout our evolutionary history. Essentially, equilibrium does not connote change, a term that is strongly glued to other associations: uncertainty, difficulties, and stress. The perception of these concepts surely has its reasons: they are even reflected in everyday speech and this not only in the English language. For instance, not only humans but even nonliving objects of any kind are said to “enjoy” one property or another. We surely do not mean to thereby assign feelings to physical systems characterized by certain properties. Similarly, the same (nonliving) systems are said to “suffer” transformations or changes of various kinds. Again, we do not mean to transfer our worries to a quantity of water that suffers a phase change. Indeed, for us humans, change, in its most general sense, can be difficult (Peck 1978). Change can imply suffering; it can imply novelty, and novelty can involve situations we might not know how to tackle. The fact that we choose to use such words seems to point to deep-seated convictions about the benefits of equilibrium. If this is the case, it is easier to explain why even in the third millennium there is a widespread mentality centered on the following two ideas: (i) Equilibrium is the normal state of the systems around us. (ii) Departures from the state of equilibrium must be caused by disturbances that are external to those systems. Far from representing a mere curiosity, such a view has major implications for the way we see and understand processes in the environment. Even more importantly, this viewpoint can be decisive for the manner in which we choose to act in our relation to the environment – even with the best intentions. One may wonder to what extent an error of such proportions can be rooted in the way we have been taught about this issue. Notwithstanding the value of equilibrium as a useful simplification that is applicable in the real world, when properly framed, it is important for students to understand the meanings of equilibrium when the processes involved are nonlinear (which is often the case). In this context, scale as size and scale in time have a unique role to play. As Per Bak, the author of the concept of “self-organized criticality,” pointed out, if the systems around us truly were in equilibrium, even for a fraction of a second, the world would end; nothing would happen, and there would be nothing to make anything happen thereafter (Bak 1996). Theories centered on the idea of equilibrium as the default state of systems are incapable of explaining the dynamic world around us: much of what we see is dominated by phenomena that cannot be reduced to unlikely fluctuations in a system in equilibrium. The latter would only admit small disturbances, the effects of which are promptly squashed.

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Here we will consider equilibrium in the environment mostly in relation to scales that are accessible to human perception, both in space and in time. In what follows, we should keep in mind the above discussion on the vagueness of the concept: the word equilibrium should thus be read every time as if it were written between quotation marks. In order to circumscribe the concept and to highlight the role of spatial and temporal scales, several points must be made. (a) Equilibrium only occurs for a limited amount of time: it is a temporary state of reality. (b) Equilibrium occupies a limited region of space: it is a local state of reality. (c) Equilibrium only reaches a certain degree – even in the limited niche of space and time where it occurs; it is not “complete”: it is a theoretical state of reality. The fact that equilibrium is never perfect, that it only occurs in limited regions in space and over limited intervals in time, should not make the concept less relevant. Even if it is a theoretical abstraction never fully embedded in reality, it is an important intellectual instrument in our understanding of reality, provided that it is properly understood and applied. More importantly, we should be careful to avoid shaping our expectations based on its “reality.” It is its status as an abstraction, only partially reflected in the surrounding world, that reveals the particular role of scale as size – in space and in time – when the concept of equilibrium is applied. With the above points (i) and (ii), as well as (a) to (c), in mind, it is straightforward to see that the identification of equilibrium can directly depend on the scale – in time and in space – over which the studied system is identified or over which a problem is studied. For example, from the point of view of the spatial scale, an ecosystem may look robust and unchanging on a large scale, while strong fluctuations and dramatic changes could take place in smaller areas, where species compete and some species might be wiped out locally. Similarly, in terms of time, on a scale of decades, a forest ecosystem may be in robust equilibrium, while on a yearly scale, major shifts could occur in the populations of large predators and of herbivores, stimulated by yearly fluctuations in vegetation cover due to weather variability. Conversely, a lake ecosystem studied on a scale of months and years can have all the signs of equilibrium, but on a scale of centuries, the lake may be shrinking and eventually disappearing. There are two sides of the coin to be considered in this context in relation to equilibrium. On one hand, it is true that equilibrium is imperfect and depends on the analyzed time – and space – scale. It is also true that large fluctuations leading to major transformations in the system can take place on various scales, as a consequence of the complex and nonlinear nature of the interactions among system components. The system can change  – even dramatically so  – without external perturbations. Examples range from self-organizing processes in the laboratory to interactions between species in an ecosystem. In such cases, the expectation that the system should be in equilibrium because it is “natural” and the conclusion that imbalances must be the outcome of an external disturbance can be both incorrect and severely misleading.

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On the other hand, it would also be wrong to assume that changes in the system must be endemic, that no external cause is involved when change occurs, simply because the system is made of complex components interacting in a nonlinear way. In some cases (in fact, when it comes to the state of the environment, in many cases), disturbances may indeed be the main causes of change, and the existence and the effects of external disturbances must be investigated.

3.6 Synchronic and Diachronic Perspectives Imagine standing at the top of a hill, looking at the landscape in front of you. You see so many shapes, and colors, and objects. You can spend time to distinguish more details in one area or another. If you really want to “take it all in,” you need some time to do it. Then you hear someone singing a song not far away. In order to hear the tune and to know what she is singing about, you also need some time. Both situations require time, but they differ significantly from each other. In the first case, you can acquire more or less detail and look away from the landscape anytime you want. In the second case, either you hear the song or you don’t. Whatever you don’t hear becomes inaccessible; it is simply lost. In the first case, you can look closer or farther; you can change the scale of your view as you please. In the second case, you can do nothing about scale in time; had it been a recorded song, you might have been able to change the playback speed, but the constraints of listening in time would have still been there. The distinction between spatial and temporal aspects of scale and perception has direct implications for our understanding of the environment, in terms of both the role of scale in general and the role played by time and time scale in particular. A major aspect of the deep nature of this difference was revealed in the thirteenth century by St. Bonaventure (1259/1978): by referring to sources of beauty, he distinguished objects perceived all at once, such as those that are seen, from objects for which perception can only occur in time, and such as those that are heard or smelled. It is in the same realm, of esthetics, that this distinction was further developed later on by Lessing (1766/1994). Lessing vividly highlighted the contrast between works of art that offer us a single, frozen glimpse of the world, which can be rich in details (like paintings), and works of art that present us with action followed in time (like poetry). Later, in the twentieth century, Ferdinand de Saussure applied the concepts of synchrony and diachrony in his posthumously published Course in General Linguistics, referring to two complementary approaches to language. For our endeavor, it is the distinction that emerged due to St. Bonaventure and Lessing that is the most important one. Their work focuses on an essential albeit limited portion of our universe – beauty and the work of art. However, their distinction, especially the one insightfully presented by St. Bonaventure, provides the basis for a clarifying classification of practical value for our understanding of the environment. Ultimately, both the synchronic and the diachronic perspectives designate views, ways of approaching the world. What Bonaventure subtly noticed

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more than 700 years ago was not only that different types of view exist but also that certain categories of objects impose (or encourage) certain types of view. Lessing deepened this idea by referring to the different types of works of art. It only takes one more step to apply this observation to objects in our environment. We can then notice right away that there are broad classes of objects that can be assigned either to a synchronic view or to a diachronic view, and in some cases to either of them or to both. In this way, the act of assigning different approaches to our objects of investigation can gain clarity. We can thus apply well-defined criteria to guide the choice of methodological approaches to classes of objects, with immediate implications concerning scale in space and/or scale in time. Table 3.1 shows how this classification can be operational in a broad context. We can see that there are classes of objects that require a certain view, while others allow us to choose the view that works best for the questions we ask in our study. Synchronic perspectives must be applied to maps and a variety of images, ranging from those produced by remote sensing to graphs and diagrams. As it is typical for the synchronic view, this means that there are virtually no constraints affecting our choice of focusing on one part of the object or another, on larger or smaller portions of it, on a lower or a higher level of detail. Most importantly, the spatial scale of our analysis can be freely modified. We can choose to move back and forth in scale space, zooming in and out of certain portions of the image. At the other end of the spectrum, the table shows that among the objects that require a diachronic view, some are “time-step-locked.” In their case, changes to the temporal scale unavoidably produce distortions in the system image: by choosing a “reading pace” or a “speed of flow” that are significantly slower or faster, we modify speech or music in ways that can make them unrecognizable and even useless. In contrast, “time-step-free” objects still require time for their “reading,” but changes to the time scale can be made – and are often made, potentially with positive results. For instance, time series can be scanned at different speeds, and cellular automata models can be run with different time steps, in order to search for specific aspects of their pattern. In between, there is a class of objects for which both approaches are applicable.

Table 3.1  Categories of objects associated with synchronic or diachronic perspectives

Object category Scale Scale change Examples

Objects that require a synchronic perspective Spatial Yes, freely Remote sensing images Graphs

Objects that invite a synchronic and/or a diachronic perspective Spatial and/or temporal Yes, possibly with constraints Time series Three-dimensional objects Manuscript images

Objects that require a diachronic perspective Time-step-­ Time-step-free locked Temporal Temporal Yes No Time series Model outputa

Music sSpeech

For certain categories of models, such as those involving computational irreducibility

a

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Even without identifying groups of objects linked to certain views and without explicitly assigning synchronic and diachronic approaches based on sets of object properties, the two approaches – the synchronic and diachronic ones – have been applied in numerous fields, well beyond those in which they initially emerged. In most cases, one or the other was seen as the appropriate instrument to be used, even if no comparison was presented concerning their strengths and limitations in a given context. Among the two, it was mostly the synchronic perspective that enjoyed implicit recognition and validation among scholars, especially in the natural sciences. After all, synchronic perspectives are primarily associated with sight, rather than sound and smell, and sight is by far the main source of perception linked to our scientific understanding of the environment: no matter what instruments we use, in the end, most outcomes of our investigations include key phases that are visually interpreted. The difference between the two approaches reaches, however, much farther than that. On one hand, it was easy for the synchronic view to be perceived as “the” objective view, the one that offers us “the whole picture.” And while it never offered the whole picture, as no single view does, it still presented us with a valuable opportunity: to assess the image in any way we wanted. Indeed, facing the whole picture at once does not mean simultaneously taking it all in, as the name “synchronic” suggests, but it allows us to freely explore the image, to spend any amount of time by considering it at any scale, by looking at its parts, in more or less detail, in any order. In contrast, the diachronic perspective resonated more with subjective impressions, with individual experiences – with all the positive or negative connotations that those implied for the various scholarly fields. On the other hand, the synchronic view represented a welcome response to our need for fragmentation. Once we understood how systems can be identified and endlessly subdivided, and how well they seem to respond to our process of analysis, fragmentation became a ubiquitous phenomenon (how deep the implications of this all-encompassing phenomenon can go was authoritatively spelled out by David Bohm 1980). The synchronic view is always ready for dissection. We can isolate parts of it, study them separately, and break up the parts in their turn. The diachronic perspective is different. Most importantly, in order to access it, we need time. We must walk through the material in order for the picture to emerge. While fragmentation is possible in this case too, it involves a higher risk of losing the coherence of the picture: here we do not have the means offered by the synchronic view to instantly switch back-and-forth between a fragment and a larger part that includes it, or between fragments and the whole. Each assessment, whether the piece is large or small, needs to be “listened to” – it needs time. Unlike the fragments in a synchronic perspective, the ones isolated from a diachronic perspective cannot all be accessed at the same speed (virtually instantly): larger (longer) fragments need more time than shorter ones. As a matter of principle, for the diachronic perspective, there is no shortcut. This is particularly important in nonlinear science. For certain categories of models (such as cellular automata), the diachronic perspective is the only one that can be applied.

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To conclude, the synchronic and the diachronic perspectives are different from each other in terms of the way they involve scale. The synchronic perspective starts with a certain scale in space, and then it can iteratively use progressively smaller scale sizes, as it focuses on subsystems of subsystems. The scale in time does also intervene, but it can be freely applied to any phase of the investigation. The diachronic perspective also starts with a certain scale: a scale in time. In many cases, if fragments of the picture are analyzed, the time sampling must be kept unchanged. “Listening” to the emerging picture at a different speed than the proper one, which corresponds to the actual process that is studied, may produce distortions, pushing the researcher away from the place where the understanding could happen. As a metaphor for this category of objects, one can think of listening to music or to speech at a very different speed than the one at which the sample was recorded. In other cases, an intentional change of “listening” speed can be applied, to serve the goals of the research; however, once again, the sampling rates and thus the time scales cannot be arbitrarily chosen – they are determined by the system image, by our goals, and by the methods applied in the investigation. In principle, movement – and, in a broad sense, change – must be addressed in a diachronic perspective. However, the mentioned tendency towards the spatialization of time has encouraged and supported a range of methodological approaches to change, which offer synchronic views of the objects’ dynamics. Phase portraits of dynamic systems (Strogatz 2015), for instance, represent static images that show the trajectories of the system in a different kind of space, state space: the latter is defined by the characteristics associated with the analyzed system, so that any state of the system corresponds to a point in that space. Even in one and the same perspective – the diachronic one – objects may be seen as a function of spatial scale. A striking example is offered by foamy sea waves seen from a distance  – for example, from an airplane: the waves look as if they were painted on the surface of the sea. They do not seem to move. This is quite an intriguing image, even if the explanation is straightforward: seen from high above, the distance traveled by the waves is very small, and during our observation time, it looks insignificant. In spite of its simplicity, the phenomenon has the merit of illustrating the fact that, for an observed part of the environment, movement and scale are coupled. While movement in physical space and its coupling with spatial scale can provide interesting perspectives on the studied systems, another category of movement – movement in scale space, which enables “motionless travel” – can also offer intriguing insights.

3.7 The Time Scale Bias Even in absence of time sampling effects and the impact of changes in the sampling rate, time scales are used in ways that may produce a misleading picture of the process under investigation. Everyone is surely familiar with statements such as “this is the largest number of lost pets in this city, since 2011,” or “the highest number of

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volcanic eruptions in this region since 1983,” or “the lowest number of visitors of this museum since 2018.” To go even further, perhaps we are in fact witnessing these days “the highest number of mass-media-highlighted records since […]” (one can fill in the blanks). There are two sides to this situation. On one hand, all such “records” can actually be true (and even if some of them are not, they could be, without changing the nature of the problem). On the other hand, their abundance as reflected in many ways by the mass media can easily create the impression  – an incorrect impression – that we are living in times of records. Everything seems to happen right now. It looks as if all the highest and all the lowest numbers have decided to accumulate right in our time. This decade. This year. This month even. This impression is created by the relation between event sizes and time scale. Let us consider a random variable, which could stand, for instance, for the number of umbrellas sold every week by all the stores in a neighborhood of a city (Fig. 3.8 – the variable was “normalized,” so that the average is zero for this graph, and we can better notice positive and negative “records”). The abscissa represents time (for simplicity, given in number of events, not in temporal units). The ordinate can reflect the mentioned number of umbrellas, as well as another possible variable we might be interested in. We can see that for any of the randomly chosen points in time, the value of the variable (marked by a filled circle), there is a time interval over which it is either a maximum or a minimum value. The intervals involved are marked by rectangles A to E. In some cases (e.g., E), the time interval for which our

Fig. 3.8  Time (shown as event number) vs. event property (in arbitrary units). For any event in the series, there is a time interval preceding the event, over which the event property represents a record value (a maximum or a minimum). Five examples are marked on the graph (A to E): the dots specify the events, and the rectangles show the time interval over which the corresponding event dominates as an extremum

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point is a record is quite modest. For others (interval C), it is much larger. The key issue here is that a record-supporting interval can be found for any point on the graph – even for a random variable. This link between time scale and the magnitude of variables can thus justify any number of records, and one can simply pick the records that match the intention of the communicator – to instill hope, or fear, or concern, or to stimulate the audience to do something as concrete as buying an umbrella. In the context of this chapter, the more relevant part is, however, that we can see here scale as size in time directly dependent on – and selected through – the momentary values of a certain variable. The availability of such a virtually endless source of records goes over and beyond all aspects of unquestionable major change (population growth in certain cities, cardiovascular diseases as a global cause of death, worldwide number of cell phone users, etc.). Knowing that, regardless of what happens, records can be found buried in the history of almost any quantity that varies in time must be reassuring for the media: there is always something to be brought up on a slow day. A (possibly unintended) negative consequence of this productive approach to news generation is non-negligible, however. Being immersed in a news environment in which record-­ breaking events are, paradoxically, part of the normal landscape, one becomes progressively less able to appreciate the meaning of records that should definitely stand out. Such records may refer to events that signal major long-term threats – and not for the local area only, but on an extensive scale, both in space and in time. Outstanding examples refer to processes endangering peace, or signs of climate change. In the latter case, we do not even have the excuse of not knowing most of what is going on. Evidence of climate change and explanations regarding its implications are out there, at our fingertips, rigorous and detailed, and retold for all audiences; the news offer abundant information on the recent findings and  – at an alarming rate – data on more and more broken records interpreted in the light of climate change. When these waves of news are embedded in the pattern of “there is always a record somewhere,” they often receive less attention than deserved. Erroneous perception has erroneous action (including lack of action) as an immediate consequence, and this happens precisely when acting is critically important, at different spatial scales  – ranging from the municipalities and rural communities (Burn et al. 2021) to the scale of the planet.

3.8 Scaling Up and Scaling Down 3.8.1 Dimension and Changes in Scale Performing major changes regarding the scale on which one operates can be tremendously helpful, but also an intricate operation. The concepts of scaling up and scaling down can be encountered in a variety of fields, and they are often defined in ways that are narrowly dependent on context – in computing science and data science, business, manufacturing, agriculture, and public services (e.g., health

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services), or climate science. Here we are particularly interested in the scientific meanings of the terms. It is for this reason that a useful starting point is the concept of dimension – considered, as usual, based on the meaning that is mostly relevant to common experience. The simplest way of intuitively defining dimension is probably the one referring to the number of coordinates needed to specify the position of a point. In this sense, since in order to define the position of a point along a line we only need one number – a distance – the line is one dimensional. We indicate the position of a point on a plane by using two coordinates, and thus the plane is two dimensional, etc. The word dimension is often used interchangeably with “size,” which is fine as long as the space-defining meaning of dimension does not play any role; otherwise, confusion is unavoidable. Rigorously speaking, dimension should be reserved to its geometrical meaning that refers to the type of spatial creature we are dealing with (this is particularly important when we address objects characterized by a dimension given by a fractional number, as shown in Chap. 7). For other uses, “size” might do. If someone asks about the dimension of a refrigerator, the correct answer is always three, since no one would use, for instance, a flat refrigerator drawn on a wall. Scaling up and scaling down make one think of changes in size and of properties of objects that would change with size. As Geoffrey West shows in insightful ways in his monograph on scale in the living world (West 2017), one can be easily misled in assessing implications of scaling up or scaling down. For instance, when one reads about the features of minuscule organisms and the way in which they would look if enlarged many times, to reach our size, utterly incorrect conclusions can be reached: grasshoppers seem to leap over distances equivalent to several city blocks, and ants – which can carry the weight of hundreds of other ants – look incomparably stronger than humans. The key behind the problem of such changes of scale is the role played by dimension. If a one-dimensional feature is increased n times (such as the side length of the building in Fig. 3.9), the area of the object (e.g., the outer surface to be painted, or the area of the windows) increases n2 times, and the object’s volume becomes n3 times larger. This dramatic discrepancy between changes in length and those affecting the other dimensions is a major source of misevaluation in imaginary scaling operations. For instance, the beams shown in Fig. 3.9 lying in front of the little house and those in front of the large house are different in length (the large one is 6 times longer); however, there is a much larger difference in volume, and therefore in mass: the large beam is not 6 times but 216 times heavier than the small one. As Galileo insightfully noticed, the strength of such elements is proportional to the area of their cross section: since the volume (and thus the mass) increases proportionally to the cube of the linear size, while the areas only increase proportionally to their square, the discrepancy between size and strength grows fast when size is increased – by scaling objects up, their relative strength decreases. In other words, the “scaled up” and the “scaled down” Alice in Lewis Carroll’s book would look (much) weaker and (much) stronger, respectively, than the original one. As Galileo carefully stated, were objects to be increased “inordinately” in size, they would end up crashing down as a mere impact of their weight (West 2017).

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Fig. 3.9  Increasing a linear size n times leads to an area that is n2 times larger and to an n3 times greater volume: this applies both to the building and to the beam that lies in front of it. Strength is proportional to the cross-section area (e.g., of the building’s beams), and not the volume: a proportionally larger building will not be as strong as the smaller one

The idea of the impact that a change in a one-dimensional feature can have on the volume of a three-dimensional body was used in a paper (Liao et al. 2012) that suggests possible solutions to be considered, in order to support sustainability and combat climate change. Some may call the paper controversial, while others may think that “controversial” is an understatement  – not without reason. The authors  – Matthew Liao, from the Center for Bioethics at New York University, and Anders Sandberg and Rebecca Roache, both in the Faculty of Philosophy at the University of Oxford – suggest, among other things, that we could reduce human height. The means indicated by the authors for this purpose are not of direct interest here, but the bottom line is that an irreversible decrease in size would be obtained. Smaller people would have a smaller volume and a lower mass. Since the change in surface grows in proportion with (or “scales with”) the square of the reduction in height, they would need significantly less fabric for their clothes. Volume and body mass, on the other hand, scale with the cube of the height reduction, and smaller people would thus have lower energy needs. Transportation would imply less fuel. The authors even mention a lower wear-out rate for shoes, carpets, and furniture. Overall, this is expected to help us to address our sustainability challenges. Could such a change be considered, at least in principle? While the dimensional aspects of such changes are undeniable, linking them with the idea of sustainability is, in fact, quite strange. As it is often the case, this is a matter of scale. Performing the suggested one-time human height reduction would imply indeed the benefits listed by the authors, but these effects would be quickly

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absorbed and made insignificant in the context of fast growth: persistently encouraging humans to consume more, to replace their technical devices increasingly frequently, to change their possessions faster in order to follow trends, etc., not to mention population growth, are powerful factors that count incomparably more in the long run than a one-time height reduction. In other words, such a solution would not be able to work. If the suggested pattern would have been “reduce the size by n percent every k years,” the solution might have worked, at least mathematically, depending on the values of n and k and on a number of other factors. However, the authors do not suggest such a pattern, as it would obviously not be applicable. The ethical dimension of the proposal is not part of our topic – and we should keep in mind that the paper was written by scholars who are ethics experts themselves, so we will only add a brief note on this issue. According to the paper, accepting such a form of human engineering would be a voluntary decision, and subjects could be offered tax breaks or free health care. It would thus be the world’s poor who would end up accepting the height-changing actions. Therefore, even if the authors do not explicitly state it, their proposal is apparently designed to lead to a world with two categories of humans: the large people and the small people, with the line being drawn as a function of their income. In what way and from what angle this solution can be considered ethical is hard to tell.

3.8.2 Downscaling in Climate Change Studies As long as one properly considers the implications of changes in scale, such operations can be highly valuable, especially when results obtained on one scale interval can be – carefully – used to make inferences regarding a different scale range. This is the case in a wide variety of fields, far beyond merely geometric considerations, ranging from computers and computer networks to economics, ecology, and the physics of the atmosphere (Mucina 2019; Vo and Silva 2020; Erlandsen et al. 2020). The word “carefully” highlighted above is meant to draw attention to the fact that cross-scale inferences can fail for a wide variety of reasons, and one should not expect to come up with a generally applicable checklist that would make scale transfers safe. Downscaling is particularly useful in the study of future climate. This is the case when one focuses on regional implications of global climate models (which are highly useful but have a relatively coarse resolution  – of hundreds of kilometers horizontally and dozens of layers vertically). In fact, important physical processes of interest, such as those regarding clouds and various forms of feedback, for instance, are associated with smaller scales. For studies on climate change impact and adaptation strategies, information on finer scales is essential. Currently, downscaling methods can be broadly divided in two categories: statistical downscaling and dynamical downscaling. Statistical downscaling relies on the construction of links between information on a global scale and information on the local scale. To accomplish this, it uses

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statistical methods to establish correlations between large-scale atmospheric variables and certain variables measured on local scales. The relations thus identified in this phase are then applied in the global circulation models, in order to arrive at projections on finer scales. The combined use of both model projections and actual measurements contributes to the strength of the method. Moreover, this approach is computationally efficient, and it allows the use of large sets of global circulation model runs. Given its high dependence on observational data, statistical downscaling requires long records (30–45 years) of high-quality data, which are available for some regions but not for many others (Erlandsen et  al. 2020). Results strongly depend on the proper identification of underlying assumptions, uncertainties assessment, and bias correction (Maraun and Widmann 2018). Dynamical downscaling hinges on a model on a different scale: a regional climate model, which benefits from a finer resolution. The model uses boundary data that are produced by circulation modeling on a global scale: in other words, the regional climate model is nested in a global circulation model. One of the key strengths of this approach is that it captures the physical processes occurring in the studied volume of the atmosphere, with information on all the needed atmospheric variables. As is the case in statistical downscaling, the reliability of this approach is dependent on the accuracy of assumptions and the proper statistical identification of relationships. Compared to statistical downscaling, its output is more comprehensive, and it benefits from a higher resolution, both in space and in time; however, it is computationally costlier: it cannot be applied to the large sets of global circulation models that are used in the statistical downscaling approach (Erlandsen et al. 2020). Both approaches have led to significant progress in our understanding of climate change and its implications, with improved outcomes regarding uncertainty and enhanced levels of detail. Both are confronting challenges implied by the nonlinear nature of changes in physical processes due to climate change, changes to feedback processes and their impacts, and changes in the occurrence of extreme events. Along with observational efforts, renewed energy is dedicated to the development of novel effective methodologies, capable of better addressing the multifaceted complexity of phenomena involved in climate change.

3.9 Scale as Size, Context, and Scale Change In every investigation – whether in the natural sciences, social sciences, or humanities – we focus on the relevant systems without completely cutting them out of their environment. Our study and our interpretations never occur in void space. On one hand, links between the object of investigation and its surroundings are taken in consideration. The question is: How far should we go when we extend our research to the object’s surroundings? On the other hand, we study the object in a certain theoretical framework, using a range of instruments. Questions arise regarding the proper backdrop against which the study should occur, the methods to be applied, etc. All these questions point to an important albeit often neglected aspect of

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research: context. Ignoring or neglecting context only means handling it without realizing that one is doing it, which can easily lead to suboptimal results. Context usually refers to the circumstances, the conditions, in which something – the process of interest – occurs. In terms of the system definition from Sect. 2.1, no matter how we define the system, there is always the environment that is surrounding it. In most cases we cannot simply ignore all interactions between the elements of the system and the environment, neither can we include the system’s environment “in its entirety.” Once we decide to select a portion of the environment surrounding the system and include it in our investigation, the need for the concept of context has already become palpable (context, however, offers much more than this, as shown below). An example regarding an environmental investigation is provided in Fig. 3.10, which shows a symbolic representation of a river (in the center), the river banks, and – on both sides – agricultural land. When a large number of dead fish shows up in the river, the river water is tested – area A is investigated. The analysis indicates the presence of an unusual pollutant in the river. The investigated area A is thus extended upstream along the river, now covering area B. Water tests suggest that the pollutant reaches the river all along a segment, so the riverbanks – its immediate spatial context – are included in the study (area C). Pollutant traces can be found along the banks, between the river and the agricultural land, along the suspected segment. Consequently, the area is again extended (area D). At that stage, investigators find that an uncommon toxic fertilizer had been applied to the field, and – on top of that – the furrows, instead of being parallel to the river, were perpendicular to it, so that rain and irrigation water from the field flowed directly into the river, carrying the toxic substances with it. This story, inspired from a real case, shows that the choice of spatial context can be essential for reaching the goals of research. As a side note, we should add that in this case the exact way in which the system is defined – where the system ends and where the context begins – is not essential. Fig. 3.10  A, B, C, and D are four stages in the selection of the spatial context involved in an environmental investigation. The represented features are, from the center towards the edges, river, river banks, and agricultural field

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What counts is our awareness of the part of the environment that is subject to our study and of the processes and interactions we decide to take into consideration. Historically, more attention has been paid to context in the humanities than in science. However, the fact that its function and impacts are more obvious in fields such as art history does not lighten our responsibility to properly consider it in other domains (which can be as different as volcanology and urban geography). One of the main roles of context is to help us to avoid a mentality that is (understandingly) affecting especially young and unexperienced scholars: the unwavering assumption that by focusing on our studied object we end up with a firm and unique view, which is science’s “answer” to “the” right question. In the simplest cases, this might even be an operational assumption; however, most situations we are addressing in research are by principle not simple. It takes a different level of understanding to realize that what may seem to be a compact answer has instead deep ramifications, linking it to other features of the studied problem, and that those links may only become partially elucidated. More often than not, pondering the answers makes new questions rise at the horizon, and those may require changes to the way in which we circumscribe our studies, changes to the methodological arrangement, etc. – changes to context. Since it is profoundly embedded in such choices, scale can be contemplated, changed, and readjusted in more than one way. For these reasons, it is necessary to have a clear picture of the meaning of context, its potential structure, the way in which it arises and in which it is being used, and especially its relation to scale. While it can rely on insights from various domains, the resulting picture should not be restricted to a narrow disciplinary niche: it should rather enjoy a broad area of applicability. For instance, concepts and principles that were developed to support interpretative acts in art theory and literary theory (Bal and Bryson 1991; Culler 1988) prove to be particularly effective in many other fields, especially in the natural sciences. A well-defined distinction can be made between two ways in which context intervenes in an investigation. Bal and Bryson (1991) call them “context of production” and “context of interpretation,” respectively. The context of production refers to the circumstances in which the studied object was produced (rather than talking about objects, the authors refer to the signs that these objects represent). The context of interpretation refers to the elements that play a role in our observation and investigation endeavor. An example is provided in Fig. 3.11. The object, shown in the center of the figure, is a piece of lava that was ejected by the Kilauea volcano in an eruption that occurred in 2013. The context of production includes a range of volcanic processes (which are responsible for the rock composition, for the eruption, etc.), rock weathering processes that occurred over time, and others – it is clearly indispensable for the scientific understanding of the object. The context of interpretation is also essential, but it is completely different. It concerns the way in which we address, analyze, and interpret the object. This includes factors such as our professional background, the instruments we use, the methodological framework we apply, the scientific model we have in mind, information we acquire from publications while studying the object, the dominating interpretative approach in our

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Context of interpretation

Context of production Fig. 3.11  A simplified diagram regarding the two types of context – in this example, a piece of lava produced by the Kilauea volcano in Hawaii. Both types imply decisions regarding scale and scale change concerning the investigation. (Photographs courtesy of USGS)

(scientific) community, and even interpretative predispositions based on our individual experience. There are multiple reasons why paying attention to the two types of context is important. One of them is the fact that some of the factors of the context of interpretation are quite rigidly prearranged, but others can be deliberately modified. On the other hand, the context of production can also be changed – not by influencing past processes but, for example, by making new choices regarding its delimitation, both in space and in time. Another reason is the fact that the two types of context can be scrutinized and adjusted in specific ways. However, both of them have scale at their core. Figure 3.11 presents a simplified diagram of the relations between the two types of context and the object we study, based on an example. It shows that the context of production includes factors such as the physical processes involved in the volcanic eruption, as well as other processes that have occurred over time. The context of scale in space can be more or less comprehensive, depending on the choice of the researcher. Certain processes might only be understandable if the spatial scale is wide enough; too large a scale, however, can dilute the research focus and imply a waste of resources. We should also keep in mind that spatial scale does not necessarily refer only to distances on the surface of the Earth. The third dimension is essential in this example, and the extent to which we choose to enhance the volume of the context has direct implications for the physical processes that can be studied,

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from the dynamics of the Hawaii hotspot all the way to the development of fractures at the surface and the lava flows. Choices regarding scale in time must also be made here. Certain processes, which are important for the studied object, have been occurring for a long time before the eruption took place. Time scale decisions can have major consequences: How far back in time is it necessary to go? Beyond space and time, there is another dimension that is often useful for establishing of context. It can be called “scale of investigation depth,” and it concerns the level of detail to be considered: it is linked both to the scale in space and to the scale in time, but its meaning extends beyond them. Like spatial and temporal scales, its choice can have drastic effects both on the effectiveness of the endeavor of understanding and on the resources that are needed in research. All three scales of context – in space, in time, and in investigation depth – must be decided from the onset of the study. This is not to say that they cannot change. On the contrary, all the three scales are expected to be updated, especially as a consequence of the evolving context of interpretation. We should underline at this point that the three scales mentioned here in relation to context are not new types of scale. All three are scale as size. Spatial and temporal scales are simply of the type scale as size, associated with units, as expected, while the investigation scale belongs to scale as size “beyond units,” which will be discussed below. In this example, the context of interpretation includes all our knowledge about the region and its geological history, the available instruments, the familiar methodological tools, etc. For these reasons, the context of interpretation must be set up after choosing the context of production. The commonly accepted latest knowledge about the studied issue, increasingly well crystallized in interactions between researchers, also plays an important role. However, some of these factors can be changed. Some can be explicitly addressed by the researcher  – for example, by assimilating a new methodology for data processing, or coming up with a new hypothesis that can imply a cascade of new or transformed aspects of research work. Other changes occur less explicitly, and they can gradually shift the research perspective based on the growing understanding that is reached over time. The context of interpretation applied when one writes a scientific paper on the performed research can be significantly different from the one that dominated the early stages of the study. Nevertheless, since context can play a substantial role in the performed study, all the features of context, especially their scales and boundaries, must be explicitly documented in scholarly communication, including publications. A warning regarding the use of context concerns the virtually unlimited nature of the procedure involving its increase in scale. When we “step back” to get a broader picture, there is no reason – Bal and Bryson (1991) point out – not to take a step back over and over again, in a process “without brakes.” It is hard to tell how tempting or probable such a possibility is, in the circumstances these authors had in mind (related to the interpretation of art). However, in science, an iterative process of context scale enlargement is not only acceptable but also possibly very useful (an example was presented above; see also Fig. 2.13), and the mentioned danger of unstoppable scale growth is low. On one hand, every time the spatial scale is increased, this happens for a well-defined reason. If the solution cannot be found,

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and the scale is increased repeatedly without success, we are not going to indefinitely increase scale. At some point, based on our scientific background, we realize that increasing scale even more is unlikely to help; alternative action must be taken – for instance, considering the question(s) from a different angle, devising new methodological means to approach the issue, etc. In the mentioned example, if researchers do not find the origin of water pollutants after the illustrated steps, sooner or later they realize that they must go back to the drawing board and make another plan for their investigation: there is no danger that they will look for the pollution source on the Moon. Notwithstanding the great value of identifying context and thoroughly deciding on its scales, our view of context can also undermine our understanding, instead of driving it further. Much like the illusion of a once-for-all stable and fixed context can be detrimental to interpretation in the humanities, so can the view of a rigid, unquestionable context undermine research work in science. This can easily happen when one mistakenly perceives context as if it were a reliable landmark standing firmly in the field of investigation. It is true that context helps us to avoid gliding and drifting in our search by anchoring our position in relation to other elements that reliably converge towards enhanced clarity and coherence. And yet, one should not forget that context is not “given” but also produced (Culler 1988) – that it is not always simpler or easier to grasp than the actual object that we study. Even the context of production may have to change, based on a better understanding achieved in one area or another. In fact, the context of production looks objective and reliable because it refers to the conditions that produced and shaped the object  – mostly physical processes, and yet, the context of production does not consist of the conditions surrounding the production per se, but rather of what we know and understand about those conditions. Since our understanding can and does evolve, the context of production also changes. Like virtually any powerful tool, context represents a major help in research, as long as it is properly identified – and applied.

References Bak P (1996) How nature works – the science of self-organized criticality. Springer, New York Bal M, Bryson N (1991) Semiotics and art history. The Art Bulletin 73(2):174-208 https://www. tandfonline.com/doi/abs/10.1080/00043079.1991.10786750 Barbour J (1999) The end of time. The next revolution in our understanding of the universe. Phoenix, London Bergson H (1889/2013) Essai sur les données immédiates de la conscience. Presses Universitaires de France, Paris Bohm D (1980) Wholeness and the implicate order. Routledge, New York Bonaventure, Saint (1259/1978) The soul's journey into God; The tree of life; The life of St. Francis (trans: Cousins E). Paulist Press, Mahwah, NJ Buhusi CV, Cordes S (2011) Time and number: The privileged status of small values in the brain. Frontiers in Integrative Neuroscience 5:67. https://doi.org/10.3389/fnint.2011.00067 Burn CR, Cooper M, Morison SR, Pronk T, Calder JH (2021) The Canadian Federation of Earth Sciences Scientific Statement on Climate Change  – its impacts in Canada, and the critical role of earth scientists in mitigation and adaptation. Geoscience Canada 48:59–72 https://doi. org/10.12789/geocanj.2021.48.173

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Canales J (2016) The physicist and the philosopher: Einstein, Bergson, and the debate that changed our understanding of time. Princeton University Press, Princeton Carathéodory C (1909) Untersuchungen über die Grundlagen der Thermodynamik, Math Ann 67:355–386. https://doi.org/10.1007/bf01450409 Culler J (1988) Framing the sign: Criticism and its institutions. Norman, London Descartes R (1641/1996) Meditations on first philosophy. With selections from the objections and replies (from Latin trans: Cottingham J). Cambridge University Press, Cambridge Eco U (1991) The limits of interpretation. Indiana University Press, Bloomington, IN Erlandsen HB, Parding KM, Benestad R, Mezghani A, Pontoppidan M (2020) A hybrid downscaling approach for future temperature and precipitation change. Journal of Applied Meteorology and Climatology 59(11):1793-1807 https://journals.ametsoc.org/view/journals/apme/59/11/ JAMC-­D-­20-­0013.1.xml Google Wind Data Collection Project (2011) RE 1) or a contraction (r  1) or lower ones (when k  1 (INGV 2018). Only aspects of the time scale variation in relation to the time of event occurrence and magnitude will be addressed here. To get a general picture of the succession of earthquakes during this time interval, we can plot the event magnitude as a function of time (Fig. 5.15a). We will call this representation “the time view.” Three elements stand out in this series: (1) the earthquake from April 6, 2009, (2) the earthquake from August 24, 2016, and (3) two earthquakes that occurred on October 26 and October 30, 2016, respectively. The latter two cannot be discerned from each other on this graph, because they are so close to each other in time. Given the large number of events that occurred during the represented time interval, many of the earthquakes (especially those that occurred during a short time period) are unavoidably indistinguishable in the time view. However, we can also represent the events in another way. We can produce the temporal informational backbone of the series of seismic events. In this case, the successive samples with identical value correspond to time intervals with zero events. In the temporal informational backbone, all the time intervals with no earthquake in the dataset, i.e., with a magnitude lower than then the data threshold M1, are suppressed. The only remaining samples in the time series will be those corresponding to the seismic events in the studied dataset. We will call this representation “the

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Fig. 5.15  Two different views of earthquakes with magnitude M > 1 in Central Italy, January 2005 to December 2016: (a) time view, based on event distribution in time; (b) event view – based on the temporal informational backbone. Remarkable events: (1) April 6, 2009; (2) August 24, 2016; (3) October 26 and October 30, 2016. The latter two can only be distinguished in the event view. Note the gradual decrease in minimum event magnitude after large events, which is only visible in the event view

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event view.” The outcome of this operation can be seen in Fig. 5.15b: in this representation, the relation between earthquake events and time is fluctuating. Time intervals are thus seen from a new perspective, and some of them become better distinguishable than in the time view, which can be particularly important during time intervals of increased seismicity. A striking result is the fact that we can distinguish now in Fig. 5.15b the two events under point (3), which are lumped together in the original representation (the time view). One may initially suspect that the new representation, the event view, with its variable temporal scale change, can hardly reveal reliable information on the earthquake set. And yet, when we look closer at Fig. 5.15, we notice a significant difference between the two views. The event view corresponding to the TIB offers insights that are not identifiable at all in the original time view. We can see how after each of the major events, the value of the minimum event magnitude is significantly higher than otherwise (it starts from M  =  2 or higher). This minimum value is slowly decreasing over the following 5000 to 10,000 events or more, until the earthquake succession becomes similar to the pattern that can be seen farther away from the large events. This diminished number of small events, which decreases in time, can be noticed after each of the earthquakes marked with (1) and (2), as well as after the second (largest) event in the pair denoted by (3). While the event view does not replace the time view, it offers useful information, which is embedded in the same data but which is not discernible in the time view. Although in the event view the scale factor varies over time, this does not make the time-scaled view less fruitful.

5.8.5 Time Scale and Auditory Information: Sonification When scale change in time turns datasets into an auditory form, it can produce records which, although they do not provide additional information, can be interpreted in productive ways. Sonification – the conversion of various forms of information in a way that makes it available to auditory perception – makes use of the superior performance of hearing regarding pattern interpretation (Worrall 2019). Sonification offers a very different experience of earthquakes and earthquake sequences: earthquake recordings include a broad range of waveforms, which can be shifted into the audible range when they are time-compressed. Thereby, earthquake patterns can be approached in different ways, beyond the analysis methods applied to data in numerical form. Sonification of earthquake signals has been used for a long time – since the early 1950s. By applying not one but a series of different values of the scale factor R, to produce a series of different sound representations of the data, the transformation helps the listener to perceive various features of the seismic signal patterns, which, in many cases, would not have been noticed on their graphical representations. For example, when compressed in time with a factor R of 30, 50, and 100, records of the magnitude 9.0 Tohoku-Oki earthquake from March 2011 provide a distinctly

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different experience for every compression factor; in each case, there are novel aspects to be uncovered (Peng et al. 2012). Due to repeated listening, the detection and interpretation of the sound patterns can gradually lead to new findings. Humans are able to accurately detect certain aspects of earthquake rupture processes in ways that are different from those of conventional analysis methods; by learning from human experience, it is then possible to improve the processing algorithms for seismic data (Paté et al. 2017). Thereby, both of the two ways of applying time scale change in research, mentioned in Sect. 5.8, can be successfully addressed: experiencing and studying the phenomena on a range of different time scales and improving the analysis methodology based on human performance regarding the detection patterns, by listening to the data in sound form. Similarly, ground-penetrating radar, which is used for the identification and evaluation of buried features, produces signal patterns that can be brought into the audible frequency domain. Sonification offers thereby additional means of pattern interpretation (Kaplanvural and Livaoğlu 2021). It can become part of an improved way of operating ground-penetrating radar equipment. If correctly evaluated (Dubus and Bresin 2013), sonification based on time scaling proves to be useful both for educational purposes and as an additional approach to the study of various physical phenomena (Peng et al. 2012). Sonification can also be intriguing, when it is applied to areas in which we have little or almost no expectations regarding temporal patterns. For example, NASA’s Juno mission produced a record of electric and magnetic fields during the flyby of Jupiter’s satellite Ganymede, which was then shifted into the audio range. The resulting audio clip features distinguishable frequency shifts, which correspond to the various regions of the satellite’s magnetosphere (NASA 2021). Time scale change and sonification are also used in many other fields. In medicine, for instance, they are applied to brain monitoring based on electroencephalography (Sanyal et  al. 2019), or to support monitoring and diagnosis regarding myocardial infarction using electrocardiography (Aldana-Blanco et  al. 2020). Added to auditory signals from other monitoring devices, such as pulse oximeters, these systems contribute to a new soundscape in the hospital, which changes the access to information arriving from multiple sources. This change involves significant advantages. Being capable of detecting a problem based on changes in the sound pattern only, without having to visually observe one more instrument, allows the physician to concentrate on key tasks without interruption. There is, however, a limitation to this approach. Having to monitor and adequately respond to a number of sound patterns in the hospital soundscape can defy the purpose of the shift to sonification. The existence of a large number of sound-­ based signals has been shown for some time now to contribute to “alarm fatigue” – a diminished capacity to promptly identify and to properly respond to alarm signals (Solet and Barach 2012). A rich and complex soundscape can exacerbate the cognitive load of medical staff (Aldana-Blanco et al. 2020). The challenge related to this new face of datasets, transformed through time scale change and turned into audible signals, can be viewed in the light of our earlier considerations regarding temporal experience (Chap. 3). There are elements in the

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environment that require distinct approaches in terms of time: synchronic or diachronic. Those approaches to reality often have critically important implications for our further explorations and for our understanding of the environment. In this context we can see that the abovementioned cognitive load is not merely increased due to (admittedly significant) additional elements that compete for a person’s attention. A qualitatively new aspect of information overload emerges due to the use of sound. Sound is different. It requires a diachronic approach, in contrast to most of the other – visual – sources of information, which can be handled in a synchronic way. For example, while a glance at a set of lights associated with each patient in a ward can immediately lead to the conclusion that no critical issue is occurring, or that a certain patient is experiencing one problem or another, sound patterns usually require listening time. An exception is represented by sounds that are alarm signs (which can be identified in a very short time). Otherwise, diachronically addressed signals produced by sonification require a minimum time interval, below which they cannot be used. Therefore, in this context the sonification-generated information overload is not just overload in general, due to the large number of stimuli, but also, specifically, a form of time density overload. This effect produced by the unavoidably diachronic nature of certain signals acts in addition to other challenges, which were identified in relation to heavy sensorial input. For instance, dividing attention to multiple information sources can lead to distorted signal perception (Zimmermann et  al. 2016). Therefore, a careful balance must be sought in relation to the number of information sources and the synchronic or diachronic nature of experience that they all involve. Fruitful studies involving time scale change are also performed in the field of animal communication. These investigations use time compression (Simmons 2005), time expansion (Newar and Bowman 2020), or both (Ter-Mikaelian et  al. 2013), providing new perspectives on sound patterns. Taking time scale change one step further, it is possible to combine visualization and sonification  – for instance, to time-compressed earthquake records. Indeed, humans are highly capable of visually assessing spatial patterns, while their auditory system can offer them reliable access to temporal patterns. Using these two forms of information together, on a series of different time scales, leads to an enhanced perception of complex patterns, supporting both the understanding of earthquake dynamics and the generation of superior pattern analysis methodologies (Paté et al. 2021). At the same time, the possibilities of presenting information in multiple forms are highly valuable from the point of view of accessibility. Selecting and combining different modes of communication, through sonification, visualization, or use of haptic devices, the diversity of individuals (including those who are deaf, hard of hearing, blind, etc.) can be addressed in unprecedented ways, starting with accessible learning. In the context of the enormous diversity of individuals who live on the planet, this creates better opportunities for everyone to develop and manifest their special strengths and to follow their professional aspirations.

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5.9 Subjective Time Scale and Learning About the Environment 5.9.1 Change and Subjective Time: The Old Masters Present attention transfers the future into the past. —St. Augustine (fourth century/1998)

It is not hard to imagine that great minds of the past were struck by the elusive nature of time. What might be less expected is their penetrating insight regarding time perception and its change: time compression and expansion. It is indeed helpful to take a look at the temporal landscape in which certain ideas have evolved. The cascade of conceptions regarding subjective time perception is particularly relevant to time scale change. As will be shown in Sect. 5.9.2, experiments and models indicate that time compression and expansion are related to attention and, in particular, to the amount of information handled per time interval – the information processing rate. Remarkably similar insights have emerged long before such investigations have even begun. One may be surprised, for instance, to see how Kant naturally states that “time is nothing but the form of inner sense” (Kant 1781/2007) and that its perception depends on “modifications of the mind.” In fact, so natural is this statement for Kant that he does not feel the need to explain it at all. The meaning of such statements may be quite intriguing, especially since time and space are presented as a priori concepts, which precede experience and play a crucial role in its shaping. However, what may first be taken for vagueness gains in clarity when seen in the light of a longer line of ideas on time perception and time scale change. Indeed, at the time, the connection between time and an “inner sense” had been firmly established. If we go one step back, to Hume (who’s work held a prominent position in Kant’s eyes), we see that for him “from the succession of ideas and impressions we form the idea of time” and when perceptions “succeed each other with greater or less rapidity, the same duration appears longer or shorter” (our emphasis) (Hume 1740/1985). Hume’s view in this regard is not new either – in his work, he promptly points to his source: John Locke. Half of a century earlier, Locke (1690/2008) analyzes this issue in great detail. He relates the perception of time with the succession of ideas, including the connection between the density of ideas and the feeling that time slows down, or that it moves faster. He also makes the following interesting observation: “There seem to be certain Bounds to the quickness and slowness of the Succession of those Ideas one to another in our Minds, beyond which they can neither delay nor hasten” (the choice for lower- and uppercase words, and the use of italics, belongs to the original text) Locke (1690/2008). With one longer leap in deep time, to the fourth century, we can hear St. Augustine not only declaring himself perplexed about the nature of time but also boldly facing the intricacies of time flow. He prudently begins by assessing time flow in terms of the duration of uttered words (Augustine fourth century/1998). Then he advances, step by step, reaching the conclusion that “it is in you, my mind, that I measure

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periods of time.” He even notices, more than a millennium before Locke, the effect of having one’s mind “distracted” on the accurate perception of time interval length. Before reaching the last station in time teleportation, let us consider another stop: in the third century, with Plotinus. He produced work whose light inspired St. Augustine in many ways and influenced other major scholars in the centuries to follow. Plotinus not only explicitly mentions the role of time associated with uttered words, but he also establishes the relation between time flow and an internal sense of time passing: “Time is the soul’s passing from one state to another.” Finding in Aristotle definite references to the perceived flow of time might be surprising. On one hand, he writes half of a millennium before Plotinus, not to mention the enormous temporal distance by which his work precedes the other intellectual creations mentioned above. On the other hand, this topic is quite uncommon to the context in which Aristotle writes, since for the Greeks, subjectivity and consciousness were not at the center of their intellectual field of exploration (Gadamer 2003). And yet, in his Physics, Aristotle associates time with conscious experience: according to him, time is, in fact, an experience of the soul (Aristotle, fourth century BC/1984), which brings us back to the beginning of this section’s journey, highlighting the pursued intellectual thread regarding the perception of time and of time scale change. The above condensed outline stopped with Kant – we chose to leave out later, highly relevant insights, such as those starting with Husserl and Bergson, who both have focused on key aspects of the experience of time. Otherwise, the richly ramified intellectual landscape that later evolved on this topic would have threatened to invade the discussion space of scale. We may, however, conclude that our awareness of temporal experience and its change in scale is not new: it follows a long and coherent history, even if the objectives as well as the means of investigation are so different nowadays.

5.9.2 Change and Subjective Time: Today’s Science Duration is in the eye of the beholder. —Julian Barbour (2000)

While Barbour’s statement sounds very much like an echo to Henri Bergson’s thoughts on “la durée” and the meaningfulness of individual time awareness, it has a different origin, and it goes in a different direction altogether. As a physicist, Barbour is interested in duration and time flow from the point of view of the role it plays in reality, as reflected in physics. It is from this perspective, taking “objective” time as a reference, that changes in time flow – subjective and temporary as they may be  – are addressed in this section. The phenomenon of temporal change of scale is relevant for our interaction with the environment, especially for the way we learn about it. Processes of temporal scale change can be very different in terms of their triggers, the sensory organs and the many interacting subsystems of the organism that

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are involved, the scales in space and time on which they operate, etc. Scale change processes are important even when they do not imply a constant – even less so a predictable – scale as ratio in time. To some extent, studies on time perception are made more difficult by the fact that one cannot have direct access to someone else’s way of experiencing time. However, significant advances have been made in this field, based on scientific experiments and models (Eagleman 2008; Ueda and Shimoda 2021). Subjective time compression and expansion can lead to massive changes to the perception of time. Certain features of time perception and time compression/expansion are time scale-dependent (whether sub-second intervals, intervals of seconds to minutes, or longer intervals are involved). High scale factors can be applied, which may significantly vary, in their turn, over time. For example, time expansion can abruptly set in, especially during traumatic events, when the elements in the surroundings appear to be in slow motion. This was interpreted by some scholars as potentially beneficial, allowing the subject to have better opportunities to react in critical situations (see, for instance, the review in New and Scholl 2009). A variety of factors have been found to be related to the subjective perception of time flow and changes to time scale, such as the complexity of the action performed by the subject, or the complexity of the evaluated stimulus. Nevertheless, an outstanding role has been recurrently assigned to attention. Numerous investigations have been dedicated to the relation between time scale perception and attention. Many of the studies point out that the perception of the length of time intervals changes if, while the subjects perform a certain task, their attention is diverted to another task (possibly a nontemporal one). The duration of unexpected and infrequently occurring events is consistently perceived as longer than for other events of equal length, which is explained by their implications for attention (New and Scholl 2009): a stronger time expansion is particularly related to so-called oddballs, i.e., objects that are new and different from those with which the subjects are familiar. Such objects are found to capture the subjects’ attention and imply an increase in information processing. When stimuli imply change – especially if they are complex and variable and involve motion, change in brightness, flickering, etc. – their subjective duration is longer (Clarke and Porubanova 2020). Most importantly, it has become increasingly clear that time scale change is not an effect of selective attention, but rather one of an increased global level of attention and increased arousal. For example, when time dilation is reported in traumatic events, it is related to the subject’s visual experience as a whole, and not to specific objects or processes, not even those considered to be responsible for the event (New and Scholl 2009). In general, changes in the level of attention prove to be strongly correlated with changes in time scale perception (Roseboom et al. 2019). Perturbations of time sense can also occur due to disease: people suffering from depression are known to experience time as passing more slowly. Superposed on this tendency, however, both time compression and time expansion can occur: usually, the experience of time is dilated for short time intervals and compressed for longer intervals (Kent et al. 2019).

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Not only the abovementioned complex, flickering, and variable elements in the environment can expand the perceived time scale. Observed relative motion also leads to time expansion: events with the same duration seem to last longer if the visual stimuli are moving, instead of being stationary. An important exception occurs if the observing subject is the one who moves, rather than the observed object. Although the relative motion between the observer and the object is one and the same, time expansion fails to appear in this latter case. This is a form of suppression of self-generated behavior. In other words, it is a way of the organism of taking into account its own motion, in order to better observe motion in the external environment, an important element in the process of exploring and assessing the environment (Lo Verde et al. 2019). Studies on the occurrence of time scale change associated with our exploring of surroundings provide valuable insights into the way in which we perceive and learn about the environment. For instance, during the observation of visual stimuli, time compression consistently takes place close to the moment of occurrence of microsaccades (small involuntary eye movements, which happen up to several times per second), as they do for voluntary, larger eye movements (Yu et al. 2017). Perceived time is thus shorter when eye movement takes place, compared to the time intervals during which the surroundings are observed from an unchanged “point of view.” In spite of the elusive nature of time, our information sources on temporal aspects of the world, especially on the changes to perceived time scale, are not limited to visual and auditory senses. For example, time compression expressed as acceleration has a direct impact on the perception of tactile stimuli (Tomassini et al. 2014). In fact, in the nineteenth century, the eminent physiologist and physicist Hermann von Helmholtz made it clear that our sensory perception is deeply connected with our movement. Although it is not one of the traditionally recognized human senses, the sense of balance is also essential to our interaction with the environment. Beyond offering a sense for equilibrium, it is also responsible for the perception of spatial orientation, direction, and – last but not least – acceleration. The latter is directly related to time scale, given its role in the detection and assessment of time compression through accelerated movement (Kniep et al. 2017). We should thus expect to see future progress in research regarding the relation between body movement, tactile stimuli, and time compression. Moreover, our senses do not work independently. Multiple fields of science study sense interaction, including the ways in which perceptions provided by certain senses have an impact on those obtained by other senses, or in which perceptions from multiple senses lead, together, to outcomes that are altogether different from each of them taken separately. Such interactions are not details of secondary importance: they are essential features of our endeavor of understanding the environment (Briciu and Briciu 2020; Efstathiou 2020). Scale change in time, therefore, is increasingly studied in a more comprehensive context, including more than visual and auditory stimuli, and focusing on the links between processes that were studied in the past in separation.

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We should also add that phenomena of scaled and variable time perception are not limited to humans. Although we do not dwell on them in this context, aspects of such manifestations have been observed in a wide diversity of species, from invertebrates to birds and mammals. Such a brief overview concerning temporal scale can offer a glimpse of the questions that are asked and of insights that have been gained in this regard. An already impressive and fast-growing area of research focuses on the problems concerning time scale change and its role in our understanding of the environment. Aspects of scale as ratio in time offer valuable information regarding the changes in time scale, whether compression or expansion, which take place, and the mechanisms that are involved in these transformations.

5.9.3 Large-Scale Time Compression “Time is a gentle deity,” said Sophocles. Perhaps it was, for him. These days it cracks the whip. —James Gleick (1999)

On a very different scale, both in space and in time (obviously, scale as size is the one meant here), a significant and persistent process of time compression has been noticed in a wide variety of circumstances. The feeling that time is, somehow, accelerating may have become increasingly intense, but it is certainly not new. The “most brisk and giddy-paced times” were mentioned four centuries ago, by none other than Shakespeare (in his Twelfth Night). On the other hand, various versions of Earth history projected into an interval of 1 hour, 1 day, 1 year, etc. offer suggestive views of the extent to which event density is increasing in time. For example, if we choose the 1-year version, we have the planet history beginning on January 1st. Life only appears in April. It develops slowly, but at an accelerating pace: the Cambrian Explosion, involving a large diversity of complex forms of life, takes place mid-­ November. Early December sees the first reptiles, and dinosaurs appear around December 12–13 and mammals – “soon” – afterwards, December 14–16. Primates can be found in the last days of December. Homo sapiens is not present before the last 30 minutes of the day. And what we call “human history” is all compressed in the last several minutes. It is hard to ignore the accelerating rhythm of change on such an extended time scale. Comparing the actual values of the scale factors that are involved requires data that are not always available. However, when such data exist, scale factors can be of foremost importance to investigations on acceleration processes occurring in various fields. A wave of scholarly works discussed the issue of compression, in the context not only of time but also of both space and time, especially towards the end of the twentieth century and the beginning of the twenty-first (Harvey 1990; Lash and Urry 1994; Urry 2003). The seminal book of the geographer David Harvey, The Condition of Postmodernity: An Enquiry in the Origins of Cultural Change (1990), was not only a bestseller – it had a remarkable impact on the broader scholarly environment.

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Harvey pointed out that our views of space and time have changed in radical ways. Time-space compression is, in fact, occurring, and it is nonlinear: it is becoming more intense in times of crisis. Major subsequent changes have been found to affect relationships between individuals, organizations, countries, etc. Technological innovations, their quick development, and their prompt and wide adoption have been seen as the triggering factors for a wide range of acceleration phenomena. Such innovations include increasingly fast means of transportation and spectacularly accelerated means of communication, as well as consistently accelerating processes of production and exchange of goods. The much faster environment required new ways of addressing social processes, taking in consideration the importance of “mobile connections”; in this context, it is the flow of information that should be placed at the core of our vision (Urry 2003), since it is enhancing the rhythm of change in many, interconnected ways. Most importantly, the ongoing transformations rely on a fast-increasing shift from material to informational values, in which the entities involved in flows tend to be signs, as opposed to material objects, and this shift contributes to even faster flows (Lash and Urry 1994). An increase in the extent of compression, and thus a decrease in the value of the scale factor in time, is widely reported. Few novel approaches to such complex, all-encompassing processes have been as deep and as effective as the concepts and theories concerning networks. Networks offer indeed powerful instruments for approaches to scale, and particularly to changes in scale, both in space and in time. They can lead to an enormous compression of spatial distances and of subsequent temporal changes. Insightful observations regarding the potential role of networks in the study of social processes were provided early on by Craven and Wellman (1973), but a wider development of this field only occurred later. Castells (1996) turned the concept of network into a core instrument in the analysis of societies: digital information networks were recognized as the warp of the complex social system. A large amount of scholarly work has been pursuing this direction. The concept along with the methodological framework that has been developing very quickly proved that networks can offer highly effective ways of approaching complex systems, as shown by Barabasi (2002). In the current context of change, networks are particularly important to phenomena of time compression. The emergence and growth of networks, materialized in a variety of systems, along with the rapid transition towards an information-­dominated environment, have changed the social and economic dynamics from small to large scale. In these networks, interactions occur faster, involving not just material objects, money, or elements of culture. They also propagate critical states (Suteanu 2005), which represent circumstances in which “avalanches” of events are triggered in domino-like processes, acting on “all” scales (Bak 1996), as discussed in detail in Chap. 8. With no apparent connection to the idea of a large-scale temporal acceleration in the history of the planet, a pervasive compression of time is perceived these days by individuals in the most different fields. On one hand, a variety of processes that affect everyday life occur, objectively, faster. On the other hand, people’s expectations have been changing accordingly. Patience has been taking a different shape.

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Endless series of examples keep popping up. Our devices, from computers to dishwashers to microwave ovens, are expected to work faster. Levine mentions the proposal to measure the “honkosecond,” the time between the moment the traffic light changes and the one when drivers begin to honk their horns to put pressure on those in front of them (Levine 1997). The very symbol of time collapse, the Internet, is both becoming faster and expected to become even faster. No surprise then that dubious slogans appear, such as “Everything worth doing is worth doing faster” (from an Internet provider’s billboard ad). Temporal acceleration takes a variety of forms. A study of Léveillé Gauvin (2017) highlighted a trend in pop music towards an increased tempo over the last 30 years; moreover, the length of the “intro” – the time interval before the lyrics start – was found to shrink fast, from over 20 seconds in the 1980s to an average of 5 seconds today. Classical music is also affected: a study found that works of Johann Sebastian Bach have been interpreted increasingly fast, their time length having decreased by 30% in the last 50 years (Solly 2018). We are witnessing time compression with a gradually decreasing scale factor, which has reached values as low as R = 0.7. Similar changes have been identified in other areas as well. For example, the average shot length in movies was found to decrease relentlessly over 80 years of film history; this corresponds to a time shortening that reached a ratio as low as R = 0.4 in 2010 (Cutting et al. 2011). Interestingly, the different forms of acceleration are often explained by distinct factors assumed to be responsible for the change, as if this broad phenomenon of acceleration occurred in various areas due to unrelated causes. Some of these hypotheses are easy to dispel. For instance, the idea that the Internet and people’s extensive online experience have led to the substantial shortening of film shots proved to be wrong: the mentioned study of Cutting et al. (2011) shows that the drop in shot length followed the very same power law between 1930 and 2010. Arguments of such clarity are not available, however, for all the manifestations of acceleration. In this quickly and perpetually renewed context of our relation to time, the value of the “now” has been rising. Most importantly, the concept of “now” has not been simply shrinking, but rather evolving, becoming more comprehensive. It is pondered in relation to time compression, and it is flexibly considered depending on context (Coleman 2020), which both involve a range of time scales: whether it hinges on a second, or it extends over hours or days, the “now” is arguably more significant than ever before. Time compression is expected to have a range of implications, some of which may come as a total surprise. Clearly, we are not only affected at the individual scale, since “time is the very cornerstone of social life” (Levine 1997). Before closing this section, we should mention that a considerable reaction has been developing against this massive accelerating trend. Born towards the end of the twentieth century, during the times of increased concern regarding temporal acceleration, the “slow movement” has been growing in an extensively ramified fashion. It insists on the need for “slowing down” on many levels, including “slow art,” “slow fashion,” “slow travel,” “slow food,” etc., all the way to “slow science.” In particular, “slow science” (Stengers 2018) is shown to have important implications

5.10  Time Scale in Reflection and Narration

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regarding the role of scholars in society and their responsibility towards the public, which tend to be distorted by the current tendency to publish in a fast-paced environment.

5.10 Time Scale in Reflection and Narration When changes of scale in time are associated, on one hand, with the external world and, on the other hand, with our perception of reality on different time scales, they seem to happen independently from us. Nevertheless, we can  – and often do  – change scale in time ourselves, whether or not we are aware of it. Here we focus on changes of scale in time involved in narration. Time scale change in narration is of interest in various areas of the humanities, in human geography, in psychology, etc. With almost no exceptions, narrations perform changes to the time flow of events that are narrated. We can thus distinguish occurrence time (based on the original set of events in reality) from narration time (the “mapped” set of events in the representation). When we relate something that we performed or witnessed, we apply time scale change. As in all other cases of time scale change, we refer here to the scale factor defined in Eq. 5.6, where one unit of narration time corresponds to N units of occurrence time. Temporal compression (R 322

Potential damage(selected examples) Minor damage. Peels surface off some roofs. Branches broken off trees. Shallow-rooted trees pushed over. Wooden houses damaged. Unprotected mobile homes or trailers may sustain moderate to serious damage Moderate damage. Roofs severely stripped. Mobile homes overturned or badly damaged. Loss of exterior doors. Wooden houses badly damaged Considerable damage. Roofs torn off from well-constructed houses. Foundations of frame homes shifted. Mobile homes completely destroyed. Large trees snapped or uprooted. Wooden houses completely destroyed. Cars lifted off ground Severe damage. Entire stories of well-constructed houses destroyed. Trains overturned. Trees debarked. Heavy cars lifted off the ground and thrown Devastating damage. Well-constructed and whole-frame houses completely leveled. Some frame homes may be swept away. Cars and other large objects thrown and small missiles generated Incredible damage. Well-built frame houses destroyed with foundations swept clean of debris. Steel-reinforced concrete structures are critically damaged. Tall buildings collapse or have severe structural deformations. Cars, trucks, and trains can be thrown more than 1 km

(in this case, two-tier) approach, as well as general properties of this type of scale. The main additional step beyond the general picture outlined in Table 6.2 consists of bringing in detailed information about the structures affected and the kind of damage that can be identified. Indeed, without specifying what kind of building lost its roof or its side walls, it is difficult to make an objective and useful evaluation of the causal factors that were involved. The EF scale therefore relies on a description of damage, which is specified for 28 types of structure. Examples include features such as “1. Small farm or farm outbuildings,” “6. Motel,” “10. Strip mall,” “16. Junior or senior high school,” “28. Trees – softwood,” etc. For each of the 28 types of structure, a degree of damage (DOD) is assigned to various forms of damage (McDonald and Mehta 2006). For example, the DOD values for the structure type “28. Trees – softwood” are shown in Table  6.3, where DOD can have a value between 1 and 5 (for other structure types, DOD can reach values as high as 8). As we can see, significantly higher wind speed values are involved when trees are snapped rather than uprooted and when they are debarked rather than snapped. As was the case for Table 6.2, here again, the EF category for a given location is assigned based on the highest wind speed that is estimated for any of the affected

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Table 6.3  Example of the degree of damage (DOD) tables: type “27. Trees – Hardwood”: oak, maple, birch, ash. Wind speed values are shown in km/h for 3-second gusts (NOAA 2021b) Degree of damage (DOD) 1 2 3 4 5

Damage description Small limbs are broken (up to 1″ diameter) Large limbs are broken (1–3″ diameter) Trees uprooted Trunks snapped Trees debarked, with only stubs of largest branches remaining

Wind speed – expected 97

Wind speed – lower bound 77

Wind speed – upper bound 116

119

98

142

146 177 230

122 150 198

190 216 269

structures. For instance, solidly rooted trees, which still stand after the tornado, found debarked represent a more relevant sign than the fact that other trees nearby were uprooted. The tornado path on the ground can have a width ranging between meters and kilometers. However, the area on the ground characterized by extremely high winds is rather well delimited in space. Tornado damage is strongly variable in space: buildings that are drastically affected can be located quite close (e.g., tens of meters) to others that sustained much less destruction. Therefore, in a process of assessment, one starts with the structures that indicate the strongest tornado effects in the analyzed location. Damage to buildings is usually highly informative with respect to the tornado event that caused them. Where no buildings are present, one may be able to focus on vegetation. To illustrate the application procedure, we will now complete a simple, practical assessment exercise. We start from the photograph shown in Fig. 6.1. The strongest effect we find there consists of hardwood trees that have their trunk snapped. According to Table 6.3, this corresponds to a DOD of 4 and thus to a wind speed between 150 and 216 km/h (with an expected wind speed of 177 km/h). We look up this interval in Table 6.2, and we see that it overlaps with most of the speed interval for EF2, which indicates that this is the most appropriate category. The expected speed value from Table 6.3, 177 km/h, lies at the boundary between EF1 and EF2, which does not contradict the initial assessment. We will thus assign a tornado category EF2. In practice, this assessment is not performed for one structure only, but rather for many structures for which such information is available. This can substantially narrow down the uncertainties involved by the imperfect match of the established interval for the selected DOD type with the interval for the tornado category. Moreover, based on the extent of damage in the tornado aftermath image, evaluators can also decide whether the estimated wind speed should be closer to the lower or to the upper bound specified for a given structure type.

6.2  Scale as Rank at Work

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Fig. 6.1  Tornado impact on vegetation: snapped trees. (Courtesy of USFWS)

Fig. 6.2  Estimating the tornado intensity on the EF scale, based on the damage to the analyzed structure and the corresponding degree of damage. The thick arrows reflect the steps followed in the example described for structure “27. Trees – hardwood.” The operations included in the accolade must be repeated for multiple structures in order to reach a decision on the EF category

To summarize the assessment process, we can distinguish (Fig. 6.2) (a) a first phase, in which a damaged structure is recognized and assigned to one of the 28 structure types; (b) a second phase, in which the damage to the structure is assessed and allocated to one of the DOD categories; and (c) a third phase, in which the wind speed interval indicated by the DOD category is used to find the proper tornado

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a) Environment Measurement & observation

Data (units). Map Abstract only Both abstract and concrete

b) Category (rank)

Data (units)

Environment Measurement & observation

Categorization

Fig. 6.3  Distinction concerning abstractness between scale as size and scale as ratio, on one hand, and scale as rank, on the other. Maps involve an intermediate phase spent in the abstract realm, while data (possibly associated with physical units) can be directly connected to the material environment. In contrast, the resulting category in scale as rank is purely abstract

intensity category. As mentioned above, phases (a) and (b) are repeatedly applied to various elements in the aftermath image. This evaluation process and its illustration in Fig. 6.3 may both suggest that we are applying a three-tiered scale as rank. However, this is not the case. Phase (a) is not a type of scale as rank: while different categories exist in this phase too, they are not ranked in relation to each other. The numbers assigned to the types of structure and the way they succeed each other, from 1 to 28, have no ordering meaning – they are arbitrary. In contrast, phase (b) and phase (c) are operations corresponding to scale as rank, in accordance with the requirements discussed so far. Therefore, the scale as rank we use in this case is two-tiered. While the assessment framework applies reliable criteria, which are able to constrain the answers at each step as objectively as possible, this scale still implies a degree of subjectivity. At a certain level – where the degree of damage is estimated for every category and then matched to an EF interval – some decisions with a subjectivity component must be made. The methodology is not capable of directly and objectively turning a set of diverse items (whether barns or towers or buildings of every other kind, each with its individual characteristics and a virtually infinite number of ways of being damaged) into a single, well-defined tornado intensity value. In spite of this unavoidable room for subjectivity, evaluation uncertainties are kept in check due to the hierarchic character of the estimation procedure, where at each level the selection space is strictly limited. Since categorization represents the engine that operates at the core of scale as rank, it is not surprising to see its strengths and its limitations being reflected in this type of scale. Focusing on categorization also provides insights into the functioning of scale as rank and into the meanings of its outcomes. It allows us to more reliably identify fake or incorrect forms of scale as rank and to avoid errors in its construction, use, and interpretation.

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6.3 Scale as Rank and Categorization 6.3.1 The Power of Categories Scale as rank can involve a variety of processes, but they all have a key feature in common: they are based on categorization. No surprise then to see that most of the general properties, the strengths, and the limitations associated with categorization can be found in scale as rank as well. A privileged starting point for the examination of the nature of scale as rank is thus offered by the theory of categories. If we liken the theory of categories to a tree, scale as rank only consists of one of its branches – one with its own ramifications and twigs. As we will see from this brief investigation, including practical examples, scale as rank also incorporates fibers found in the stem of categorization in general. However, it also exhibits important features that distinguish it from the other branches of the greater category tree. Categorization is generally understood as a process of assigning elements (material objects, processes, ideas, etc.) to a limited number of categories, using a set of rules. It is often based on similarity among objects, resulting from the adopted criteria. Categorization is profoundly involved in our endeavor of understanding the world. So deep is it rooted that we can recognize it even in the early stages of human learning. This author witnessed categorization at work when, in a park, a girl less than 2 years old saw two brothers, whom she didn’t know, approaching her baby carriage and standing next to each other. One was about 12, the other around 16. The girl pointed to the first one and said “Baby”; she then pointed to the other one and said “Man.” The genuine impulse to categorize was more surprising than the result of categorization itself, which might have been flattering for the older boy and no so much for the younger one. The vital role played by categorization in our interaction with the environment might look less surprising, but not less important, when we see that its application is not limited to humans: categorization and its role in learning were identified in the animal world as well, and this not only in primates (Aust 2017; Fugazza and Miklosi 2020). Categorization and classification are sometimes used interchangeably. While some authors see critically important differences between them, when those differences are described in detail, one can often find the same properties being assigned sometimes to one, sometimes to the other of these two notions. In this book, we will not distinguish categorization from classification, a choice made by many scholars, including Hjørland (2017), who draws a detailed picture of the long and highly ramified history of these concepts. Not surprisingly, this history starts with Plato, who mentions in his Statesman the process of assigning objects to categories of similar characteristics. More interestingly, in Phaedrus, Socrates recommends “to cut up each kind according to its species along its natural joints, and to try not to splinter any part, as a bad butcher might do” (Plato fourth century BC/1997). By referring to “cutting” and “butcher,” Plato is the first one to highlight with such clarity the contrast between the whole of

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reality and the forceful, artificial operation of fragmenting it for the purpose of object identification and classification. While burgeoning in Plato’s writings, categorization blooms due to Aristotle (fourth century BC/1984), who treats it in detail and with utmost rigor. In fact, with his work, in which the theory of categories (the first part of the Organon) plays a prominent role, Aristotle lays the foundations for a major framework of thinking. The shape of this groundbreaking endeavor has been recognizable in the work of scholars ever since, whether they were “with him or against him” (Jaspers 2003). The topic of categories is recognized as essential and addressed by many prominent thinkers, from René Descartes and John Locke all the way to Bertrand Russell and Karl Popper. It also enjoys an expanding interest among contemporary scholars in a growing number of fields, including mathematics, physics, and computer science, which have a fruitful relation to category theory (Marquis 2009). Categories were recognized as an invaluable guide in the process of our understanding the world, well beyond the fields of mathematics and the natural sciences. A well-known example is offered by the prominent Swiss art historian Heinrich Wölfflin (1915/1950), who established “fundamental categories” of artistic description. These categories were then applied in what he called “art history without names,” an approach to art that changed its focus from individual creators to the discovery of the essence of styles. This endeavor became possible when he discovered that the various modes of artistic expression can be captured in a small number of categories (Cassirer 1944/1992). Using this approach, Wölfflin identified “general structural patterns” that transcend historical phases, uncovering deep common features of artistic creations from different historical epochs. His approach was then absorbed into literature theory as well. The developments in the advancement of the theory of categories can be followed – a luminescent thread – moving through time and crossing disciplinary boundaries. In particular, Kant (1787/2007) had a profound impact on the way we think about categories. In his vision, without categories, there is no understanding of the environment; there are no concepts; there are no elements to be distinguished, named, and recognized in the environment; in this sense, there is no environment to talk about. As soon as we move past the very first representations due to perception, as soon as we identify an object, categories can already be identified as being fully involved. There is nothing we can call “object” that does not have thought imprinted on it – and thus categories inherently rooted in the concept. When we look around, if we move past the chaotic avalanche of sensorial input, if we begin to make sense of our environment, categories of thought are at work. There are major differences between the categories envisaged by Aristotle, along with the range of developments that are directly reflecting them, and more recently evolved ones that are dedicated to mathematical objects – as well as to an in-depth (re-)evaluation of the foundations of mathematics (Lambek 2017). In particular, as we will clearly see below, by discussing here elements of the theory of categories, we do not mean to explore the ample field of “category theory” in mathematics. The latter refers to general abstract structures and focuses on relations between objects and elements, rather than on the elements in sets (as set theory does). Nevertheless,

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“category theory,” as applied in mathematics and computer science, shares core features with the multitude of domains of categorization, broadly considered. They all imply a high level of abstraction, coupled with an extraordinary degree of generality. They operate with objects identified according to a certain “theory” (for our purpose, we can consider them here as being identified based on some consistently applied criteria), and they apply transformations involving these objects. The degree of generality of the theory of categories – and of scale as rank, as its offspring – should not be surprising, when category theory can be seen as “a template for any mathematical theory” (Mazur 2008). Quite remarkably, the vast degree of generality of the theory of categories does not undermine its precision and its practical applicability.

6.3.2 Categorization: Basic Principles In the broadest sense, classification (or categorization) includes two main construction phases and a third working (application) phase: (i) Establishing a set of ordered bins – “categories” or “classes,” ordered according to a chosen criterion. The categories are generated in abstract space, in a way expected to be appropriate for the elements of interest, as well as for the goal of the endeavor. This phase includes selecting the number of categories and setting up the boundaries that separate them. (ii) Creating a set of rules regarding the distribution of the elements of interest in the available bins. It is this phase that most clearly distinguishes various types of scale as rank from each other. This phase also has a critical impact on the scale’s degree of objectivity, as discussed below. (iii) Performing the actual transformation operation: passing the world through the assessment framework thus prepared and ending up with abstract signs in abstract bins. This phase culminates in a relation between the actual (possibly multifaceted and complex) process or object in reality and an abstract place assigned to it in the generated ordered category framework. In practice, one can often witness these three phases being applied together, in an amalgamated way, in which they can hardly be distinguished from each other. Such an approach can work when a well-tried-out scale as rank is applied to a well-known type of elements. In all other cases (and, to some extent, even in this case), it is useful to be aware both of the actual steps we are applying and of the reasons why those particular steps are chosen. The most obvious advantage of proceeding in this way is an enhanced capability of detecting and preventing possible errors in the classification process. There is, however, an additional benefit to such an analytical methodological clarity. By carefully looking at the different phases, by questioning the way we implement them, we can discover novel perspectives and thus new ways of implementing the categorization – or even of redesigning it. Stimulating methodological innovation is no small achievement in research, whether we are exploring

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new paths in partly uncharted territory or approaching familiar problems. Looking at things in a different way can be more than a spark in the dark – more than an isolated incident. It can be part of a general attitude towards problem-solving, kindled by fascination for the beauty and intricacy of reality. The three phase framework discussed above implicitly suggests that, in principle, the same elements can be categorized in more than one way. This is a potentially important property of the objects surrounding us in reality. It is directly opposed to the tendency of repeatedly applying the same categorization process, the same scale as rank, safely walking the old road and thus potentially missing new insights about “well-known” issues. However, whether old or new, an approach to the virtual assigning of elements to categories is governed by certain rules, which can be decisive for the outcome of categorization. Among the main conditions to be observed according to the so-called classical view of categorization, the following are particularly relevant for scale as rank: (a) Categories are discrete and they have clearly defined boundaries. (b) An element must belong to one, and only one, category. (c) All the elements in a category enjoy the same “status” or significance; regardless of their original condition, they all belong to the same extent to their category. In some cases, the categories can simply be defined based on numerical values assigned to bin boundaries, which separate objects characterized by a certain quantity. This is often the case, for instance, of scale as rank in the geosciences, as will be shown in this chapter. In other cases, the definition of categories is less simple; they may involve a series of tests to be applied to objects, along with an interpretation scheme, in order to unambiguously project the results of the tests into the appropriate categories. These differences will be illustrated below. The three phases and the three conditions mentioned above may be apparently disregarded when categorization occurs spontaneously, in our everyday interaction with the environment. And yet, if we pay close attention to the classification we instinctively perform, we can find that in most cases those phases are indeed included and the conditions are fulfilled – or mostly fulfilled. One way or another, our capability of applying categorization to our complex and diverse environment has fundamental implications – both positive and negative – for our understanding of reality, as discussed in the next section.

6.3.3 Positive and Negative Effects of Categorization Categorization is meant to be – and often is – highly effective. Carving out objects from their place in reality, and assigning to each of them appropriate and time-­ proven action, in ways that are clearly described and easy to implement, how can this not be helpful? Doing so can make the difference between being successful in a complex, often hostile, environment and succumbing to its challenges. Having to

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assess a continuous spectrum of reality, an infinity of nuances of phenomena, and having to incessantly find a proper way of acting and reacting in the world can take time – perhaps more time than might be available in a critical situation. Moreover, this might not lead to suitable results either, as the individual is expected to spontaneously discover the appropriate course of action while benefitting from a reduced experience. On the other hand, if the ready-made bins, together with the best ways of handling those situations, are established, accumulated, and conserved in time, some of them even refined and communicated from one generation to another, they become collective wisdom, they can be quickly applied, and they work effectively. There are thus two distinct types of effect that act synergically when categories are used: massively saving time due to the limited number of circumstances to be considered and having ready-made answers. The latter incorporate concentrated thought that was elaborated over time, by a diversity of individuals, gradually adjusted over time. On the other hand, as Plato states in the cited fragment, categorization may not be as innocent as it seems. One of the disturbing effects of our current way of exploring the environment is a deep, pervasive fragmentation of reality, which occurs in all areas of human activity. This approach can have a profound impact on our world – and its future, alarmingly emphasized and meticulously analyzed by David Bohm (1980). In his vision, the most obvious process of fragmentation begins with detecting an object and distinguishing it from everything else – which might work well, as long as we remember that the fragmentation is only a methodological step. It is this step that led us to the use of systems and of system-centered methodology, which proved unquestionably powerful. However, such an approach can be productive as long as we do not end up believing that the fragmentation, which we ubiquitously produce in concept, is an actual property of reality. When carefully considered, while the subject-object splitting sets fundamental limits to our approach to reality (Jaspers 2003), categorization has its own major contribution to the process of fragmentation: it takes it to a higher level, in a widely applied, generalized manner. Furthermore, having ready-made answers  – modes of interpretation or procedures to follow in our actions – means that errors in classification can be deleterious. This happens especially when we promptly assign situations or people to categories, employing ready-made answers, which represent preconceived ideas and lead straight to prejudice. One is thus challenged to distinguish between out-of-the-box meanings or action packages expected to be useful, on one hand, and those that need to be reviewed, drastically questioned, and reconstructed, on the other hand. This is particularly the case when it comes to ethical reasoning and ethical decisions. Quickly making such distinctions, while peer pressure exerts powerful effects on the individual, is not easy. But then, it might not even be possible for humans to move forward without overcoming certain limitations, without taking some vitally important steps, without truly achieving certain powers they do not currently possess. Cultivating the capacity to make such distinctions is probably one of them.

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6.3.4 Scale as Rank Versus Changes in Categorization When we generate a concept, the object seems to already be projected into a designated place in abstract space. While this image is simplistic, it might not be entirely useless. We must carefully make the distinction between the crystallization of concepts, along with the reflection of objects in concepts, on one hand, and the creation of categories and the allocation of objects to categories, on the other hand. This difference is easy to overlook (as we will show below), but ignoring it may hamper the path to understanding. With his background in mathematics, Edmund Husserl played a key role in a major process of questioning and redrawing some of the defining fibers of logic, which had not experienced renewal in a long time. He was the first one to so highlight clearly the distinction between categories of meanings, the outcome of our perception, and categories of objects, those that are perceived (Husserl 1913/2000). Most importantly, he thought that (i) both types of categories emerge from our act of thinking and that (ii) there is a correspondence between the two sets of categories, so much so that one can learn from one by relying on the other (Thomasson 2019). This may be considered a statement with heavy consequences. Succeeding in employing the two sets of categories, those of the “abstract” and those of the “concrete” elements, and safely moving from one to the other, is a valuable skill. It appears to bring certainty to the process of understanding reality. Such a perspective may even look too good to be true – it can raise suspicions. How is it possible to claim that there is such a solid and extensive relationship between “all” our meanings and “all” the objects in reality, allowing us to travel between the two realms – not over one bridge, but over any bridge of the innumerable ones leading from all points on one side to all points on the other? How do we know that the two realms are connected with bridges hanging everywhere, like spider webs filling the space between the two sides of a fracture? Sadly, this is not even what Husserl meant. If we look more closely, we notice that the objects he discusses are not “all” the objects that exist – or ever existed or will ever exist – in reality. True, he refers to categories of meanings and categories of objects. But the reference to “categories of objects” implies that the objects have already been categorized. If they are categorized, this means that they are recognized and thus conceptualized. In other words, categories of objects refer only to those objects for which meanings are already in place. Therefore, it is not inconceivable to have such extensive and reliable relations between the two realms, as they involve meanings of objects, on one hand, and the objects corresponding to those meanings, on the other hand. One may object that even so, the reasons for our reliance on the relations established between the two sets of categories – meanings and objects in reality – do not seem to be rock-solid. They imply at least the existence of stable and generally recognized structures of the categories. What happens if those structures change? In other words, in a more general sense, what are the consequences of change in the case of categorization?

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An approach to this question hinges on the answer to another question, which is often considered to be implicitly asked – and answered. It is safer, nevertheless, to bring it out in the open: When a categorization structure is given, is it at all possible for another one to exist? The question may pose difficulties if launched in a wider space of categorization, but it is faster to find the answer in the limited subspace in which scale as rank dwells. Examples that illustrate the affirmative answer are easy to find. Such an example is shown in Table 6.4, which presents side-by-side three implementations of scale as rank, all with the same purpose: to indicate tropical storm intensity. As noted in Table 6.4, the presented scales are region-specific and are applied by different agencies. To make the three scales comparable, the table shows all the wind speed values in the same units, kilometers per hour. Even the wind speed recording time interval lengths are different: according to the different agencies, Table 6.4  Comparative perspective of three different scales for tropical storm intensity (WMO 2017; Bloemendaal et  al. 2020; Padgett 2020)  – recording time conversions are presented for indication only Wind speed (km/h, 1-minute sustained) 252

60–69

Deep depression

70–100

Cyclonic storm 70–100

101–131

Severe 101–117 cyclonic storm 118–131 Very severe 132–153 cyclonic storm 154–177

Category 5 major hurricane

132–153 154–177 178–180 181–208 209–227 228–240 241–252 >252

National Hurricane Center Central Pacific Hurricane Center c India Meteorological Department d Bureau of Meteorology (Australia) e Fiji Meteorological Service a

b

IMDc Depression

Wind speed (km/h, 1-minute sustained)  10), but not the index values themselves. The next example used to emphasize the difference between scale as rank and pseudo-scale as rank refers to the assessment of one and the same phenomenon in two distinct ways, one of which corresponds to scale as rank, while the other does not. We will consider the size of seismic events, given on one hand as earthquake magnitude (in this example, using moment magnitude) and on the other hand as earthquake intensity (using the Modified Mercalli scale).

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Earthquake magnitude refers to the size of an earthquake from the point of view of the physical processes that were involved: the magnitude characterizes the event, and not the effect of the event on a given location. It is considered to be the most reliable way of referring to the size of an earthquake (Bormann et al. 2013). The seismic moment represents the work involved by the inelastic deformation due to the slip on a fault on a rupture area A, over an average distance D, when the shear modulus (characterizing the rock response to shear stress) is μ:

Mo = µ D A

(6.4)

Based on the seismic moment, the moment magnitude MW is (Bormann et al. 2013):



MW =

2 ( log M0 − 9.1) 3

(6.5)

Equations 6.4 and 6.5 are only presented here in order to show the way in which we obtain a value for earthquake magnitude, as a function of the physical quantities that are involved. As discussed above, this process of obtaining a number characterizing the event size does not involve scale as rank. It is also a type F situation, rather than IF. There is no categorization operation at work. Obviously, there is no reason for the resulting values to be integers either, not even due to rounding, as had been the case in the preceding example. It should be mentioned that magnitude uses a logarithmic scale – as we can see from its definition, it can be considered proportional to the logarithm of energy. In contrast, what one often calls earthquake intensity, determined on the Modified Mercalli (MM) scale, characterizes a seismic event from the point of view of its effects on humans and on human-built structures. Its value is thus location-­ dependent. Unlike earthquake magnitude, earthquake intensity established on the MM scale comprises a limited number of categories  – 12  – written as Roman numerals (which helps avoiding confusion between these two measures of earthquake size). Categories are assigned to perceived events based on a description of each of the sizes, from I to XII.  Table  6.8 shows several examples of such descriptions. In this case, we can see an actual scale as rank: the categorization phase is clearly visible; its directedness and the granularity and abstractness of the outcomes are evident. There are no physical meanings attached to the categories. We can also notice that the IF questions to be asked are explicitly stated in the description of each category. To conclude, we can see that quantification frameworks can at times be similar to scale as rank. Nevertheless, the distinction can confidently be made by considering the main characteristic features of scale as rank, starting with the categorization operation, which must rely on IF-type, not on F-type transformations, and continuing with its core properties.

6.6  Histograms at the Core of Scale as Rank

191

Table 6.8  Examples of descriptions regarding earthquake size categories according to the Modified Mercalli scale (Musson and Cecić 2012) MM intensity III

X

XII

Description Felt indoors by several, motion usually rapid vibration. Sometimes not recognized to be an earthquake at first. Duration estimated in some cases. Vibration like that due to passing of light, or lightly loaded trucks, or heavy trucks some distance away. Hanging objects may swing slightly. Movements may be appreciable on upper levels of tall structures. Rocked standing motor cars slightly Damage serious to dams, dykes, embankments. Severe to well-built wooden structures and bridges, some destroyed. Developed dangerous cracks in excellent brick walls. Destroyed most masonry and frame structures, also their foundations Damage total – practically all works of construction damaged greatly or destroyed

6.6 Histograms at the Core of Scale as Rank Up to this point, we have studied the transformation from one realm to another, drawing the arrows from individual elements in the environment into the proper bins in the abstract categorization framework. We will now shift our focus to the overall outcome of this operation, when it is applied to a whole set of elements that are categorized and represented on a scale as rank. Whether we represent such an outcome in graphical or numerical form, we end up with a histogram. This might not always be obvious, but it is almost always the case: the result of applying scale as rank is a histogram. Histograms are an important instrument, and this is true not only due to their role of scale as rank. Map representations often rely on histograms, even if the presence of scale as rank is not necessarily obvious in this case. The properties of histograms, including the ways in which they can lead to biased views, are also directly reflected in maps. One should thus be aware of the main features of histograms and of ways in which our decisions regarding scale as rank influence the picture of reality we wish to generate. We will present these main features below and address their implications for maps in the next section. A histogram is a way of seeing the distribution of a set of numerical values, by dividing them up in a number of intervals. This wording is meant to avoid the impression that a histogram actually represents the distribution of those values, which has a different mathematical meaning. In fact, even for one and the same set of values, a histogram can lead to distinct perspectives that are very different from each other, depending on the way in which the bins are designed. We will not go into the details here regarding the construction of a histogram: the histogram is a common instrument in a wide variety of fields and thus familiar for many; furthermore, the construction of the histogram has much in common with the process of categorization already discussed above  – it actually coincides with this process, except where the outcome of its application is concerned. More importantly, we will pay attention to the implications of category design for the overall outcome of categorization and for certain applications, including the production of maps.

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A series of recommendations are offered regarding the design of categories. The usefulness and the limitations and possible undesired effects of the various choices of class design are established by histogram users based on their experience. The strengths and weaknesses associated with those choices are often linked to the main areas of application. Any assessment in this regard must start from the question: “Why are we using a histogram?” The usual answer is that we are trying to better make sense of data that are numerous and diverse, and we want to go “below the surface.” By “the surface,” I mean a global characterization with the help of measures of central tendency (like the mean, the median, the mode, etc.) and measures of dispersion (like variance and various statistical moments). One way of diving deeper into the structure of the data consists of producing their distribution. Therefore, when we have a number of numerical values we wish to assess, creating histograms may help us grasp aspects that are not offered by their global assessment and might even be surprising. Histograms can help. However, as is the case with any instrument, the help they offer depends on the way in which they are applied. It is in this context that we will briefly review the most commonly mentioned methods for data bin design. (i) Creating equal intervals. Choosing categories of equal width is often perceived to be the simplest and also the sensible thing to do. It seems to involve the lowest degree of “perturbation” to the data and to represent an “objective” approach to them. This view is partially justified. Any departure from an equal interval splitting can be regarded with suspicion, as a way of manipulating the data: let us add here that suspicion regarding the manipulation of data, and in particular the use of histograms, is motivated by many real-life examples when this actually happens (this issue will be addressed further below). On the other hand, one must be aware that any categorization/any histogram creation is unavoidably a form of “perturbing” the data, which do not look the same anymore – nor will they ever look the same as before (aspects of irreversibility were mentioned above). The histogram is an instrument that works upon the data and changes them – the point is to make a proper use of this instrument. As to the issue of objectivity, no histogram creation method can guarantee it: all the methods enumerated here can support objectivity or undermine it, depending on the way in which they are applied. Last, but not least, choosing equal intervals is not the only element that can have a significant effect on the results, as discussed below. (ii) Molding the bins to include an equal number of elements in each category. In the language of statistics, these equal portions of data are quantiles. This is a choice that also looks objective and balanced and which may thus be seen as justified. The outcome of the categorization will obviously be very different from the first case above. Each category will have a different width. The graphical representation looks unusual to those used to equal interval frameworks. On the other hand, certain data processing methods are designed to be applied to data distributed in equal-size categories, rather than on histogram bins of varying widths. The method seems to assume that the time of data input has

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passed. Clearly, this is not always the case. When new data come in, they must be added to some category, and this will unavoidably – at least temporarily – change the existing bin structure, as the category framework will no longer include an equal number of elements in each bin. Alternatively, we must apply such a bin selection method when sufficient data have been collected, to ensure that their distribution is, at least to some approximation, accurately captured and that the subsequently added data will not produce a major change. (iii) Using the so-called natural breaks in the collected data. According to this method, categories are separated from each other by cuts performed for data values for which the number of elements is lower than for other values, in “valleys” of the data landscape with the peaks standing out in the middle of each category. This approach seems to follow the “natural” tendency of data to cluster, and the mere use of this phrase is enough to raise at least some suspicions in a scientist’s mind. Declaring data structures to be natural should only be acceptable when the statement can be backed by some objectively established measure (rather than by visually judging the data and ruling where the “natural” breaks should be). Of course, the method does not mean that there are zero elements where the cut is performed, so there will still be elements that are very similar to each other yet fall in distinct categories. However, the desired effect of this approach to categorization is to minimize the variation inside each of the categories and to enhance the contrast between each category and its neighbors. The method may be applicable with more success in some cases than in other ones, depending on how clearly the different clusters can be distinguished from each other. The disadvantages mentioned above in case (ii), where category widths differ from each other, are also at work in this case. (iv) Applying other “artificial” interventions. This implies that one has a good understanding of the data and knows that some breaks between categories or some other rules applied to establish the width of the bins are expected to work better than others. At first sight, this seems to be the most suspect method of all. Not being constrained by objective procedures, this approach seems to offer a lot of dangerous space to subjectively maneuver the data evaluation process, with outcomes condemned to have their value diminished by the embedded elements of subjectivity. However, this does not have to be the case. A thorough understanding of the data – coupled with a thorough study of the processes reflected in the data – may indeed lead to conclusions regarding an adequate histogram design, which is different from those listed above. An example refers to the evaluation of size distribution of objects for which the number of objects decreases with their size (the details are not essential here: this can be a “fat tail,” distribution like a power law, or another type of function). Such distributions are found in many cases  – for instance, for rock ­fragments produced by impact in laboratory experiments, in space in the collision of asteroids, etc. If we use an equal-sized bin distribution, we will see that the number of small fragments will follow the same pattern quite consistently; however, the large fragments, which are very few, will make the last bins (for the largest sizes) be more or less populated, with large fluctuations from one

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experiment to another. In this case, to avoid the problem of having a small number of elements in certain bins, and thus histogram instability, towards the tail, we can apply bins that grow from one category to each other – for instance, according to a power law. This case is not covered by methods (i) to (iii) mentioned above, but it can serve the purpose of a study without inserting artificial effects of subjectivity: the same law is used throughout the histogram generation process, in the absence of any subjective choice. When we described above several approaches to category design, we did not mention an aspect of histograms that is often significant: the number of categories to be used. There is no general rule that would specify the best number of bins or intervals. Textbooks mention sometimes “rules of thumb,” generally meant to keep the histogram clustered in a number of bins that are not too large (examples include making the number of bins equal to the square root of the number of elements, using less than 10 bins, using less than 20 bins, etc.). Without a theoretical justification, it is hard to accept such “rules,” and daily practice indeed proves that certain rules can work well in some cases, and not so well in others. An important reason for this situation is the fact that histograms can be applied to a wide variety of data distributions; the distributions can drastically differ from the point of view of the effect of bin number on the resulting histograms. We must again underline in this context that what we are discussing here is not the topic of distributions and distribution functions, which is a different issue altogether. In the case of scale as rank, we assign the data to frameworks usually consisting of a relatively small number of “wide” bins, which are related to distributions (they represent coarse views of those) but are a different kind of object. This is actually the reason why we discuss the problem of the number of bins to be selected – a problem that would not show up in the case of distributions. Even histograms produced with a similar number of categories can look different from each other and influence the interpretation in different ways. Figure 6.5 shows an example of a set of 50 values from a pseudorandom distribution, which is reflected in histograms produced with 16, 17, and 18 bins, respectively. As we can see, there are significant differences between them, while the bin sizes are almost the same in all cases. None of the three “truly” represents the dataset. There are authors who insist that a high sensitivity of the histogram with respect to the number of classes is the result of an incorrect choice of that number. According to that view, if by using different bin sizes the picture of the data varies as much as it does in Fig. 6.5, this means that the number of bins should be lower. However, this is not what one can see in research, neither in synthetic data produced according to various algorithms nor in many applications in Earth and environmental sciences. We can look at the data shown in Fig. 6.5 by applying a much lower number of classes. According to a rule of thumb mentioned above, the number of bins for 50 data should be around 7. Therefore, histograms using a number of 6, 7, and 8 classes, respectively, are presented in Fig. 6.6. As we can see, the three new histograms are also significantly different from each other. There is no “clustering” towards a stable form of the histogram when we

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Fig. 6.5  One and the same set of data seen through the lens of histograms that use a different number of classes each (16, 17, and 18 classes, respectively)

Fig. 6.6  The same set of data shown in Fig. 6.5, reflected this time in histograms with a much lower number of classes (6, 7, and 8 classes, respectively)

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decrease the number of classes. The challenges of interpretation we encountered by looking at the three histograms in Fig. 6.5 have not vanished: histogram dependence on the number of classes persists. The class number sensitivity of histograms can have significant implications for studies that include this approach to data processing and interpretation. Rigorously collected data, correctly analyzed using the appropriate tools, can lead to misleading results due to the “mere” category dependence of the histogram. In the example shown in Fig. 6.6, if the categories represented age group intervals and the number of cases (the column heights) indicated the number of individuals found to be affected by a certain infection, we would reach different conclusions based on such representations. The six-category histogram indicates that a certain age group (the fourth column) dominates in terms of infection cases. In contrast, the seven-­category histogram shows that two age groups stand out as being more affected than others (columns 4 and 7), whereas the eight-class histogram makes these two age groups stand out even more among all others. As a consequence, there are two main messages that emerge from such challenges. First, one must always tread carefully when histograms are used and check the variability of histogram representations as a function of the number of classes. Second, one must be aware of the uncertainties that are involved and address them. Having a large number of elements in each bin helps; however, increasing the number of elements per class is not always possible. Sometimes there are no additional data available. On the other hand, increasing element density per column by decreasing the number of classes is not always enough, as we could see in the examples above. Most importantly, depending on the data distribution and the goal of the study, applying well-chosen statistical tests can have a decisive positive effect concerning the uncertainties of the interpretation. A simple, yet elegant way of circumventing this problem consists of leaving the initial form of the histogram behind and producing a different kind of object: a cumulative histogram, i.e., representing the number of elements that has a value larger than each bin edge, instead of the number of elements between the edges of each bin. The same operation can be performed by taking cumulative sums from lower to higher values or from higher to lower values, thus producing left and right cumulative sums, which are chosen depending on the actual data and on our research objectives. This coarse data integration has a powerful smoothing effect: it squeezes the variability in column height, which is the source of difficulties such as those outlined above. However, while the method is useful in many practical applications, the outcome may not look like a scale as rank; furthermore, it may not support comprehension of the studied system image in the way in which scale as rank is defined and discussed here. On the other hand, cumulative histograms have their drawback: they smoothen the shape of the histogram so much that any structure in the data is almost completely erased. This may hide possibly useful properties of the data distribution, which might be visible in noncumulative histograms, such as relationships between the succession of peaks (Suteanu et al. 2000).

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6.7 Scale as Rank and Maps The problem of class number sensitivity can be highly relevant when the histogram that results from scale as rank is translated to maps. In this case, the limits of bins that separate the categories decide upon the classes to which various areas are assigned. Therefore, the way in which bin sizes (and thus also bin boundaries) are selected can have an even more outstanding impact on the resulting maps, than on histogram shapes. Instead of having columns shown side by side, with heights that differ depending on bin choice, we can see large changes in the map areas marked by one symbol or another. All the points covered above with respect to histogram classes are also relevant in this case, as the maps are directly reflecting their underlying histograms: the distinctive difference here is the fact that maps are often more sensitively dependent on changes in class number and class limits. The number of classes is a reflection of the property of granularity (Sect. 6.4.3). However, as we could see above, histogram sensitivity concerns more than the size of bins: changes in bin sizes also imply shifts in bin boundaries, with effects that are illustrated in Figs. 6.5 and 6.6. Therefore, we will explore here choices regarding granularity and interval size distribution, and subsequently we will see how intentional changes in bin boundaries can lie behind distorted images of reality based on maps. Figure 6.7 presents an example of a map based on a coarse-grained histogram that includes only three classes. The fourth category symbol does not count as an actual class, as it refers to missing data. The map shows the percentage of adult individuals found to suffer from hypertension in the state of Montana, USA. The widths of the bin intervals are different from each other: 11%, 3.2%, and 3.6%,

Fig. 6.7  Coarse-grained histogram with unequal bin sizes. Prevalence of hypertension among adults 18  years and older, in Montana, USA. (Behavioral Risk Factor Surveillance System, Montana Cardiovascular Health Program, August 2020, courtesy of CDC)

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respectively. In this case, the bin boundaries are justified by numerical values obtained in a different manner: they represent the average percentage of adults with hypertension in the whole state of Montana (29%) and in the USA (32.3%). This choice is meant to be an objective way of selecting class boundaries, which are otherwise difficult to establish. The first bin is thus meant to highlight all the counties in which the percentage of people with hypertension is equal to the state-wide average or lower. The second bin includes counties with hypertension percentages that are higher than the state average but lower than the national average. The lowest percentage value (the lower bin limit for the first bin) and the highest value (the upper bin limit for the last bin) are usually the lowest and the highest values, respectively, which were found in the study. Equal interval histograms are often applied, but the bins located right at the ends of the scale, or even those that are close to the ends, may represent exceptions from the interval size uniformity. An example is presented in Fig. 6.8, which shows lava flows from Mauna Loa in Hawaii, according to their age. Almost all intervals are 1000 years wide. Only the intervals that reflect the most recent and the most remote times have a different size. Choosing bin boundaries based on round numbers is quite common: this approach relies on our perception of round numbers as somehow being not only more meaningful but also easier to handle (Jain et al. 2020). Here, the round values encourage viewers to quickly figure out distances in time between the various flow events. The relatively small bin size offers a high temporal resolution view of processes extending over more than 20,000 years (with the limitation involved by the fact that newer flows cover older ones and thus events that occurred farther away in time can only be grasped if they have not been hidden by the presence of newer ones). In some cases, interval sizes change consistently from one class to another – for instance, according to a power law, as mentioned in Sect. 6.6. In other cases, a pattern for class size increase is followed only for a specific set of classes. An example of such a histogram implementation is shown in Fig. 6.9, which presents the geographical distribution of precipitation rates at a given moment in time. Scale as rank is represented by the set of categories corresponding to precipitation rate intervals: the first six intervals are of equal size, after which they grow faster. Along the second half of the scale, the growth follows, with some approximation, an interval doubling rule from one bin to the next: the widths change from 1 mm/hour to 2, then to 5, to 10, and to 30 mm/hour. From this and other images produced at various moments in time, one can conclude that the increased bin size for large precipitation rates is meant to enhance the visibility of such phenomena on the map: if the class width had been kept unchanged, the areal extent with high precipitation intensity would have been very small and in most cases hardly noticeable. If we had the means to shift the class boundaries, we could see how areas marked by certain colors quickly increase or decrease in size with changes in the bin limits, possibly producing an altogether different picture of the represented phenomena – in this case, precipitation rates. This is one of the reasons why it is good to remember that such representations are nothing but this – representations, not the actual phenomenon. None of the series of maps that could possibly be produced based on

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Fig. 6.8  Partially equal interval histogram representation: the age of surface lava flows (years before present) from Mauna Loa, Big Island of Hawaii. Departures from time interval uniformity occur towards the ends of the scale, as is often the case. (Courtesy of USGS)

one and the same precipitation information would be “the” real picture of our environment. They all capture aspects of the investigated phenomenon, seen in different ways. In some cases, the histogram to be transferred to the map is produced for a very large number of elements and also uses a very large number of classes. The number of classes can be so large that it may not even be recognizable as a discrete set of intervals anymore. The transition from one class to another along the scale looks unquestionably continuous. Examples include maps showing topography, soil moisture, atmospheric gas composition, land surface temperature, etc. An example of a histogram that applies a large number of intervals is shown in Fig. 6.10, representing a view of the topography of planet Mars. The graphical representation of the scale as rank does not present series of distinct bins. When such scales are applied, maps cease to support the expectation that one can choose a certain spot on the map,

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Fig. 6.9  Histogram using intervals of varying sizes: precipitation rate on November 21, 2021, 1:30 am UTC, based on the Global Precipitation Measurement satellite constellation. (Courtesy of NASA)

identify its pattern and color, compare those with the scale as rank, and assign them to one of the classes along the scale the exact value assigned to the corresponding interval is not determined. When one faces an enormous number of color shades, it is virtually impossible to find an exact equivalent of the one hue associated with a

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Fig. 6.10  Categorization using a large number of categories: false color topographic views of Mars. The vertical resolution is on the order of meters, and the number of categories is so large that their succession on the scale looks continuous. (Courtesy of NASA/JPL)

point of interest. One must rely on approximations. Nevertheless, it goes without saying that any coarse-grained scale as rank implies approximations in its turn, as it assigns one and the same interval to a whole range of actual values of the studied variable – which, in fact, is a key principle of scale as rank. Therefore, whether an interval-by-interval readable scale as rank with a relatively low number of classes should be used, or if a scale as rank comprising a high density of intervals, perceived as almost continuous, is preferable, is a decision to be made on a case-by-case basis. Interestingly, it is the coarser version of scale as rank, which is more frequently applied with the goal of distorting the resulting image offered by the map, as we will show in the next section.

6.8 Uncovering Map Manipulation As Sect. 6.6 has shown in detail, the construction methods involved in scale as rank and even the principles that lie at its core leave room for a range of different images to emerge from the same set of data. The main parameters that can be tuned are the number of classes (or, equivalently, the size of the classes) and the actual position of the class boundaries on the scale. The values of the boundaries between classes change unavoidably as soon as the number of classes and their size are modified;

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however, those boundary values can also be changed independently. As we could see in the examples shown above, a class boundary change can be performed, for instance, with the purpose of enhancing the clarity of the resulting image (Fig. 6.9) or of anchoring the classes to relevant and reliable values obtained through other independent processes (Fig. 6.7). The problem is that the same method of modifying class boundaries can also be applied for dishonest purposes. In principle, one can shift class boundaries around until the final picture looks closest to the goals of the manipulator. However, we can also uncover methods designed to distort the image produced by the map, which go beyond mere shifts in category boundaries. As is often the case, the most dangerous ones are the methods that still rely on the proper data without changing these and without committing any error in the processing of numbers, so that the authors of such operations could, in principle, still claim that theirs is a correct assessment. The best way to shed light on such a procedure is to present an example. The following example was inspired by an actual international state of the environment report from the past. The numerical values, however, are only meant to illustrate this approach to creating a distorted picture based on accurate data. The goal of the state of the environment analysis was to check to what extent the air pollution levels had changed compared to a baseline from 8 years before. It appears that after the first steps of the investigation, the desired conclusion was not supported by the data: the air pollution levels had not improved as much as one had hoped. On the contrary, the decision was thus made to present the data by district, referring to the total geographical area in which air pollution levels had changed in one direction or another. Three types of area were identified: one in which pollution levels had increased, which represented 26% of the area of the region; one in which pollution levels had basically stayed the same, which covered 61% of the area of the region; and finally, one in which the situation had actually improved, as some inefficient energy plants had been replaced with other plants located elsewhere, and the total geographical area where pollution levels had improved covered 13% of the region’s area. These results are represented in graphical form in Fig.  6.11a. They were not considered satisfactory, and thus a different way of representing them was chosen. The two categories “lower pollution level” and “pollution level unchanged” were amalgamated. Together, they now represented 74% of the region’s area. Based on this result, the agency that produced the report proudly declared that in 74% of the area, the pollution levels are “better or equal” compared to the chosen reference point in the past. Only in 26% of the area had pollution worsened. These results, represented in Fig. 6.11b, looked much closer to the desired outcome, and they were, in fact, accurate. No data were changed, hidden, left out, etc. What the new picture does not say, of course, is the extent to which the higher level of pollution is actually higher. Some emissions could be 100% or 500% higher than in the past, and the resulting image would still not change. We can also see the process from a geographical perspective and look at the way in which the change in interval boundaries  – not by shifting the limits between classes, but simply by amalgamating two categories – is reflected in the map of the

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a)

b)

Lower

Pollution level unchanged

Higher

13%

61%

26%

Pollution level lower or unchanged

Higher

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Fig. 6.11  Histogram outcome manipulation by class amalgamation. By shifting the view from (a), based on three categories, to (b), where two categories have been amalgamated, a different image is obtained. Although more than a quarter of the area is affected by a higher level of pollution and only over 13% of the area has the situation improved, in the new view, three quarters of it are said to enjoy a situation that is “better or the same.” The change in the histogram outcome was produced without distorting the data, and thus it can even be claimed that the image is “accurate”

region (Fig. 6.12). We can indeed notice that after the category amalgamation, only five districts are affected by pollution levels that are worse, while all the other districts enjoy the “same or better” status. These changes were made through mere amalgamation and can always be shown to rely on the real, undistorted data; of course, this does not mean that other means of “lying with maps” using scale as rank cannot be applied, starting with shifts in class boundaries. It is not straightforward to discover and expose manipulation based on the abovementioned methods or on other ones (since the imagination of fraudsters might always discover new paths that they can follow). One of the reasons for the difficulty involved in checking the data representation is the fact that, as mentioned above, one of the properties of histograms is its irreversibility. One cannot retrieve the initial data from datasets that are already categorized. Therefore, in order to perform a thorough verification, one must to obtain access to the initial data; for this reason, those who publish them must be compelled to make the initial data available. We are witnessing a positive trend in this regard: an increasing number of scholarly publications apply the requirement that the data involved in the analysis must also be published. On the other hand, as a simple rule, it is usually a healthy habit to become suspicious when results of studies include amalgamated categories, with labels such as “same or better” or an equivalent wording. Based on the considerations presented above, we can see that properly recognizing and applying scale as rank are not always straightforward, and it is easier to err here than when using the other two types of scale. The most common errors include employing a procedure that is applied to some quantitative value describing the element to directly find a value on the scale (an F-type transformation) and applying a categorization framework that is flat and not ranked.

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Fig. 6.12  The change in histogram boundaries presented in Fig. 6.11 reflected in the map of the region: (a) The initial image, corresponding to Fig. 6.11a, comprising three categories. (b) The new image, produced after the amalgamation of two categories. Now, most of the districts, with a total area of 74% of the area of the region, are assigned the label “same or better.” The new image suggests that an improvement can be seen in terms of air pollution levels, although this is not the case

Scale as rank can be correctly recognized as such if all the elements to be subject to the scale are assigned to one single category among those in the ranked set of categories.

6.9 Scale as Rank: Inside and Outside Scale as rank is not only very useful: akin to the other two types of scale, it is indispensable. As shown above, a series of principles and rules are in place, which can be used to safely and effectively apply scale as rank to real-world objects. The latter is achievable, as long as the elements to be transformed through the scale as rank are indeed transformable. Not every element can be successfully processed by a scale as rank. Moreover, there are no exact rules that would tell us which exact objects can or cannot be submitted for processing by a particular implementation of scale as rank. Therefore, it is our responsibility to decide and describe what objects are “inside” its realm and which ones must remain “outside.” There are multiple reasons why elements must be considered to belong “outside.” We have shown above that even objects that seem to be of the same kind as those routinely classified using a certain framework can have a negative impact on the whole categorization system. In other cases, objects can be unrankable because their properties locate them far away from the boundaries of the scale as rank that

References

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we want to consider. Such “outliers” are often not considered worthy enough to be the reason for a change of the categorization framework itself, but in other cases outliers may not be ignored; then, a different, properly adapted scale as rank must be designed. Finally, there are objects that are so different from those for which the scale as rank was designed, that they should never be brought “inside.” It is useful to keep in mind that certain elements might be unrankable. In fact, sometimes the reason why certain results of categorization and of scale as rank are perturbed is precisely the presence of elements that should not be assessed in that way. If the scale as rank machinery looks clogged, this might happen because it is actually clogged with elements that should have been kept out of it.

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McDonald J, Mehta KC (2006) A recommendation for an Enhanced Fujita Scale (EF-Scale). Wind Science and Engineering Research Center, Texas Tech University, Lubbock Met Office (2021) What are the National Severe Weather Warning Service Impact tables? https:// www.metoffice.gov.uk/weather/guides/severe-­weather-­advice Accessed 4 November 2021 Musson RM, Cecić I (2012) Intensity and intensity scales. In: Bormann P (ed.) New manual of seismological observatory practice 2 (NMSOP-2). Deutsches GeoForschungsZentrum GFZ, Potsdam, p 1-41. https://doi.org/10.2312/GFZ.NMSOP-­2_ch12 NOAA (2007) Enhanced Fujita tornado damage scale. https://www.spc.noaa.gov/efscale/ef-­scale. html accessed 30 October 2021 NOAA (2021a) The Enhanced Fujita Scale (EF Scale). https://www.spc.noaa.gov/efscale/ accessed 30 October 2021 NOAA (2021b) The Enhanced Fujita Scale  – trees: Hardwood https://www.spc.noaa.gov/efscale/28.html accessed 30 October 2021 Padgett G (2020) A comparison between the Australian Tropical Cyclone Intensity Scale and the Saffir/Simpson Hurricane Intensity Scale. In: Australian Severe Weather. https://www.australiasevereweather.com/cyclones/tropical_cyclone_intensity_scale.htm Accessed 28 October 2021 Plato (fourth century BC/1997) Phaedrus. In: Cooper JM (ed.) Plato: Complete works. Hackett Publishing Company, Indianapolis Stieb DM, Burnett RT, Smith-Doiron M, Brion O, Shin HH, Economou V (2008) A new multipollutant, no-threshold air quality health index based on short-term associations observed in daily time-series analyses. Journal of the Air & Waste Management Association 58(3):435-450. https://doi.org/10.3155/1047-­3289.58.3.435 Suteanu C, Zugravescu D, Munteanu F (2000) Fractal approach of structuring by fragmentation. Pure and Applied Geophysics, 157(4):539-557 Thomasson A (2019) Categories. In: Edward NZ (ed.)The Stanford encyclopedia of philosophy, Summer 2019 Edition. https://plato.stanford.edu/archives/sum2019/entries/categories/ Trieu J, Yao J, McLean KE, Stieb DM, Henderson SB (2020) Evaluating an Air Quality Health Index (AQHI) amendment for communities impacted by residential woodsmoke in British Columbia, Canada. Journal of the Air & Waste Management Association 70(10):1009-1021. https://doi.org/10.1080/10962247.2020.1797927 WMO (2017) Global guide to tropical cyclone forecasting. WMO-No 1194. World Meteorological Organization, Geneva. https://library.wmo.int/doc_num.php?explnum_id=5736. Accessed 25 February 2022 Wölfflin H (1915/1950) Principles of art history. Dover Publications, Mineola

Chapter 7

Scale, Patterns, and Fractals

Abstract  This chapter applies the concepts associated with scale to a broader context. It introduces fundamental concepts regarding invariance, patterns, and pattern analysis from a perspective centered on scale, presents the main types of symmetry, and highlights the role of symmetry in our exploration of the environment. In particular, scale symmetry is shown to have scale as ratio operating at its core. Given its relevance for our understanding of the natural environment, scale symmetry is further addressed in more details. Fractals and fractal dimension are introduced and defined. An effective method for fractal analysis is described step by step, and the key role of scale as ratio that lies at the core of the method is discussed. Practical aspects regarding the application of the method are described and explained. Scalebound and scale-free patterns are defined, explained, and illustrated with examples.

Keywords  Scale · Pattern · Invariants · Pattern analysis · Environment · Scale invariance · Scale symmetry · Fractal · Self-similar · Self-affine · Power law · Fractal dimension · Box-counting method · Scaling regimes · Multifractals

As we go along we see patterns and put theories together; a certain clarity comes and things get simpler. —Richard Feynman (1985)

Now that the streams of scale are better disentangled, it should be helpful to follow scale at work in a wider diversity of real-world circumstances. In particular, we will focus on applications that concern important areas of human-environment relations and advance our understanding of the environment. As we would expect, scale as size seems to dominate the landscape. Nevertheless, scale as ratio plays a central role in an impressive diversity of situations. Even if this may happen in the guise of scale as size, the true nature of scale as ratio can still be uncovered. Compared to the ubiquity of scale as size and scale as ratio, the occurrence of scale as rank in this context occupies a lower place, as its applications are clustered in more limited areas of human activity compared to those of the first two.

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7.1 Scale and Patterns 7.1.1 Scale and the Reality of Relationships Imagine a room with all its furniture piled up in the middle. Imagine then the same room with the furniture tastefully arranged in space, says John Deely (1990). “What a difference,” he adds. There is indeed a huge difference for us, for whom the scale of the room is the familiar scale in our environment. However, this would not be the case for all “points of view.” For instance, on a larger scale, on which the whole room would look like a dot, the change in the furniture arrangement would not make any difference. What about a nonspatial perspective? If the room was a wooden box prepared for an exhibition and carried by a crane, the mass hanging from the hook would be the same, regardless of the distribution of the objects inside. However, most rooms are not hanging from a crane. In most cases, we are interested in the spatial perspective, and Deely’s note is relevant: while in both types of arrangement the objects are the same, the difference consists of the relationships between objects. Most importantly, this difference is real: it is “physical,” which for Deely means “independent from human thought.” Relationships enjoy a status of full reality. And yet, we cannot act directly upon relationships: it is only by acting upon objects that we can change relationships, as real as they may be. We can make the distinction, however, between “being real” and “being relevant.” Among all possible relationships, spatial relationships are not only real but also particularly relevant to us. Ever since Descartes wrote his “inflammatory work, igniting geometry and algebra by the expedient of combining them” (Berlinski 2002), we have known that positions in space can be translated into sets of numbers representing spatial sizes. Undoubtedly, scale plays a role, and often an essential one, in relationships that are part of reality. In fact, scale can be essential in defining and characterizing relationships. On the other hand, as we will show below, all three topics in the chapter title – patterns, symmetry, and nonlinearity – are centered on relationships. We can thus state here already that scale plays a crucial role in relation to patterns and pattern analysis, various manifestations of symmetry, and the study of nonlinear processes.

7.1.2 Distinguishing Patterns Experience is the result of active exploration by the organism, of the search for regularities or invariants. —Karl Popper (1974)

The links between “scale” and “pattern” are deeply involved in our understanding of the environment. Not only does scale play an irreplaceable role in the recognition and analysis of patterns. For an entire class of patterns, one that is ubiquitously present in our environment, scale represents the core element, which, by unfolding, is capable of producing a wide variety of complex forms of manifestation, both in

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space and in time. Understanding scale and its role in natural and human-made patterns stimulates and supports our exploration of reality. Our visual system is remarkably capable of assessing patterns. We can recognize patterns, even if they are complex and strongly irregular. There is a striking discrepancy between our ability to identify patterns and pattern change and our capacity to make sense of what exactly we detect when we face patterns. While explaining how we recognize a pattern is still a matter of active research, it is difficult to even describe the pattern itself. How would we describe the image in Fig. 7.1 to someone who is not there? We would normally tend to recall in the other person’s mind memories of similar features, rather than providing an actual description. In this case, mentioning the keyword “clouds” would not be enough because there are so many different cloud patterns. We might try to awaken in other’s mind images of some other object, such as fluffy cotton. Thus, rather than depicting the pattern, we are more likely to evoke it, as the latter is easier to accomplish than the former. Figure 7.2 would also be difficult to describe. We would hope that the other person has already seen something similar or knows, at least, the concept of a braided river. These irregular undulations are easy to detect, even if they get fairly complicated; however, conveying the image to somebody else is another story. And yet, in spite of this embarrassing difficulty regarding pattern description, we are very quick when it comes to pattern recognition. What is it that we perform so effectively? What do we look for in a pattern? Answers to such questions are important when we try to uncover ways in which we understand certain aspects of the environment. They are also valuable when we want to characterize patterns in

Fig. 7.1  A cloud pattern that we can easily recognize, even if it does not rely on any actual “repetition”. (Courtesy of NOAA)

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Fig. 7.2  A river pattern: a natural color satellite image of the Yukon Delta, Alaska. (Courtesy of NASA)

science – rigorously, quantitatively. The answers are expectedly quite complex and imply highly ramified roots in different domains; this exploration is not meant to investigate them. There is, however, one specific question that we will address: among all the aspects of pattern processing, is there an outstanding one, which is decisive for a successful approach and which is so general that it can be found in the majority of cases of pattern assessment? Everyday experience often relates patterns to repetition. Repetition can indeed be involved in what we call a pattern, but is this helpful enough? Not only does repetition not have to be present in every pattern. Even when it is, the main problem to elucidate is this: What kind of repetition is involved in a certain pattern, which makes it a pattern? The mere fact that repetition is present does not offer us a reliable guide in the process of making sense of patterns – recognizing them, establishing similarities between them, distinguishing them from each other, and noticing pattern change. Figure 7.3a shows a simple pattern example, which was commonly used as a decorative border in ancient Greece, on garments, pottery, temples, etc. – a so-called meander. We can easily recognize it even if it is shown in another color (Fig. 7.3b) and even if the intersections of segments look different (Fig. 7.3c). Indeed, there is repetition in each of these cases, but what we detect as a pattern is not the mere

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Fig. 7.3  A simple pattern (a) and several modifications after which it is still easily recognizable (b, c) or not (d)

repetition. What the first three images in Fig. 7.3 have in common is a certain relation between the successive segments: vertical and horizontal. The angles in each instance of the repetition, in each loop, are always the same. What we detect here is a certain kind of invariance. Here, invariance concerns angles and segment lengths or rather ratios between segment lengths. The same repetition is present in Fig. 7.3d: however, because in this case some segment portions are missing, we cannot find the same relation between the segment lengths, and we may consider this one to be a different pattern. What we seem to look for is not repetition, but rather a certain rule (or set of rules) that is consistently applied and which we can recognize. In some cases, we can spell it out, which enables us to clearly describe the pattern in words or using the language of mathematics. In other cases, we can perceive the presence of a rule, without being able to express it in a transmissible form (as it happens in the case of clouds or of braided rivers). The fact that we cannot dissect a pattern, pointing to several features that define it, does not mean that we are not capable of “assimilating” it, getting thoroughly acquainted with it, and confidently recognizing it.

7.1.3 Defining Patterns What one actually sees is determined somehow by the abstraction of what is invariant from a set of variations in what is seen. —David Bohm (1965)

A meaningful step in this exploration consists of focusing on the definition of patterns and pattern analysis – not with the goal of reaching intellectual comfort, but of generating a safe path to move forward. The way in which we define these terms should ideally be accurate, appropriately comprehensive (encompassing all relevant aspects of patterns in the environment), and operationally useful. Such a definition of patterns is difficult to find. Most existing definitions are severely limited with

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respect to the variety of expression in patterns – insisting, for instance, on the idea that repetition is their main feature or in terms of the form in which patterns are found – referring, for example, almost exclusively to visual patterns and often to simple ones. Such definitions do not do justice to the diversity of patterns we can encounter, nor do they support actual pattern analysis. As we know from experience, patterns do not have to involve repetition – and definitely not simple or rigid repetition (Fig. 7.4). Nor do they have to be visually represented on a piece of paper or on a screen. We can recognize, unmistakably so, patterns of clouds in the sky, sound patterns produced by an animal, or patterns of behavior and possibly a change in such patterns, in the case of one person or another. We can identify patterns in an abstract realm, and not only in mathematics: we can refer to something as vague and subtle as patterns of thought. Many of these examples would be difficult to capture with definitions such as those mentioned above. Based on some of the ideas reached in the previous section, we will thus define a pattern as follows: Given a set of elements, a pattern is an ensemble of relations between these elements, which are associated with invariants. This definition includes the following important points: 1. The main characteristic of patterns is the existence of invariants. 2. The invariants refer to relations. 3. These relations are some of the relations that exist between elements in a set (see the selection problem and the guided cut in Chap. 9). According to this definition, all the examples mentioned above are naturally accepted as patterns; however, the definition also captures other forms of patterns, whether they are found in a digital photograph, in a real-world forest, in data regarding discharge fluctuations of a river, in a speech, in a piece of classical music, in a

Fig. 7.4  The same meander pattern from Fig. 7.3 can be recognized even if it is placed in a different position (a), when it is differently rotated and shown in other colors and against a different background (b), or when it is placed on a plane surface that is seen in a different perspective in space (c)

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mathematical construct, etc. Based on the definition of patterns, we can also define pattern analysis: Pattern analysis consists of the identification and characterization of invariants in a pattern. Both of these definitions aim to be operationally useful. According to them, pattern analysis focuses on invariants. For instance, finding a measure of central tendency and a measure of dispersion, such as the mean and the variance, does not represent pattern analysis. Those statistical measures can be useful ways of describing sets of data, and they can be incorporated in pattern analysis, but they are not pattern analysis per se. Only where invariants are pursued can we talk about pattern analysis. Data analysis is not the same with pattern analysis: it includes pattern analysis as a special subcategory of analysis. According to these definitions, patterns exist, whether or not somebody is able to detect them. Invariants are objective properties of the analyzed configuration. Some methods may be more capable than others to identify them or to capture some of their properties. It is possible to grasp patterns in sets of data that had formerly been extensively analyzed, without any trace of patterns being found: new approaches – new angles of attack – can uncover previously “hidden” patterns, sometimes with utmost clarity. This observation should cast doubt on definitions that consider the “recognition” of patterns by a human being to be a condition for patterns to exist. Nevertheless, humans have highly developed skills of recognizing invariants in visual information. The role of invariants in the process of understanding the environment has been explored for a long time. When George Berkeley (1710/2009) discusses our capacity of capturing a variety of individual shapes, all different from each other, and grasping them in one single concept, which comprises them all, one can recognize in his approach what today we would call the identification of invariants – or “the unity of the rule of change” (Cassirer 1929/2021). The elegant book of Peter Stevens (1979), which highlights striking similarities between patterns that are otherwise extremely different in terms of spatial scale, processes that are involved, etc., strongly, albeit implicitly, emphasizes invariance. More recent research established the pivotal role of invariant detection in visual information, as well as the remarkable capability of humans (and nonhumans too) to recognize objects in the environment in ways that are heavily relying on invariants. In his monograph on the special theory of relativity, David Bohm (1965/2006) dedicates a special section to the capability of the eye-brain system to detect invariants. More recent research (Ellis 2018) confirms these conclusions. It is therefore not surprising that patterns can be identified so well: in fact, we are proficient at picking patterns, although many of them are part of a complex environment. With his “constructal theory,” Adrian Bejan takes a leap forward by not only detecting invariants in a wide diversity of patterns, in the most different fields. He also identifies their key properties, and based on many examples, which are presented in the book by Bejan and Zane (2013), he shows how they can be explained by physics with the help of general optimization principles.

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7.1.4 Patterns and Scale In spite of the ongoing debate on the appropriate scale on which geographic processes should be analyzed, a widespread agreement exists that explanatory variables for a given phenomenon change as the scale of analysis changes. —Gibson et al. (2000)

The above definitions also highlight the critical role of scale in relation to patterns. Since the role of invariants is essential to the definition of patterns and invariants characterize relations between elements, the question is: What are the “elements” to be considered? Deciding what the elements are must be paramount to the search for invariants manifested in their relations. In mathematics, the elements involved are usually unambiguously defined. In the surrounding reality, in the environment, this is rarely the case. There are no “elements” out there, much like there is no “system” either. As discussed in Chap. 1, systems are defined by us, the observers, and we can do that in many different ways: the choice depends on our actual question and, to some extent, on our chosen approach (which may be subsequently adapted, in turn, to the system selection). Similarly, the elements in a pattern are the product of a selection process. This time, however, the choice is not as explicitly performed as it was in the case of the system. Our perception, strongly rooted in our experience and background and also related to our goals and expectations, seems to confidently and promptly decide what the elements should be. In the case of the pattern in Fig. 7.3, the elements are the straight segments that compose the broken line: it is in the relations between these elements that invariants can be identified. Their relative positions make it possible for us to detect the pattern, regardless of its color, regardless of its size, whether it is carved in stone, weaved in garments, or painted on pottery. If this basic pattern were to be used in a larger pattern, consisting of spirals with several arms, in which each arm is a thick line made of a meander pattern, we might consider the “elements” to be those larger pieces, the spiral arms. The way we decide what the elements are depends on scale. The fact that the outcome of the system observation depends on scale is well known, but now we can also see it anchored in the definition of patterns. An example is shown in Fig. 7.5: we can notice different patterns on this photograph, some of them more clearly than others. The dominant pattern is the one of the meandering Colorado River: it involves shapes that are difficult to describe in words (they are difficult to describe mathematically too, for that matter). Nevertheless, we would always be able to detect this pattern in other images – even in those of other rivers. On the other hand, careful observation can reveal differences in the river pattern along its different segments. On the other hand, if we zoom into the image in Fig. 7.5, we can suddenly see new aspects of the pattern (Fig. 7.6): a tributary of the Colorado River, which joins it along the Lake Canyon, can be seen at a level of detail that reveals its own meandering pattern. The shape of the small tributaries suddenly gets new relevance, as does the scale of observation. In other words, the elements that we use in order to look for patterns are different now: they consist of curves and arcs that are tiny

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Fig. 7.5  A variety of distinct patterns can be distinguished on this image of the Colorado River. The arrow and the white rectangle indicate the area in the image shown in Fig. 7.6. (Courtesy of NASA)

compared to the initial ones. This image illustrates a sharp difference between artificial (human-designed) patterns and those found in the natural environment. We cannot benefit from clearly separable segments, as we could in the case of Fig. 7.3. The so-called elements are not even explicitly “separable” from each other – we cannot, and we do not try to imagine cutting the curves into distinct pieces; and yet, we can always tell at what scale we are looking at a pattern. In the same spirit, pattern analysis methods can rigorously take in consideration scale, without imposing a fragmentation process on the analyzed system. If we consider our visual analysis of a pattern in more realistic terms, we realize that even if we pay special attention to a certain spatial scale, we do not focus on that scale alone. When we look at a photograph like the one in Fig. 7.5, we are struck, first of all, by the main river – the dominating feature that crosses the image from top to bottom – but then we look at smaller parts of the image, only to return to the larger picture from time to time. This approach to interpretation was theorized almost two centuries ago by Friedrich Schleiermacher, who  – in very different

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Fig. 7.6  An enlarged fragment of the image shown in Fig. 7.5. Here, we can distinguish a river pattern that was hardly noticeable in the initial image. Although we can see now something new, the meandering shape of this tributary river matches the pattern of the larger river, the Colorado river, itself. (Courtesy of NASA)

circumstances  – noted that in our endeavor of understanding, we keep moving between the smaller parts of the picture and the larger ones, or “the whole,” producing temporary and provisional versions of comprehension throughout this process. It is noteworthy to see that a whole generation of pattern analysis methods is applying this approach of moving across scales, as we will show below. From images like the one in Fig. 7.6, we can see that when the scale (as ratio) decreases, new details pop up, as expected, but also that the pattern can be and is established based on other elements than those initially considered. More interestingly, we notice that by looking at a smaller scale and by relying on spatially much smaller entities, we may uncover a pattern that is the same with – or very similar to – the pattern that was initially established.

7.1.5 Patterns and Scale Change In this context, it is natural to ask whether or not (and in what way, if that is the case) scale change has an impact on invariants. It would be best to address this question in terms of spatial configurations. Clearly, if the patterns have a geometric nature, scale change could affect them. It is useful to remember that while they are associated with objects, patterns are not objects themselves. In fact, when scientific literature uses the word “object” in the context of pattern analysis, it usually refers to the “pattern” instead: it is either a mathematical object, which overlaps with the pattern itself, or a concrete physical object that is the support of the pattern.

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If what we are interested in is the pattern, and not the object that carries it, then we can recognize it, whether it is large and traced in the sand, or much smaller, cut into a silver necklace. The size should not matter. A change in scale should not affect the relations between the sizes that are involved, nor should it distort angles (we cannot say the same about affine transformations in general, which do not guarantee that patterns are not changed – for instance, they do not preserve angles). It may thus appear that under these circumstances, a mere change in scale does not involve a pattern change. However, when we think about the material environment and we relate the pattern question to the scale of observation, we see immediately that scale is not pattern-neutral. Considered on certain scale ranges, some patterns may be differently identifiable than on others, and there are scale ranges on which they are not identifiable at all. A symbolic example is shown in Fig. 7.7. The same pattern of the meander from Fig. 7.3 is shown here after two extreme scale changes: towards the very large and towards the extremely small. Both cases can make it difficult or impossible for the observer to properly detect the invariance properties and thus the pattern. In terms of time, we can think of a process that occurs over a time interval that is very long, compared to the available time of observation: this may make it impossible for researchers to detect patterns, or, if patterns are presumed to exist, based on theoretical reasons, it would be difficult to confirm them with the desired level of certainty. At the other end of the scale change spectrum, the pattern can occur on temporal scales that are so small, that they might not be, in practice, accessible to researchers. Scale change can remove a pattern from our scale window of observability. Much like our vision covers only a limited range of the electromagnetic wavelength spectrum, our capacity of discerning patterns has limits in space, as well as in time. On the other hand, as we can expand our visual spectrum with the help of instruments, there are means that can help us to extend the range of scale of analysis in space, as well as in time. The possibility of scale extension for the observation and analysis of patterns can be particularly important, since patterns are typically manifested on

Fig. 7.7  Scale change can make the pattern indistinguishable: a symbolic representation of the scale problem, using the pattern from Fig. 7.3: (a) pattern greatly enlarged, compared to the size of the observer (standing at the top of the first shape); (b) pattern greatly decreased in size (located in the center of the circle)

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a very wide range of scales: potentially meaningful results may only be accessible if the proper scale extension is applied. A traditional view on the relation between scale and patterns insists on the major differences between our views on distinct scales of observation: from particles at nanoscale to the tip of a needle, a raindrop hanging on a leaf, etc. all the way to photographs of the Earth and image representations of the Solar System, of the Milky Way Galaxy, and of galaxy clusters. The feeling that scale change equals pattern change has dominated popular literature but has also often been applied in scientific research. To some extent, the idea that different patterns can be observed at different scales is certainly true. However, it is not generally valid. What is missing from this explosively diverse series of scale-by-scale pictures is another relevant property of natural patterns: their invariance with respect to scale (Lovejoy 2019). Numerous natural and human-generated patterns are scale invariant. Scale invariance of patterns is not a mere curiosity of nature. It can be important for many reasons. First, since it is an outcome of the processes that produced the pattern, it can offer theoretically and practically valuable insights into the mechanisms at work. Second, it reveals potentially important properties of the systems in question, especially regarding their interaction with other systems, with their surroundings. Third, knowing about such invariance can effectively guide the generation and the application of effective tools and methods, which support our further understanding of the studied system. It follows that even from this simplified point of view, we can distinguish patterns using scale change: certain patterns remain unchanged over a range of scales, while others do not. It is worth noting that even the patterns that enjoy scale invariance only have this property on a limited range of scales. In their case, pattern analysis usually focuses not only on the characterization of the various manifestations of scale invariance but also on the scale limits between which the established characteristics are valid. These aspects of patterns will be further explored below.

7.2 Scale, Symmetry, and Pattern Analysis 7.2.1 Symmetry: Old and New There was no debate between Parmenides and Heraclitus. Instead, we created such a debate – virtually: we pitted the worldviews of giants against each other. The towering figures of Parmenides and Heraclitus, who lived approximately at the same time (sixth–fifth century BC) at the Eastern and Western ends of the ancient Greek world, respectively, have probably never met. There was no back and forth sending and receiving of arguments and answers, as it had been the case (somewhat indirectly) with Leibniz and Newton. It is often stated that the two Greek philosophers had opposite positions on the subject of change. Parmenides affirms that no change can actually occur: any change that we think that we witness is an illusion.

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Meanwhile, Heraclitus apparently only sees change in the world, and steadiness is an illusion – flames seem to have the shape of objects, although they are made of movement only. There are reasons for this interpretation. However, Gadamer (2003) warned against such an oversimplification, especially regarding the views of Heraclitus, to whom he dedicated deep and extensive inquiries. Heraclitus emphasizes, in fact, the idea that the diversity of manifestations in the world is deceptive. In truth  – he claims  – they are all the same. A significant number of his sentences, such as “changing, it is at rest,” “the way up and the way down are one and the same,” etc., point to a conception in which all the different phenomena of change are, in fact, a manifestation of an underlying unity. Underneath all the noise, there is deep, all-­ encompassing symmetry. Considering the degree of generality implied by this view, one could argue that no thinker in pre-modern times has ever been so close to our current understanding of symmetry in physics. The ancient Greeks did not have the same concept of symmetry as we do today. Symmetry had two dominating meanings, one applied in mathematics, “having a common measure,” i.e., “commensuration,” and one used for the real-world objects, i.e., “well proportioned” (Hon and Goldstein 2008). However, had they meant by “symmetry” the same thing we do today, they would have probably recognized it in the thought of Heraclitus. If Parmenides and Heraclitus had met, if they had walked together along the seashore, talking about change, they might have agreed with each other. It could have been a kind of discussion that Gadamer calls “successful”: one in which the disputants may start from conflicting viewpoints, but in their dialog they move farther and farther together into new insights, and they do not even remember what had caused the initial divergence. Parmenides at the level of the visible and Heraclitus at the level of the deepest roots of reality both point towards an all-encompassing invariance. These views do not clash with today’s understanding of the world provided by physics. Richard Feynman is profoundly intrigued by the lack of answer to the question “why are all the theories of physics so similar in their structure?”. He suggests three possibilities (Feynman 1985): (i) “the limited imagination of physicists” makes them think repeatedly in the same way; (ii) “nature has only one way of doing things, and She repeats her story from time to time”; and (iii) “things look similar because they are aspects of the same thing”: there is one broad underlying picture “from which things can be broken into parts that look different, like fingers on the same hand.” It is interesting to see a physicist like Feynman suggesting that we may be projecting our way of understanding on the structure of reality – and to such an extent that our scientific results may be dressed in the cloak of invariance. One would perhaps be allowed to presume that this first suggestion was not Feynman’s favorite. As to his second and third suggestions, they are not only compatible with each other: they might even be seen as part of one and the same view of reality. Here, invariance in general and scale invariance in particular would be expected to play prominent roles in the material environment. As, in fact according to everything we know, they do. What does this tell us about the nature of reality? Explicit answers to this question are difficult to find. But this does not mean that they will not emerge,

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sooner or later, in a myriad of small waves of understanding, punctuated from time to time by giant waves of insight. We may witness significant revelations based on – and involving – scale. Feynman’s second and third suggested answers are surprising, so much so that we would almost feel tempted to check the cover of the book and see if we were reading Feynman or Spinoza. They also vividly remind us of Heraclitus’ view. In fact, in spite of major differences in many respects, one can notice certain common shades, sometimes subtle, but still distinguishable, in ways of thinking we can identify across space and time. From the views of Heraclitus (Gadamer 2003) and of thinkers in ancient China and India (e.g., Wilhelm and Wilhelm 1956/1995; Radhakrishnan 1948/2009) to the wholeness in African thought (Anyanwu 1987) to more recent thinkers and from the last three centuries to insights from physicists like David Bohm (1980) with his concept of unfolding, a landscape of lights seems to emerge, like the one seen on our rotating planet from the International Space Station. It suggests that if we traveled on an International Space-Time Station instead, we could actually identify uniquely distinctive lights, with specific spectra, across times and continents. Nowadays, in mathematics, invariance refers to the property of (mathematical) objects to stay unchanged when they suffer a transformation, i.e., when an operation is applied to them. In a general sense, even an operation of counting reveals invariance, since it produces the same outcome when it is applied to a set of elements, regardless of the order in which the elements are counted. Invariance is tightly related to symmetry, which is defined as a property of an object to be invariant under some transformation. Mathematically, it is treated using the elegant framework of group theory (Rosen’s 1983 book is still one of the most friendly and productive introductions to this topic). In spite of the apparent abyss which, in many ways, separates the nature of their worlds, physics and mathematics have similar views on symmetry. In fact, symmetry represents one of those deep and mysterious bridges, which connect the two fields. In physics, symmetry plays a tremendous role. It is not uncommon to state the idea that “today we realize that symmetry principles […] dictate the form of the laws of nature” (D. Gross, cited in Schwichtenberg 2018). Symmetry is studied on all scales, from the very small to the very large. As Weyl (1952) proves in his magnificent book on this subject, our world is literally permeated by aspects of symmetry. If there were other completely different worlds out there, maybe somebody could refer to ours by saying: “you know, the one with the symmetries!”. There is no surprise then to find that symmetry is essential to our understanding of the world as we know it. The fact that transformations that occur around us keep certain aspects of reality unchanged offers us invaluable opportunities to discover links between otherwise apparently disparate manifestations of nature. One should still keep in mind that there is a major difference between symmetry that characterizes physical laws, on one hand, and symmetry corresponding to a configuration of objects, as in an experimental arrangement, on the other hand (Zee 2008).

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Moreover, symmetry is not only ubiquitously present in the physical world. Incorporating a focus on symmetry in our approach to reality can help us in many ways, even with respect to key challenges in physics (Sundermeyer 2014), such as the unification of existing theories, guidance in the exploration of new theories, the simplification of views and formalisms, and – as expected – the comprehension of the laws and principles that govern the physical world. The most familiar types of symmetry include: (i) Translation symmetry: we can imagine, for example, a long series of identical rail wagons being moved, one wagon at a time, or a straight smooth wire that slides along its axis by any distance; the first example refers to discrete, the second one to continuous symmetry. (ii) Rotation symmetry, as in the case of an equilateral triangle, which is rotated by increments of 120°, or a circle, which can be rotated by any angle; again, we see discrete symmetry in the first case and continuous symmetry in the second one. (iii) Reflection symmetry, the familiar mirror symmetry: a butterfly is the most common image example, but we can also think of typical creations of classical architecture (when we consider such examples, we should keep in mind that, unlike mathematical objects, physical objects do not have strict geometric shapes and therefore they are never truly symmetric). (iv) Helical symmetry, which, among familiar objects, is best incorporated by a spring or a screw: the transformation consists of intercorrelated rotation and translation in three-dimensional space. (v) Scale symmetry, which involves invariance with respect to scale change. Various other kinds of symmetry can be obtained through combinations of these types of symmetry. Among all of them, it is this last type of symmetry – scale symmetry – that is the most important one for us here. Before we leave the other types of symmetry behind, let us notice that in each and every case, symmetry refers to relationships between the elements that are considered, starting from their sizes and positions.

7.2.2 Scale Symmetry We often notice invariants in natural patterns, even unconsciously. An example is shown in Fig. 7.8. The most obvious one here is the angle involved in bifurcations. Other invariants may also be used when we tend to recognize a certain tree species, knowingly or unknowingly. For instance, we may take into consideration the ratios of distances between successive bifurcation points and the angles involved in undulations of the branch segments. Remarkably though, these invariants can be noticed across different scales: the same kind of bifurcation can be identified for large branches and for smaller and smaller branches and twigs. In contrast, the body of the crow sitting on a branch does not involve such scale invariance – not in its external shape.

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Fig. 7.8  A natural pattern where we can notice invariants, in spite of the large number of elements and of their pronounced irregularity

What we can see here is invariance with respect to changes of scale or scale invariance. As was the case with various aspects of scale discussed in previous chapters, scale invariance can be found both in space and in time. In fact, a range of different meanings are more or less implicitly assigned to scale invariance (Sornette 2006): it can refer to spatial shapes, to certain physical quantities, to equations, to probability distributions, etc. Many of these different manifestations of scale invariance reveal interesting properties of the studied aspects of the environment. They point to significant roles played by scale, both in the functioning of real-world systems and in our ways of conducting our inquiries. With scale invariant patterns, we suddenly find ourselves in a novel and rather awkward situation. When we make a change  – concerning nothing less than the scale of observation – the observed reality stays unchanged. We zoom in. We move in deeper and deeper. And yet, the observed structure does not change. When we ponder this journey in scale space, we may realize how peculiar this phenomenon is. In a scale invariant environment, if we keep zooming in, it is as if we were advancing further and further, despite remaining in the same spot. Normally, the environment changes as a function of our movement; we get to see different objects from different perspectives, while we pass by them. But with scale invariance, the environment does not appear to move. This type of environment is immune to scale change. When we advance without moving, what happens is quite special. It is not at all as if we were walking towards the horizon on an endless, flat, plain. In that case, we

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would expect to notice no change for a very long time. But we do not refer here to the scale invariance of geometric objects, such as the straight line or the geometrical plane. In a scale invariant environment, there is an abundancy of elements, which we can see quite well: however, we are unable to “grasp” any of them. This may sound more like a strange dream than a mental experiment involving mathematical objects. And yet, this is exactly what happens in a perfectly scale invariant environment. Of course, a perfectly scale invariant environment does not exist. However, experiencing perfect scale invariance (albeit only virtually) is useful, at least for two reasons. First, it reveals with enhanced clarity the ways in which scale is involved in such structures. Second, instances in which the environment is almost perfectly scale invariant truly exist; in fact, not only do they exist: they are profusely present all around us. Our understanding of such instances, and of the modes in which they react to our various exploration tools, can be a decisive aspect of our understanding of reality. In brief, the way in which scale is involved in objects that are scale invariant is both unusual and important: it makes scale transparent. Every time when we discover that a phenomenon is immune to scale, whether spatial or temporal, we learn about reality. It is often the case that we learn something important about things that matter. In fact, one of the paradoxes of scale stems from the fact that being made irrelevant enhances its importance. It is thus a revealing experience when we cover a part of the environment with a “net” of scales to see it melting, vanishing in front of our eyes. When this happens, we know that we have made a significant step in our inquiry concerning reality.

7.2.3 Fractals The fact that various aspects of reality display a strange behavior when confronted with scale invariance was not revealed in a single explosion of insight. This was noticed by scholars, over time, in numerous instances. There were, however, some steps that Benoit Mandelbrot took, which turned out to be remarkable; in fact, the most remarkable feature of his work was arguably the fact that he took all the steps. In a simplified metaphor, we could first say that he noticed outstanding manifestations of reality in circumstances and fields that were painfully diverse for more typical scholars, especially his mathematician colleagues. He could safely navigate diversity by focusing on invariants – his Ariadne’s thread on this journey. Second, he took those unusual but consistent expressions of reality seriously. Rather than dismissing them to the museum of anomalies, he engaged in an in-depth inquiry, which many scholars would have hesitated to dedicate to topics that seemed so elusive. Third, and this is a very long third step, he distinguished in mathematical space a range of features that were attached by invisible strong ties to the physical world; he worked upon them, weaving a theoretical framework of outstanding clarity and beauty. There is no exaggeration in saying that whole new worlds began to open up

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after he pointed the way – and not just in one field, as it often happens with great discoveries. They emerged in physics, Earth sciences, biology, medicine, finances, engineering, and many other areas, in which humans investigate reality. After putting forward a definition for fractals based on mathematically precise concepts (Mandelbrot 1982), he found that definition to be too restrictive and proposed a surprisingly nontechnical one: “A fractal is a shape made of parts similar to the whole in some way” (Mandelbrot 1986). The generality of this latter definition makes it applicable to the unending series of practical applications in which fractals are identified, analyzed, and often fruitfully involved in broader investigations. It would be right to say that there is a special flavor associated with Mandelbrot’s books, starting with the earliest one, in French (Mandelbrot 1975), and its modified and amplified English language successor (Mandelbrot 1977). They are a blend of insight, precision, and revelation of endless capabilities of thought. The vision shared by the author would be hard to find anywhere else. With his groundbreaking books, Mandelbrot not only introduced a concept: he offered a view that looked so fresh and compelling that one couldn’t help feeling that an unexpected sixth sense had been suddenly developed. Certain aspects of the world seemed to be experienced for the very first time. Avalanches of discoveries followed, and they concerned a wide variety of scaling properties, which were identified at every step. New “objects” kept popping up, being recognized, analyzed, and characterized, where only wild and irregular features in need of drastic simplification had previously been seen. The scientific literature on the topic of fractals and scale invariance grew at a pace that can rarely be found in the unfolding of a new scholarly domain. The fact that Mandelbrot named such objects using a Latin root (fractus), which means “broken,” was significant. With this name, he pointed towards the presence of intricacies in the details which involve scale invariance properties. This was thus not about scale invariance alone. After all, a straight line is also scale invariant: it is unlikely that this was ever perceived as a mind-boggling discovery. Fractals are different. Mandelbrot insisted the existence of rich structures, of details, which are involved in scale invariant properties. According to his “rough definition” (Mandelbrot 1977), a fractal refers to “a mathematical set or a concrete object whose form is extremely irregular and/or fragmented at all scales.” The distinction between mathematical and concrete objects is explicitly made by Mandelbrot here; had this distinction always been carefully considered by those who got involved in fractal analysis and those who commented on it from a distance, a multitude of confusions and contradictions could have been avoided. Nowadays, we can even witness the persistent existence of a third category of situations which claim to be related to fractals. Consequently, we can distinguish three broad categories of features that are most often referred to as “fractals”: (i) mathematical fractals, (ii) real-world structures with fractal properties, and (iii) fractal-resembling features. The first category consists of mathematical fractals, i.e., mathematical objects, the properties of which are determined mathematically. Rigorously speaking, these are the only “true” fractals: they can be identified and characterized without the

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need of relaxing the defining conditions. An example of a representation of a mathematical fractal is shown in Fig. 7.9a: the Koch curve. Indeed, for a mathematical fractal, all we can show is an imperfect representation: the mathematical object itself can only be defined in theory. The curve in Fig. 7.9a is obtained by repeatedly applying one and the same process, starting from a straight line segment: the middle third of the segment is removed and replaced with two sides of an equilateral triangle; this operation is then applied iteratively to each of the resulting segments. The

Fig. 7.9  Examples of fractal objects: (a) the Koch curve; (b) a pattern obtained by applying the model of diffusion-limited aggregation (DLA). From the smallest scale to the scale of the whole object, parts are identical to each other in case (a) and similar to each other in case (b)

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pattern in Fig.  7.9a was obtained after five iterations. The actual fractal object requires the number of iterations to be infinite: it is not possible to draw it. However, the representation, even after a few iterations, can look so close to the actual object, that subsequent iterations would not change it to a noticeable extent. Scale as ratio is the key factor in the construction of such a mathematical fractal. Its prominent role will be recognized over and over again in this chapter. In fact, once the shaping algorithm is defined and applied for the first time, all that we do is change the scale (again, scale as ratio) and reapply the algorithm. It is due to the change in scale as ratio that the constructed fractal object comes to life. The pattern in Fig. 7.9b is different. It was obtained by using the model of diffusion-limited aggregation (Falconer 2003): virtual “particles” move randomly until they meet either the initial (central) point considered to be the “seed” of the structure or particles that are already attached to it; when this happens, the wandering particles can attach themselves to the encountered structure, with a certain probability. The results show that laws that are strictly local give rise to long-range properties, which are reflected in scale invariant patterns: details on many scales are similar (albeit never identical) to each other. The second category consists of what Mandelbrot calls the “concrete objects” which are real-world structures with fractal properties. Mandelbrot calls some of those, such as coastlines, “approximate fractal curves” (Mandelbrot 1977). Such fractals can be identified through an evaluation process based on pattern analysis methods. As we will see below, while these objects never behave like mathematical objects, and do not fully meet the rigorous mathematical conditions established in fractal theory, they are arguably the most interesting ones. An example can be seen in Fig. 7.10: zooming in this image, we can easily recognize similar (though never identical) images on a wide range of scales. Real objects are limited in size, and their scaling properties can go down the spatial scale only up to a certain scale level, beyond which the scale invariance breaks down. However, the scale invariance properties of such objects can be accurately established. Insights drawn from their analysis can open fruitful perspectives regarding the characterization of the objects themselves, comparisons between objects, interactions between objects, and mechanisms involved in their formation, along with the scale intervals dominated by certain mechanisms, etc. We assign to the third category, “fractal-resembling features,” those features that are labeled as “fractals” in a metaphorical sense, even when scaling aspects are not established, but rather suggested by some of their aspects. The abundance of occasions when such features are mentioned, especially in the social sciences and the humanities, makes it necessary for us to set them up as a category of its own, if only to emphasize that the assignation of the label “fractal” is risky and often unjustified. Sometimes, the presence of some forms of similarity is considered to suffice. In some cases, it may be enough for a system to have a distinguishable internal structure, with subsystems that are divided up in other subsystems of their own, for the word “fractal” to be applied. Clearly, in this investigation, we are particularly interested in objects in the second category, because we focus on scale in relation to our environment. As stated

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above, even if the scale range on which their scaling properties can be established is always limited, we can reliably evaluate these properties. Such objects are often called “fractals” too, with the understanding that such limitations are inherent to all concrete objects. The most common fractals are the self-similar fractals, for which parts look like the whole, when they are magnified equally along the different coordinate axes (for images, those would be X and Y axes; this is the case of the image in Fig. 7.10). In other words, these objects have a structure that is invariant under change of scale. Although the scale type is never specified in this context, we can clearly recognize it as scale as ratio. Self-affine fractals, on the other hand, are objects for which parts look like the whole if magnified differently along the different axes (again, for images, those are the X and Y axes). An affine transformation in E-dimensional space takes a point x with coordinates x1, x2, …, xE and transforms it into the point x′ with coordinates r1 ∙ x1, r2 ∙ x2, …, rE ∙ xE, where the constants ri are not all equal, as they would be in the case of self-similarity (affine transformations and their ramified implications were discussed in Chap. 4). Typical examples of self-affinity include time series, rock rupture shapes, topography profiles, etc. Of course, in most cases, we address situations in which E = 2. Whether self-similar or self-affine, a fractal object is one for which its parts look like the whole (as well as like parts of the object on different scales): a fractal object has no characteristic length scale. This property seems to be intuitively useful. However, when we said that parts look like the whole, we did not constrain the

Fig. 7.10  The same branching pattern can be found on a very wide range of scales: an image of the Amazon basin. (Courtesy of NASA)

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problem very well: What did we actually mean by “looks like”? We did not specify what kind of similarity was meant. For instance, the similarity may only be established statistically. We could concisely state the condition of rediscovering a certain structure “on all scales” by simply referring to a function as follows: Given a function f(x), if we multiply its argument x by a constant k, thus performing a change in scale, we should be able to retrieve the initial form by amplifying the outcome by a number that is a function of k:

k  R, c  k  ,

f  kx   c  k · f  x 

(7.1)



i.e., for any real number k, a value c(k) exists, such that when a scale change by a factor k is applied to the function argument x, the initial object is retrieved when c(k) is applied to the object f(x). In other words, no matter what scale we choose to dive in (or what fraction of the initial object we select), we can always find the way of blowing it up, and it will look like the initial object. As we will see further below, the power law form of f(x), which fulfills Eq. (7.1), is highly relevant, in more than one way, in the case of fractals: f  x   a  x



(7.2)



Indeed: f  kx   a  k  x  k   f  x 



(7.2′)



Equation (7.1) consists of two parts. The first part stipulates the existence of an amplification factor, and the second shows what the factor will do. When one or the other of the two parts of Eq. (7.1) is rigorously fulfilled, we can distinguish two principal types of fractal objects. Let us rewrite Eq. (7.1) in two ways, each of which will – realistically – only rigorously fulfill one of the two parts. First, instead of being able to pick any real number k and still retrieving the initial shape, we stipulate that this can only happen for a discrete number of values k1, k2, …, kn:

k ki  ,

i N,

c  k  ,

f  ki x   c  ki   f  x 



(7.3)

i.e., for any of the discrete set of numbers ki, a value c(ki) exists, so that when a scale change by a factor k is applied to the function argument x, the initial object is retrieved when c(ki) is applied to the object f(x). We can notice that this is the case of the object in Fig. 7.9a, the Koch curve. Its self-similarity cannot be obtained just for any fraction of the object: it is only valid when 1/3 of the total length is considered, on every scale. Given the fact that the condition in Eq. (7.1) is only met for a discrete set of values of k, we will call these fractal patterns with a discrete spectrum.

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On the other hand, there are patterns for which the values of k, as a real number, can be freely selected: the first part of Eq. (7.1) is thus fulfilled. However, the resulting scaling of the pattern does not exactly correspond to the object as a whole: in such cases, similarity is valid in a statistical sense only. We rewrite Eq. (7.1) to point this out:

k  R, c  k  ,

f  kx   c  k · f  x 



(7.4)

i.e., for any real number k, a value c(k) exists, so that when a scale change by a factor k is applied to the function argument x, the initial object is approximately (statistically) retrieved when c(k) is applied to the object f(x). In this case, when Eq. (7.1) is fulfilled when the factor k can be chosen from the set of real numbers, we call these fractal patterns with a continuous spectrum. We notice that the example in Fig. 7.9b belongs to this category.

7.2.4 Scale as Ratio, Maps, and Fractals Similarity across scales is a fundamental property of objects that are studied in the realm of fractal theory. Whether it is mathematically exact or statistically established, whether it can be found in a context that is one-dimensional, two-­dimensional, or three-dimensional, whether it is isotropic or associated with anisotropies, similarity on many scales is ubiquitously present. What is less visible, however, is the importance of scale – in particular, scale as ratio. It might be worth emphasizing the role of scale: Every time we mention “similarity on many scales,” what we mean is that following a pattern transformation that uses a certain scale as ratio, the outcome is similar to the initial object; this transformation can be repeated for many other values of scale as ratio, always with the same result. The implicit fact that it is scale as ratio that is at work and conveys meaning to self-similarity is often overlooked.

Since the theory of fractals and many of its applications were developed by a mathematician, Benoit Mandelbrot, who soon reached powerful generalizations, fractals were further deepened and expanded in an enormous diversity of fields. The image of the links between fractals and geography has become too pale compared to their significance. And yet, applications in the field of geography or with particular relevance for geography have been developing faster than they were even assimilated in the discipline itself. The pioneering work of Michael Goodchild (1980, 1988) has been standing out as a key reference and example for a very long time. To a large extent, it is due to Goodchild’s innovative work that fractal theory gradually began to feel “owned” by geographers (Goodchild 1999, 2004, 2011; Goodchild and Mark 1987); its theoretical developments and diverse applications took off in many areas of geographical research (Lam and Quattrochi 1992; Gao and Xia 1996; Man and Chen 2020; Encalada-Abarca et al. 2022).

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The roots of fractals have many starting points, each with its noteworthy history (an enlightening and thought-provoking overview is offered in Mandelbrot’s 1977 book). Most of them grew in the pure air of mathematics, protected from any contact with the physical world. However, one of the most spectacular and impactful roots referred, actually, to physical reality; it arose from a concrete  – “messy” even – problem encountered by geographers: the practical impossibility of reliably determining the length of coastlines. When faced with such an issue, the geographer may feel embarrassed – but also intrigued. Why are we unable to find the perimeter of an island, and what does this mean for us? In truth, we can, of course, measure the wiggly line of a coast on a map: there are different ways of accomplishing that. One can use a divider with a fixed opening, which we “walk” along the line to be measured, or an opisometer – an instrument with a tiny wheel, used to follow curved lines on a map and to determine their length. The problem is that our result will not be unique. With the divider, we obtain a total length that depends on the divider opening. In the case of the opisometer, the total length may seem to be more reliably determined, but it still depends on the map scale. Figure 7.11 illustrates the fact that larger-scale maps make more details of the measured line available and the presence of additional details leads to a larger value of the coastal length. The more we zoom in, the longer the line gets. If different values are produced depending on the map scale that we happen to use, this calls into question the very measurement process, along with its result. In Mandelbrot’s (1977) words, “the concept of geographic length is not as inoffensive as it seems.” This was what Lewis Fry Richardson (1961) pointed out very clearly, and his study

Fig. 7.11  Compared to image (a), more details of the intricate contours of the coastline can be observed if we increase the scale, as in image (b). If we measure the coastline length, the results depend on the scale on which the measurement is performed. (Images courtesy of NASA/GSFC/ LaRC/JPL)

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was effectively integrated in the impressive edifice built by Mandelbrot (1977, 1982). One and the same line, like the frontier between Portugal and Spain, showed Richardson’s study, was assigned different lengths by different encyclopedias, from Spain and from Portugal: at least one of the reasons for this discrepancy was the fact that distinct map scales were used to determine this length in the two cases. One of the beautiful findings in Richardson’s paper (subsequently identified as a defining trait of fractal patterns) was that the lengthening of the coast due to the increase in scale does not occur at random. It obeys a well-defined relation: there is a power law dependence between scale (even when materialized in the divider step) and the total resulting length. More importantly, the exponent of the power law was directly related to the irregularity, or the “roughness,” of the measured line. Before proceeding with the quantitative evaluation of irregularly curved lines and addressing their strange stubbornness, we should mention that coastlines were only the beginning of a torrent of features in physical reality, which turned out to reveal novel and fascinating properties when approached in the new and different manner based on changes in scale. Indeed, from river drainages to earthquake patterns, from contours of microscopic features in biological tissues or in geological thin sections to the distribution of stars in the universe, an overwhelmingly rich image of scale symmetry aspects in the real world grew fast after Mandelbrot’s groundbreaking publications (e.g., Takayasu 1992; Kaandorp 1994; Bunde and Havlin 1994, 1995; Kaye 1994; Turcotte 1997; Dewey 1998; Blenkinsop 2015; Liucci et al. 2015; Liucci and Melelli 2017). While the diversity of analyzed aspects of reality was immense, there is one key feature that all those characterized properties have in common: they hinge on the application of scale as ratio. Whether or not they are represented or representable using maps, they incorporate the principle of map representation, which relies on a well-defined proportional change. As we have shown above, the work performed by scale involves not only one scale value but also a set of values (whether finite or not).

7.2.5 Scale as Ratio and the Fractal Dimension In what follows, we will make a brief incursion into the role played by the concept of scale in defining and interpreting fractals and fractal pattern analysis. However, our goal is not to produce yet another introduction to this fascinating, multifaceted field. Such an attempt would not only overflow the boundaries of a book chapter: detailed presentations can be found in numerous books dedicated to this topic. “Classical” books beyond those by Mandelbrot include those of Feder (1988), Falconer (2003), Takayasu (1992), Peitgen et al. (1992, reprinted in 2012), Peitgen et al. (2004), Bunde and Havlin (1994, reprinted in 2013), and Bunde and Havlin (1995), among many others. (The first two are better suited for those mathematically inclined, while the others cited above offer broadly accessible introductions to concepts, methods, and applications.) Domain-specific material focusing on fractals can also be very useful. For instance, in the field of Earth Sciences, one can find

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insightful introductions provided by Korvin (1992) or Hergarten (1992) and the widely cited monograph by Turcotte (1997), as well as comprehensive review articles, such as the one by Kruhl (2013). A simple approach to the topic could start like this. Let us imagine that you discover a crack that crosses the concrete patch between the flower beds in your backyard. Let’s suppose that you want to cover it with identical, square tiles. How many tiles do you need? This depends on the size of the tiles. If the crack is straight and it has a length L, the answer is easy. If the side of a tile has a length s, the number of tiles is simply given by L/s. We can thus write that for a straight crack, the number N of tiles of size s is:

N  s   L·s 1

(7.5)



But what if the crack looks more like the line in Fig. 7.12? Here again, the smaller the tiles, the larger their number. But how much larger exactly? We can use a range of tile sizes; cover the crack with every size, in sequence; and check each time the outcome of our work. Figure  7.12 shows examples of the process using two tile sizes. Of course, we only count the tiles that actually cover the crack – the other squares do not matter for this purpose. If we summarize the results, we may find a relation between s and N, which differs from Eq. (7.5) in a slight yet important way:

N  s   L  sD



(7.6)

Fig. 7.12  Covering the crack with tiles of different sizes: the number of necessary tiles grows according to a power law with the decrease in tile size

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Since we are interested in the N(s) relationship regardless of the actual length L, which is a constant, we can write Eq. (7.6) in a relevant form, by retaining only the fact that the number N is proportional to s−D:

N  s  ~ sD



(7.6′)

The exponent of the tile size s is not equal to −1 anymore, but to a fractional number, D > 1. If we perform the same operation for other curves, which are less and less irregular and closer to the straight line, we may still find the same relation, but with a lower and lower value of the exponent D, until we reach a perfectly straight line, for which we retrieve the value of −1. On the contrary, for more irregular lines, D takes higher values. Would this happen to any crack regardless of its shape? Not necessarily: this behavior signals a scale invariant curve. Interestingly enough, scale invariant curves are not exceptionally rare. In fact, a wide range of cracks occurring in various circumstances, in natural as well as human-made materials, enjoy scale invariance. What we have just described also refers to a simple, yet highly efficient method of characterizing scale invariant spatial patterns: the box-counting method. More details and application examples for this method will be provided below; for now, it is important to acknowledge that the described procedure touches upon the very core of the meaning of the fractal dimension. In fact, in simplified terms, we can arrive at the fractal dimension as follows: Given a certain object (like the outline of the crack mentioned above) and a set of spheres of diameter s, we cover the object using the minimum number Ns of overlapping spheres. We repeat the operation using increasingly smaller spheres, and we see how the minimum number of required spheres varies when s is approaching zero. For a smooth line, we have, in agreement with Eq. (7.5), the following:

lim N s ~ s 1 s 0



(7.7)

If we apply the same thought experiment to a smooth surface, we have:

lim N s ~ s 2 s 0



(7.8)

There is nothing surprising about this. In each of these cases, we end up with an exponent that is equal to the dimension of the analyzed feature: 1 for a smooth line and 2 for a smooth surface. A more interesting result is obtained when we apply this approach to a pattern like the irregular curve in Fig. 7.12. In this case, we obtain:

lim N s ~ s  D s 0



(7.9)

where D is the fractal dimension associated with that object. In other words, in a mathematical sense, D is only defined at an asymptotic limit. The need to take the limit s → 0 may look confusing: What is its practical meaning when we address an

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actual feature in the physical environment? In fact, we are not interested at all in the limit where the measuring stick or sphere goes down to the infinitely small. The meaning of the sphere-shrinking operation unavoidably breaks down as soon as we reach a certain size of s that is small enough for the analyzed structure to have different physical properties than those it enjoys on larger scales. For example, in the case of the coastline example mentioned above, below a certain scale limit, we would stop seeing the undulations of the coast, and we would begin to perceive the presence of boulders, large and small, and if we would zoom in even more, we would analyze the surface texture of those rocks. This is one of the key features that distinguish mathematical fractals from those entities that we called above “real-world structures with fractal properties.” The latter are never actual fractals in a mathematical sense. The scaling properties manifested by concrete objects, which are captured by Eq. (7.6) rather than Eq. (7.9), are often highly relevant. In agreement with the definition of patterns and pattern analysis from Sect. 7.1.3, the main goal of fractal analysis is to detect and quantitatively characterize invariants. In the concrete world, invariants are expressed within certain scale intervals. Together with the scaling exponents, those intervals can be linked in important ways with physical and chemical processes that are relevant to the studied systems. In general, such findings support our understanding of the studied systems, sometimes shedding light on pattern-forming processes (Kruhl and Nega 1996; Kruhl et al. 2004; Perugini 2021). As we mentioned in the first paragraph of this chapter, one may think that the type of scale that is relevant to studies on symmetry and patterns is mostly, if not exclusively, scale as size. Indeed, in fractal theory, one keeps focusing on size – the size of tiles, the size of spheres, etc. As long as we define the fractal dimension based on the limit of a shrinking sphere, it is hard to think about any other type of scale beyond scale as size. However, if we pay attention to the process involved in the determination of the fractal dimension, we can see that what we do, in fact, is change the scale of observation. One can more easily notice this when we apply the analysis to a concrete object, as we did above to address the crack in our backyard. Choosing tiles of smaller and smaller size to cover the feature means focusing on details that are smaller: we pay attention to smaller areas, as if we were to descend from the “big picture” seen from above, coming increasingly closer to the ground. In order to use very small tiles, we need to observe the crack close up. Every change to a new tile size, which was needed in order to characterize the invariance, involved a new scale of observation and therefore relied on scale as ratio. As we have already stated above, when it comes to scale invariance, scale as ratio plays the main role at every step.

7.2.6 Scale as Ratio in Pattern Analysis Let us revisit the box-counting method, which we applied above as a mental experiment to a plane irregular curve. This time, we will specify, point by point, the stages that are involved, along with some method-specific concrete details. The method is

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not new. However, it is remarkably effective and reliable for a variety of types of patterns (Falconer 2003). Furthermore, it is pedagogically valuable, since it also offers an intuitive approach to the evaluation of the fractal dimension and to its meaning, as we could see above. The method consists of the following steps: 1. We define the area to be studied, which includes the pattern or the structure of interest (whether it is a set of points, a curve, a set of lines, etc.). Defining and properly documenting the delimitations of the study area are always important in pattern analysis. Sometimes even small variations in area boundaries can affect the results of the analysis. 2. We divide the area in a number of cells of size s (for pattern analysis in a plane, the size refers to the length of the side of square cells). 3. For the given cell size s, we find the number Ns of cells that are occupied by the analyzed structure. A cell is considered to be “occupied” if it contains at least one point of the structure. In other words, for each cell size s, we will only have two types of cells: those that contain some part of the structure and those that do not. This is a step of particular conceptual relevance: since all we “see” in one step is the set of cells that are occupied, as distinguished from all the others, this is equivalent to observing the structure with a resolution equal to the size of the cell. In other words, by using smaller and smaller cell sizes, we get access to more and more details: we change the scale of observation, as if we were to access maps at a larger and larger scale as ratio. 4. We repeat steps 2 and 3 for a range of scale sizes s, and we keep the record of the pairs s and Ns. In practice, instead of choosing the actual size of a cell, we can select the number of cells to be used in order to cover the area. For instance, if the area has the shape of a square, we may choose to divide it in 4, 16, 64, etc. square cells: we thus start with the lowest number of cells (with the largest cell size); in this case, the cell size s is not selected directly; it is a result of the chosen division process. Alternatively, we can start from the smallest cell size that we wish to apply and increase it iteratively (see point “(c)” below for details). 5. We check if the relation between s and Ns is a power law, i.e., of the form given in Eq. (7.6′), and if this is the case, we find the exponent D. This exponent usually takes a fractional value, which characterizes the structure. We can immediately see that if we apply the logarithm to Eq. (7.6), we get log(Ns) = log(L) − D log(s), which is the equation of a straight line in coordinates log(s) and log(Ns). This means that if we plot log(s) vs. log(Ns), we obtain a straight line with the slope equal to the fractal dimension D. An example is shown in Fig. 7.13, in which the method is applied to a set of points: epicenters of earthquakes on the Big Island of Hawaii. As we can see, the correlation in the log-log graph is very strong, which is typical for situations in which the analyzed pattern has scale invariant properties. The value of the exponent D is relevant, since  – unlike other numbers obtained in statistics applied to spatial patterns – it has the meaning of a dimension, as further discussed below. The described log-log representation is quick and convenient, visually pleasing, and highly useful for quick observation. However, it does not offer the most reliable

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Fig. 7.13  Box-counting analysis of a point pattern representing earthquake epicenters on the Big Island of Hawaii: (a) spatial distribution of epicenters (data courtesy of USGS); (b) the corresponding log-log graph, leading to a fractal dimension D = 1.57 ± 0.01

way of accurately assessing the validity of a power law and, in particular, the scale range of that validity. From this point of view, other methods, such as the maximum likelihood method (Clauset et al. 2009), are preferable. For the purpose of this chapter, we will still discuss here the box-counting method using log-log graphs, because they are intuitive and easy to interpret.

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Moreover, in principle, one should never exclusively rely on the numerical results of the analysis, not even if the power law diagnosis is backed up by more reliable means. One must always carefully consider both the actual data, using various graphic representations, and what we know about the concrete processes that we study; both can reveal aspects that are important to the interpretation of the results. Only in this way can properties that are as important as scale invariance be validated. While a proper identification of scaling can be relevant and support our understanding of the investigated systems, spurious results can be seriously misleading. There are currently numerous ready-made procedures that implement the box-­ counting method. Many of them, however, suffer from certain limitations, if not flaws, concerning the way in which the methodology is used. If one wishes to better assimilate the method and the meanings it unlocks for various patterns, it is strongly recommended that one writes one’s own procedures. The generation of the program that implements the methodology is a valuable learning exercise in its own right. Moreover, this can be accomplished without advanced programming skills. Some important notes must be added to the series of steps described above. Such notes can make the difference between being able to immediately implement the method in a proper way and get relevant results – or not. We will add them in the form of a series of questions and answers, which often proves to be helpful in a teaching context: (a) What if Eq. (7.6) is not fulfilled by the s and Ns data collected in the described steps? There are many ways in which this can happen or which may at least cast doubt on the validity of a power law. Some of the latter cases are exemplified below. However, if a power law is not present, we must admit that the analyzed pattern, or at least the property that is assessed, does not enjoy self-similarity over the scale range used in the analysis. (b) How do we pick the minimum and maximum values of cell sizes s to be considered in the pattern analysis? The lower and upper limits of s are important, because they define the scale range for which the analysis is performed. When we decide upon those limits, we must take into consideration the pattern features that we wish to explore, the scale ranges over which the relevant processes are expected to act, etc., but also the available dataset  – e.g., the resolution associated with the pattern, the spatial uncertainties in the data, etc. This total scale range – between minimum and maximum s-values – must be as wide as possible, to avoid spuriously detected power laws. The range should preferably be several orders of magnitude wide, but one often does not have the necessary data to even check if such a wide-scale range would apply. However, scale ranges that are less than two orders of magnitude wide (which are still applied in a large number of articles, as shown in Kruhl 2013) should be treated with caution. Especially when the scaling interval is narrow, it is important to treat the analysis in the context of our knowledge about the studied system and the mechanisms involved in pattern formation.

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(c) How do we select the actual intermediate s-values between the minimum and maximum limits established above? Since we know that the points on the log-­ log graph we use (at least for visualization purposes, if not for the analysis per se), in order to have equally spaced points along the X-axis, we must choose s-values that grow in a geometric progression. For instance, we may have s1 = smin (determined at point b); then, s2 = k s1, s3 = k s2, etc., sn + 1 = kn smin. The lower the value of k, the larger the number of points between smin and smax, so we can choose k as a function of the number of points we wish to obtain. Values of k around 1.2 work well in a variety of situations. (d) When is the correlation in the log-log graph strong enough to be considered reliable? As mentioned above, the log-log representation is not the most reliable method of confirming the power law and its scale range. However, as long as we apply this representation, we can cautiously use the coefficient of determination R2 for a general orientation regarding the underlying power law. Experience shows that for the vast majority of patterns that are known to enjoy scale symmetry, from coastlines to earthquake epicenters to fault patterns, etc., the correlation in such log-log graphs is strong: so much so that suspicion may arise as soon as R2 is lower than 0.99, and in such cases both the pattern and the graph should be carefully analyzed. However, with an R2 that is higher and which is associated with a wide-scale range (of orders of magnitude), the existence of a power law is increasingly plausible, and the resulting exponent can be determined within narrow uncertainty boundaries. Therefore, when the correlation is that strong, even small differences between different patterns can be reliably detected and quantified. Figure 7.14 illustrates two relevant situations regarding the interpretation of correlation strength. In Fig. 7.14a, the correlation is not strong because of the wide scattering of the points around the regression line. This is not how scaling properties are usually reflected in such a graph (note that R2 = 0.73). If such a situation occurs with box-counting, we are most probably not looking at a scale invariant pattern. Figure 7.14b shows a more interesting case. In spite of a very strong correlation (R2 = 0.99), observation can lead to the discovery that all the points consistently indicate a curved line, not a straight log(s) vs. log(Ns) relation, which eliminates the possibility of a power law being at work. (e) Does a pattern always show either a single scaling law or no scaling at all? No. In some cases, more than one valid scale range can be found: the distinct intervals indicate that the similarity governing aspects on various scales has distinct properties over different ranges of scale. For instance, in the case of desiccation patterns (cracked soil or cracked starch suspensions in water), one may identify distinct ranges of scale, characterized by different exponents (Suteanu et  al. 2000). It is necessary to carefully detect the presence of such scaling regimes. Figure 7.15 shows what can happen if the distinction of scaling regimes is not made: analyzing them together (Fig. 7.15a) leads to an exponent that is irrelevant for both ranges of scale, whereas an analysis that takes into consideration this distinction can reveal the exponents belonging to each scale interval, as well as the scale value that separates the two regimes. It is important to note that

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Fig. 7.14  Two common reasons to suspect that the linear relation suggested by the log-log graph does not correspond to a power law relation (here, x = 1/s stands for the reciprocal of the cell size, while y is the number of cells): (a) A wide dispersion of y-values is usually easy to spot, since it is also reflected in a decreased R2 value; here, R2 = 0.73, which might be considered high in other circumstances, but not for box-counting. (b) Graph points consistently corresponding to another pattern, not to a straight line, in spite of the strong correlation: here, R2  =  0.99, which is high enough for box-counting; we can notice the gradual decrease and then increase of log(y) values compared to the linear regression line; the R2 value alone cannot confirm the presence of a power law

in most cases, scaling regimes do not show any transition from one to another: their separation is sharp, even when we increase the point density on the graph to study the scale interval of their junction in detail. All the above notes of ­caution repeatedly emphasize how important it is to use reliable methods to diagnose the presence of a power law and its scale range.

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Fig. 7.15  The presence of two scaling regimes, as seen in a log-log graph: (a) Evaluating them together can lead to a meaningless result, even when the correlation is strong; here, R2 = 0.98. (b) Separately evaluating the different scaling regimes reveals that each of them is associated with a distinct value of the scaling exponent. In many cases, the actual presence of the scaling regimes, along with the scale that separates them, is at least as important as the exponents themselves. While the detection of two different scaling regimes can also be attempted in the case shown in Fig. 7.14b, the analysis will show that even when separated in two sets, those points will still belong to a curved pattern (x and y have the same meaning as in Fig. 7.14)

(f) Can the method be applied to patterns other than those composed of single, irregular lines or point sets? The method can be effectively applied to a wide variety of patterns, which can include mixtures of points; lines that are irregular, intersecting each other; etc. While the method was explained using two-­ dimensional examples, which are the most suggestive and the most common

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ones, it can also be used for other kinds of patterns. One can also analyze in this way one-dimensional patterns, i.e., points along a line – like events over time, three-dimensional patterns, or even features of a higher dimension, as long as the cells are appropriately designed (the key problem of dimensions will be discussed in the next section). Let us summarize the main properties of the box-counting method and their implications. Most importantly, it relies on the way in which the number of cells that are occupied by the structure depends according to a power law on (or “scales with”) the size of the cells used to cover the structure. In this case again, the wording does not do justice to the role of scale as ratio, which might not even be noticed, although it operates at the core of the method, as shown above. The method proves its effectiveness especially when confrontation with strong variability is unavoidable, as geometric simplifications (assimilating natural objects with circles, cylinders, cones, etc.) would not work. It is particularly useful when applied to a variety of highly irregular real-world systems (both in space and in time), for which it can provide a quantitative description. Thereby, rigorous comparisons can be made between different features with “wild,” irregular shapes or between different instances of the same system at different moments in time, in order to detect and characterize change. One can thereby identify situations in which details keep changing, but the pattern stays the same, or circumstances in which the pattern itself is subject to change (both can be encountered in the case of coastlines, for instance). Its capability of reliably revealing scale intervals associated with certain pattern properties can be particularly valuable. The box-counting method is remarkably capable of revealing self-similarity with utmost clarity (more so than many other methods). First of all, even when one uses log-log graphs – for instance, for an initial exploration of the pattern – the correlation of data points in the log-log representation is very strong: when it is not, it is doubtful that we are looking at a scale-free pattern. One can understandably suspect that the logarithmic transformation, with its difference-squeezing action applied to the original values, contributes to the high correlation coefficients. However, what we see in the case of such self-similar patterns usually goes far beyond the effects that would be expected from logarithms. Coefficients of determination R2 higher than 99%, over scale intervals comprising two, three, or more orders of magnitude (and, as a rule, more than one order of magnitude) usually look quite convincing. And yet, even when the outcomes seem solid, one must proceed with caution and always assess the pattern carefully (as we could see in the example from Fig. 7.14b). Depending on the analyzed pattern, the position of the image in relation to the cell grid may have a slight influence on the result, but when the pattern has strong self-similarity properties, this influence is negligible; however, to annihilate the possible effect of position dependence altogether, one can perform on each scale (for each cell size) multiple evaluations, for different positions of the object on the grid, and for different orientations and take the average result at each step. The method is simple, easy to implement (especially if recursive programming is applied), and it operates fast. Last, but not least, it is more versatile than other fractal

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analysis methods: it can be reliably applied to patterns comprising any combinations of points, curves, and various shapes. Due to its effectiveness and simplicity, the method was proven useful in a variety of fields, including geography, geology and geophysics, materials science, biology, medicine, etc. As we will see below, it was also applied to the evaluation of works of art. Last, but not least, operations of scale change are performed for every point on the graph: they all rely on the application of scale as ratio.

7.2.7 Dimensions and Their Meanings We stated above that the fractal dimension is more than a mere number, which can be compared with numbers obtained for other patterns or for the same pattern at another moment in time. Such comparisons are surely possible and relevant, but, in the case of the fractal dimension, we also benefit from a geometric meaning of the result. To put this meaning in perspective, we must mention two other dimensions – both are familiar to the reader. The first is the topological dimension DT: the dimension of the building element. Perhaps the simplest and most intuitive way of referring to those dimensions (even if not the ideal one in terms of mathematical rigor) is to think about the number of coordinates (of values) that we would need in order to specify the position of a point – let us call it the “test point.” For the point, if our test point sits right on it, we would obviously need no coordinate at all; the point is thus zero-dimensional. For a line, we would need one coordinate to refer to our test point. For a plane, we would need two. For a “space,” we would need three. Of course, spaces of higher dimension can also be considered, and patterns in such spaces might be relevant in a variety of situations. Here, we will limit our discussion to dimensions that we can readily visualize and observe. If our fractal object is made of a line, no matter how wiggly, its “building element” will still be a line, and the topological dimension DT = 1. If the object is made of a wrinkled surface, its topological dimension would be DT = 2. A fractal object made of sets of points has the topological dimension of the point, which is zero. On the other hand, the fractal object is embedded in a certain environment, and the dimension of that embedding space is called Euclidian dimension, DE. For a set of points sprinkled along a straight line, DE = 1. If the points lie in a plane, DE = 2. If they are distributed in space, DE = 3. The wrinkled surface mentioned above will also be embedded in space and have DE = 3. Fractal objects have dimensions that do not coincide with the ones listed above. An irregular line is more than a straight line: in some way, it tends to “fill” the plane, and its fractal dimension reflects that with a value that is higher than 1 – for instance, 1.3. No matter how complicated the line lying in the plane can get, it will never be able to fill more than the plane, and therefore its fractal dimension can only go as high as D = 2, when it would be plane-filling. On the other hand, since the curve is made from a one-dimensional line, no matter how it lies in the plane, its topological dimension cannot be lower than 1. The same reasoning applies to other dimensions

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as well. We can thus notice an important relation between the dimensions discussed above: DT ≤ D ≤ DE



(7.10)

i.e., the fractal dimension D must be equal to or higher than the topological dimension DT of the building element and equal to or lower than the Euclidian dimension DE of the embedding space. A series of examples illustrate this idea (Table 7.1). Table 7.1 offers us an opportunity to reflect on various situations regarding patterns, their fractal dimension, and their relation to their building element and the space in which they live. When the fractal dimension is close to the Euclidian dimension, we can call the pattern “space-filling,” in which case the space is the one in which the pattern is embedded, which can be a line, a plane, or three-dimensional space. For example, sets of points can have a fractal dimension D that can go from close to zero to a value as high as 3. A fractal dimension that is lower than 1 can belong equally well to point sets embedded either on a straight line or in a plane or in space. However, the meaning of the fractal dimension would not be the same in these three cases. When points sit on a line, a fractal dimension approaching the value of 1 would signal space-filling – in one-dimensional space, which we would thus call “line filling”; there would be no plane-filling, however, if points with the same fractal dimension lied on a plane. In the case of points in a plane, if D is close to 2 (e.g., D = 1.9), we could say that the pattern is close to “plane-filling” – and it would be more plane-filling than a point set with D = 1.8, for instance. Similarly, in order to be space-filling in three-dimensional space, their fractal dimension would be closer to 3. A curve in the plane is also plane-filling if its fractal dimension is close to 2, etc. A noteworthy implication is the fact that if the fractal dimension happens to be an integer, this does not necessarily make the pattern less “fractal”: for instance, we could see that a set of points in space can have a fractal dimension D between 0 and 3, depending on the way they are clustered in space. The spatial distribution of stars corresponds to D = 1.3: if the points of such a pattern were more clustered, they could have a fractal dimension D = 1. Similarly, if the points were less clustered (and more space-filling), they could also have a fractal dimension D = 2. Of course, it would be unlikely for a pattern that can have a dimension between 0 and 3, to have Table 7.1  Pattern examples: topological dimension, fractal dimension, and Euclidian dimension Geometric elements Points on a line Points in a plane Points in space Curve in a plane Curve in space Surface in space

Topological dimension DT 0 0 0 1 1 2

A possible fractal dimension D 0.7 1.4 1.3 1.2 1.7 2.2

Euclidian dimension DE 1 2 3 2 3 3

Real-world examples Events in time Epicenters Stars in space Coastline Path in 3-D state space Topography

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D precisely equal to 1.0 or 2.0, but in theory, this is possible, with the point pattern still being scale invariant.

7.2.8 Scalebound and Scale-Free Patterns Consider the following example, offered sometimes in introductory presentations on symmetry and scale: Ask yourself what would be the odds that you find someone with a height that is three times larger than yours. Then ask yourself about the probability for someone to have an annual income that is three times higher than yours. The difference between the two situations is, of course, unquestionable (unless, of course, you are one of those several people who are the richest ones on the planet; in that case, you would just have to think about another person in order to complete this exercise). This difference can also suggest the existence of two broad categories of objects. The discovery of a wealth of mathematical objects, as well as of numerous concrete objects, which enjoy scale invariance properties, made such distinction particularly relevant. The first category includes objects that have “characteristic elements of scale […] with a clearly distinct size” (Mandelbrot 1981): Mandelbrot called such objects “scalebound.” Scalebound objects typically have a characteristic size: in fact, if we think of a size, the more different it is from the mean, the lower the probability will be to find an object of that size. A widely encountered distribution – the normal distribution – illustrates this situation very well: in this case, the probability of finding objects of a given size drops fast, exponentially, in fact, when that size gets farther away from the mean (Fig. 7.16, inset). For instance, virtually indistinguishable objects cannot be identical: there are always differences between presumably identical car parts, but those differences are usually distributed Fig. 7.16  Examples of graphs for a power law distribution (main graph) and for a normal distribution (inset)

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according to the normal distribution. It is highly unlikely for parts to be very different from each other – and this is important, because otherwise they couldn’t be used as spare parts. There is always a small difference between the lengths of different but “identical” cars; however, again, the larger their distance from the mean, the lower the probability of their existence. The same applies to tables, rooms, and buildings. They are all scalebound. In the other category, we have the fractal objects or scaling objects, which are scale-free. In Mandelbrot’s (1981) words, “for practical purposes, a scaling object does not have a scale that characterizes it.” Power laws characterize such objects, as we could see in Eq. (7.6), and in the sections above. The major difference between the two types of distribution can be distinctly observed when we look at their mathematical form. For the normal distribution, we have:



f  x 

1

 2

e



 x   2 2 2

(7.11)

where μ is the mean or the expectation of the distribution and σ is the standard deviation. We can see that the exponential function acts on the square of the difference between a given value x and the mean. In contrast, for a power law distribution, we have:

f  x   a  xk



(7.12)

which, as we have seen above, enjoys the property of scale invariance. When x grows, the decrease in f(x) occurs less fast: the power law makes the decrease very different from the preceding situation. As we can also see in Fig. 7.8, the striking difference between a normal distribution and a power law distribution has major effects when we are moving towards their extremes or “tails.” The tail quickly becomes very thin for normal distributions, and it almost vanishes when we go beyond three times the standard deviation of the dataset. On the other hand, the tail has a sizeable thickness for power laws, even for sizes that get farther and farther out, towards extreme values; the thickness of the tail decreases much more slowly in this case. It is this tail effect that explains the discrepancy revealed by the two questions asked in the beginning of this section. For the sake of rigor, we should add that beyond “thin-tailed” and “thick-tailed” distributions, there are distributions with properties that fit somewhere between the two: this is the case, for instance, of “log-­ normal” ones, distributions that are normal if performed for the logarithm of the data, rather than for the data themselves. The existence of this and other types of distribution does not diminish the importance of the two broad categories of scalebound and scale-free objects, which are ubiquitous in the environment. Innumerable shapes in the natural environment, from clouds to mountains, from tree branches to mineral distributions in rock samples, are scale-free. They enjoy

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fractal properties; they “look the same” at various scales. Such objects stand in sharp contrast to most human-made objects, from pencils to desks, rooms, and buildings and from cars to flower beds: these are all scalebound. The size of scalebound objects can be evaluated with a fairly good approximation from their image alone. Scale-free objects often do not disclose their size unless additional elements offer a clue: it is not possible to tell the size of a cloud in a picture, unless a familiar object of known size is also present – for instance, a bird. Similarly, the scale symmetry property that is typical of topographical surfaces makes it difficult to interpret certain pictures of the natural environment, unless one adds in the image a familiar element such as a hammer or a coin. An extensive, rigorous scale symmetry in our environment, which would present us with similar shapes on a wide range of scales, would be challenging for us to handle. Orientation in a rigorously self-affine environment is not possible. When every part of the surroundings is similar to every other part, at the same and at every other scale, no element or part can be distinguished and specified. How can we (or any living species) make sense of the real, multifaceted self-affine environment? There are at least two plain answers to this question. On one hand, our natural environment’s self-affinity is neither all-­ encompassing nor endless. Some natural elements are not scale-free (tree trunks are an example), and all scaling properties are expressed over limited ranges of scale. On the other hand, humans typically create shapes that are scalebound. In our evolving interaction with the natural environment, we keep pushing scale-free shapes back while advancing with our endlessly multiplying scalebound features. An example is shown in Fig. 7.17.

7.2.9 Scaling and Spatially Variable Scale as Ratio We have discussed so far scale invariant patterns that are either self-similar, for which changes in observation scale revealed similar features, or self-affine, for which similarities can be noticed when the observation scale is different in the X direction vs. the Y direction. Other types of scale invariance have also proven to be important to our inquiries on the environment. As early as the 1980s, it was discovered that the application areas of fractals can be vastly expanded and the implications of doing so could be amplified. The new, major step was triggered by the realization that certain natural patterns may reveal their scale invariance if scale transformations are not simply confined to a constant scale (whether the objects are self-similar or self-affine). Indeed, in the case of clouds, scale changes cannot go very far before we realize that beyond a certain amount of magnification, regardless of the way we tune the X vs. Y scaling, cloud structures become too different to be “similar.” For instance, small clouds extend mostly in the vertical direction, while for larger clouds the horizontal size becomes increasingly dominant. Scale invariance seems to be confined to relatively small volumes. It was shown, however (Lovejoy 2019), that applying an anisotropic transformation, i.e., combining scale change with a change in the horizontal vs. vertical proportion,

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Fig. 7.17  Natural scale-free objects are gradually but persistently forced to make room for scalebound ones (represented here by wires and a house roof)

scaling in the atmosphere could be extended by many orders of magnitude. When applied to cloud shapes, a consistent relation between the two types of change, “blowing up” and “squashing,” had spectacular consequences: scale invariance could be recognized from minuscule pieces of a cloud all the way to the size of the planet. Soon thereafter, it was discovered that adding a change in orientation, occurring gradually while the scale was expanded and the squashing was enhanced, led to an even broader generality. The resulting generalized scale invariance could not only address real structures in the atmosphere in novel and effective ways, accounting for the horizontal stratification: it proved that it was possible to use a general symmetry principle in a way in which the scale was directly connected to turbulence (Lovejoy and Schertzer 1985; Schertzer and Lovejoy 1985). A detailed account of this fascinating discovery adventure is given in Lovejoy’s (2019) insightful book. When Mandelbrot began to uncover scaling phenomena, which were considered in the past to be anomalies and curiosities, little did he know that scale invariance would propel us to a place from where we will see the world in such a different way, unless, of course, he knew that, having perceived endless, ever branching, paths of discovery. The theory and the methodological framework of scale invariance proved capable of evolving fast, addressing more and more intricate aspects of the material environment. First, we identified self-similarity in an increasing number of cases. Then we found that by having different scaling factors associated with the two perpendicular axes, the resulting self-affinity can be applied to even more real-world patterns. Afterwards, we saw that the distinction between the scaling in the

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horizontal direction and the one in the vertical direction can be made scale-­ dependent, and if a rotation is also involved and correlated with the scale change, the world of scale invariance includes even more of our real environment. Throughout all these stages and with all these changes, scale as ratio can always be identified at the core of the evaluation process. Let us briefly mention that the intricacies of real objects stimulated further developments in our approach to scale invariance. For instance, when in the preceding examples we mentioned sets of points or of line segments, all points were equal to each other, and so were the line segments. What if the points, or the different segments, are not all alike, but have different “weights”? What if the points were shown in a variety of shades of gray, each shade symbolizing a “weight” or “depth”? What if the mass of the line segments was irregularly distributed along each of them? Such questions lead to the domain of multifractals, which has been developing for over three decades. In their case, we do not determine a single scaling exponent anymore, but a continuous spectrum of exponents. We could say about the developed instruments that they sometimes have the power of reaching into previously foggy spaces and returning from there with features shown with unprecedented clarity: an example is offered by Lovejoy’s and Schertzer’s The Weather and Climate: Emergent Laws and Multifractal Cascades (2013). Applications of multifractals concern an impressive variety of topics, including finance, earthquakes, topography, rainfall, water flow and turbulence, and galactic luminosity among many others. While fractals were presented here in a spatial context, they are omnipresent and important in time as well. When the variation of a certain quantity is shown in time, given the distinct nature of the time axis compared to the represented variable, scale invariant patterns cannot be self-similar: they can only be self-affine. An example of a self-affine curve is shown in Fig. 7.18. Similar shapes can be identified on many

Fig. 7.18  Representation of a random self-affine curve. Similar shapes can be recognized on a wide range of scales. The curve was obtained by integration (cumulative summation) of a series of numbers produced with a pseudorandom number generator

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scales by looking at the graph, from the tiniest peak groups to the whole graph. However, there are a large number of powerful methods that can be used to characterize scale invariant time series. One can still hear skeptics asking if “actual” fractals exist in the real world. Of course, they don’t. But this question is as relevant as asking if “actual” straight lines exist out there. The fact that straight lines and circles and other geometric figures don’t exist in the real world did not prevent us from building everything from tables and chairs to starships. Euclid’s geometry, which lives its pure life in abstract space, has been tremendously useful. Fractal theory and the broader field of scale invariance are proving their value, both theoretically and practically, in increasingly deeper ways, in many scholarly fields. When we mentioned maps in Chap. 1, we saw that the stage of abstraction they involve is part of a broader process of abstraction, which has been playing an essential role in the history of our becoming. It is quite remarkable in that context that after operating the major shift from real objects to abstract shapes, humanity chose to look in the opposite direction. What had emerged as a breathtaking leap forward – turning tree trunks into cylinders, and eventually into points, and rivers into lines – was later recognized as an obstacle to be overcome. In the second half of the twentieth century, cylinders, cones, and lines had to be left behind, to reveal the frightening irregularity of tree trunks, mountains, and rivers. In the beginning, such irregularity looked daunting: there was no room for it in our abstract thinking. We had neither the concepts to assimilate it, nor the proper tools capable of handling it. In this context, a remarkable change of course took place. This switch from one type of abstract thinking to another could only occur when the human environment, with concepts, tools, and all, was prepared to embrace the new type of objects. Unsurprisingly, this major breakthrough was also centered on the concept of scale.

7.2.10 Scale and Its Role in Multiscale Pattern Analysis One can sometimes encounter the idea that variable scale-based (or multiscale) analysis can only be applied to real-world patterns in a few, special cases, if at all. Here is how the reasoning goes. “Variable scale-based analysis is, after all, fractal analysis. Fractal analysis is to be applied to fractals. True fractals can hardly be found in the real world. Hence, pattern analysis based on variable scale is rarely useful to our study of reality.” This is a good example of apparently sound arguments leading to a wrong conclusion. The main cause of the distorted outcome is the statement “fractal analysis is to be applied to fractals.” Although it may sound reasonable, if one does not look carefully at the meaning of the terms, this statement is incorrect. As we showed in Sect. 7.2.3, when we analyze patterns in the real environment, we are not looking at “fractals.” True fractals only live in the world of mathematics. We are interested in “real-world structures with fractal properties,” in other words, in patterns that enjoy some form of scale invariance. Those patterns are countless. In many cases, scaling

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properties can be detected and quantified when it is difficult or even impossible to distinguish them visually. And yet, multiscale analysis is able to reliably identify pattern properties that are relevant to our understanding of the investigated system. Variable scale-based or multiscale analysis  – we will prefer the shorter form, multiscale analysis – can be used with valuable results in a vast number of situations. This is especially the case when our objects are irregular, in space and/or in time – when we wish to quantify the “irregularity” in various aspects of the pattern, to establish scale range intervals over which certain mechanisms are dominant, to compare the studied object with others, or to detect and evaluate change, occurring over time in one and the same object, etc. Even only by browsing the images in Steven’s (1979) beautiful book, one can find many examples of patterns for which such an approach can be fruitful. One can reach deeper insights into complex phenomena, including atmospheric processes on a wide range of scales (Lovejoy and Schertzer 2013). One may associate the quantified pattern properties with certain processes that are involved in their formation: for instance, Kruhl and Nega (1996) used the values of fractal dimension to characterize quartz grain boundaries in rocks and related them to the temperature of suture formation; Blenkinsop (2015) captured pattern properties revealing information on the spatial distribution of natural resources. Moreover, the objects of our inquiry do not have to be “natural.” Multiscale analysis can also be successfully applied to traffic, the structure of cities, or phenomena occurring in the financial world. Applications are literally endless. It is helpful to recognize multiscale analysis as a highly effective instrument which, if properly applied, can support our understanding of the environment even in its complex, strongly irregular forms of manifestation.

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Lovejoy S, Schertzer D (2013) The weather and climate: Emergent laws and multifractal cascades. Cambridge University Press, Cambridge Man X, Chen Y (2020) Fractal-Based modeling and spatial analysis of urban form and growth: A case study of Shenzhen in China. ISPRS International Journal of Geo-Information 9(11):672. https://doi.org/10.3390/ijgi9110672 Mandelbrot BB (1975) Les objets fractals: forme, hasard, et dimension. Flammarion, Paris Mandelbrot BB (1977) Fractals: Form, chance, and dimension. WH Freeman, San Francisco Mandelbrot BB (1981) Scalebound or scaling shapes: A useful distinction in the visual arts and in the natural sciences. Leonardo 14:43–47 Mandelbrot BB (1982) The fractal geometry of nature. WH Freeman, New York Mandelbrot BB (1986) Self-affine fractal sets. In: Pietronero L, Tosatti E (eds) Fractals in physics. North Holland, Amsterdam p 3–28 Peitgen H-O, Jürgens H, Saupe D (1992) Fractals for the classroom: Part one – Introduction to fractals and chaos. Springer, New York (reprinted 2012) Peitgen H-O, Jürgens H, Saupe D (2004) Chaos and fractals: New frontiers of science. Springer, New York Perugini D. (2021) The mixing of magmas. Advances in volcanology. Springer, New York. https:// doi.org/10.1007/978-­3-­030-­81811-­1_2, p 13–28 Popper K (1974/2002) Unended quest. Routledge, London Radhakrishnan S (1948/2009) Indian philosophy. Oxford University Press, Oxford Richardson LF (1961) The problem of contiguity: an appendix of statistics of deadly quarrels. General Systems Yearbook 6:139–187 Rosen J (1983) A symmetry primer for scientists. John Wiley, New York, Chichester Schertzer D, Lovejoy S (1985) Generalised scale invariance in turbulent phenomena. Physico-­ Chemical Hydrodynamics 6(5–6):623–635 Schwichtenberg J (2018) Physics from symmetry. Springer, New York Sornette D (2006) Critical phenomena in natural sciences – chaos, fractals, selforganization and disorder: concepts and tools. Springer, Berlin Stevens P (1979) Patterns in nature. Little Brown & Co, New York Sundermeyer K (2014) Symmetries in fundamental physics. Springer, New York Suteanu C, Zugravescu D, Munteanu F (2000) Fractal approach of structuring by fragmentation. Pure and Applied Geophysics 157(4):539–557 Takayasu H (1992) Fractals in the physical sciences. John Wiley, Chichester Turcotte D (1997) Fractals and chaos in geology and geophysics. Cambridge University Press, Cambridge Weyl H (1952) Symmetry. Princeton University Press, Princeton Wilhelm H, Wilhelm R (1995) Understanding the I Ching. Princeton University Press, Princeton Zee A (2008) Fearful symmetry – The search for beauty in modern physics. Princeton University Press, Princeton

Chapter 8

Scale, Symmetry, and Nonlinearity

Abstract  In this chapter, scale is captured while actively shaping not only our understanding of the environment but our environment itself. Symmetry types and their relation to scale are applied in art history: Classicism and Romanticism are distinguished with the help of scale and symmetry concepts. Scale symmetry is then shown at work in relation to individual artistic styles. Finally, aspects of scale in our understanding of nonlinear processes are discussed, including notions of chaos and self-organized criticality, along with practical implications of the part played by scale in each case. Keywords  Scale · Art history · Classicism · Romanticism · Symmetry · Self-­ similarity · Fractal analysis · Fractal dimension · Scalebound · Scale-free · Paintings · Nonlinear science · Chaos theory · Natural hazards · Self-organized criticality

8.1 Insights into Artistic Currents If scale is operating so pervasively in our interactions with reality, we should be able to find it in numerous other forms, also outside of science. We have seen that this is indeed the case in a series of circumstances. But does the scale-permeated intellectual space include art? After all that we have explored so far, an affirmative answer should not be surprising. What may be less expected is that together with symmetry, scale may make rewarding contributions to inquiries into art currents. In this endeavor, we can start from fruitful insights of Edgar Papu (1977, 1980, 1985), who took the understanding of art history due to Wölfflin (1915/1950) and Wellek (1964) to a whole new level. Papu identifies in artistic currents what he calls “types” or “states” of existence: we could illustrate them with an image of veins that penetrate throughout a given social and cultural universe, bringing life to it on every scale, shaping the individuals’ ways of experiencing reality, and presenting various forms of manifestation of people’s worldviews. These states of existence are not © Springer Nature Switzerland AG 2022 C. Suteanu, Scale, https://doi.org/10.1007/978-3-031-15733-2_8

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static structures, on the contrary: their inner dynamics, their processes of transformation, can be visibly observed and followed in time, and it is in this context that their characterization brings out the value of this vision regarding the history of culture. Compared to a certain “emotional attitude” (Wellek 1964), to a shape of life associated with a specific, multidimensional human context, an artistic style can be perceived to emerge as a secondary phenomenon (Papu 1977). According to this view, an artistic style is an expression, albeit a distinctive and highly significant one, of that particular state of existence. As a consequence, an artistic current (like the Baroque, for instance) can be discovered in different historical periods and geographical regions, beyond those to which it is conventionally ascribed. On one hand, “there is no important artistic style or current that appears and ends at a certain point in time, without having antecedents in the – sometimes remote – past, and without continuing through its extensions, in the future” (Papu 1977). On the other hand, similar states of existence emerge in places that can be far away from each other, both in space and in time. From this perspective, scale, symmetry, and their mutual relationship turn out to be potentially significant elements in an inquiry into artistic currents. They have been operating – more or less stridently – in many instances throughout our history. For this discussion, we chose Classicism and Romanticism, because they reveal the role of those elements with utmost clarity, without the need of diving deep into details. While both currents have an unmistakable presence in every cultural realm – in literature, painting, music, etc. – their materialization in parks and gardens offers a helpful backdrop against which the involvement of scale and symmetry can be depicted. Although the view we offer here into these different worlds is unavoidably brief and simplified, we intend to preserve some of their core features, which make them so distinctive in the universe of artworks.

8.1.1 Classicism: Scale, Symmetry, and Our Understanding of the Environment When referring to Classicism, one spontaneously thinks of the cultural currents that emerged towards the end of the eighteenth century in Europe and prevailed over several fruitful decades of the nineteenth century. However, in the spirit of the approach presented above, we can also go farther back, to the French gardens of the seventeenth century. We see Classicism leaving the Baroque style behind, to float in the past, and concentrating on solid, well-distinguishable, geometrically describable shapes, designed to convey a sense of clarity and order. In fact, when Louis XIV invited Bernini, one of the most prominent figures of Baroque sculpture and architecture, to design a new wing of his palace, he rejected Bernini’s ideas, favoring the “classical” ones of Perrault and Le Vau instead (Cabanne 1988). This major gesture can be considered a symbolic act in art history, an early signal of separation from Baroque (which was, at the time, extensively and deeply involved in a large

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part of Europe), and the opening of grand gates to Classicism. The fact that the very same Le Vau dedicated then most of his career to the design of the Palace of Versailles is particularly significant, since this palace was going to offer over time, throughout an interval of more than a century, a whole range of traits of Classicism. Classicism has an unyielding relationship to symmetry. The types of symmetry involved are mainly reflection (with multiple, sometimes majestic, manifestations of mirror symmetry), translation (with equally spaced elements of identical shape), and rotation (due to the uniform distribution of objects along the circumference of circles of various sizes), as we can see in a famous example in Fig. 8.1. It is important to note that all these forms of symmetry act on a wide range of spatial scales. They start from small scales, with individual features such as a vase, a column, or even a bush, to large scales, such as huge buildings with ample wings, orderly intersecting alleys, and enormous gardens, including a variety of elements at intermediate scales, like geometrically organized flower beds, water basins with central fountains, etc. The distinction between scalebound and scale-free objects will be useful in this context too. Most human-made objects, from flowerpots to trucks, are scalebound; they have a characteristic size. This size can always be clearly identified and measured. As we could see, scale-free objects do not have a characteristic size – they have shapes that are similar to each other on many scales: tree branches bifurcate into ever small branches and twigs, clouds have similar humps from very small to

Fig. 8.1  A view of the gardens of the Versailles Palace, which include the types of symmetry that are typical for Classicism (reflection, translation, and rotation): even natural, normally scale-free features, like bushes and trees, are turned into scalebound objects like parallelepipeds and spheres. (Photograph by Gilles Messian, https://commons.wikimedia.org/wiki/File:Parc_du_ Ch%C3%A2teau_de_Versailles_-­_Les_Jardins.jpg)

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very large scales, etc., as objects with fractal properties do. Unlike mathematical fractals, concrete objects can only enjoy self-similarity on a limited range of scales: however, this does not make them less scale-free. Scalebound and scale-free patterns are not confined to individual objects – the objects’ arrangement in space can also enjoy such properties. We can recognize, for instance, a scalebound shape when cubes of concrete are placed at equal distances from each other, delineating a rectangle on the ground. Scale-free patterns can be found in the spatial distribution of the trees in a naturally grown forest. On the other hand, scale-free objects, like trees, can be positioned in a scalebound arrangement – as we can see in the case of trees that are planted equally distanced in perfectly straight rows. As a rule, elements must be scalebound in Classicism, which focuses on order and clarity (its reasons for this tendency, including its reaction to previous currents, such as the Baroque, are not addressed here). Sets of objects as well as individual objects themselves are expected to emanate balance, discipline, and control. They must have well-defined geometrical shapes, proudly radiating their symmetry from the whiteness of their marble. Compared to human-made structures, which are easy to be confined by symmetry scale-balanced forms, natural objects may look rebelliously free of the constraints of plain geometry. They have to be bent, trimmed, and brought down from their wild attitude into the channels of the current’s beauty standards. Trees and bushes are thus turned into cylinders, cones, spheres, or even parallelepipeds, so that each object enjoys obvious  – and thus simple and easily perceived  – symmetry. Trees are, in their turn, placed in symmetric positions, like columns usually are, at equal distances along straight lines, as series of concentric features according to rotation symmetry, and – usually as part of a larger arrangement of features – in configurations dominated by mirror symmetry. Scale as size is therefore at home in every part of the Classicist garden, with the same size being repeated over and over again, in the ways dictated by the applied type of symmetry. It is commonplace among art historians to state that Classicism specifically means to emphasize the role of reason in all its creations. This is a well-justified claim. What might be worth adding is that the Classicism’s objective to highlight reason is addressed, in architecture and parks, at a relatively low level of the reason’s capacity to create and understand an environment. One could even say that this is probably the lowest level that was achievable, considering the circumstances: the goal was to create gardens that include architectural and natural elements, which would respond to the expectations of those times. Indeed, with such an abundance of symmetry, one could say that the resulting patterns are informationally quite poor. It is easy to compress the features of this kind of garden, by merely specifying in each case the symmetry and its main components: the compression can be applied from the scale of each geometrically simple object all the way to the scale of the whole garden. The Classicist garden is actually designed in this way. Surely, neither those who desired them nor those who created them would declare that Classicist gardens are meant to be informationally poor. Not only would that point of view have been unavailable to them. Even if such an approach would have

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been within reach, it would have had negative connotations that were incompatible with the spirit of Classicism. The tendency to consistently produce patterns that are “easy to describe,” suggesting that theirs is a universe that is easy to oversee and easy to control, is, however, a reasonably valid, possible interpretation. And yet, a pattern that is associated with a low degree of complexity does not make it insignificant. This is, in fact, a point where we can see the abyss between two ways of looking at information: on one hand, there is the view for which meanings are the ones that count and where the quantity of information is completely irrelevant and, on the other hand, there is the powerful approach centered on the “amount of information,” completely ignoring any meanings, as sharp and crisp as Shannon captured it (Shannon and Weaver 1949). Surely, talking about information while ignoring meanings may sound like a meaningless endeavor: this is what Classicist spirits would have probably concluded if a time travel experiment had taken place. They would have explained that low information content is not equal to low significance, that to them, the clear, scalebound, starkly symmetric shapes did not spell out simplicity. For them, such patterns radiated powerful meanings, with strong symbolic roots – some of them of medieval origin, as described by Umberto Eco (1986), others reaching much deeper in the past, as shown in Mircea Eliade’s major work on the morphology of the sacred (1958), symbols that had been connecting symmetry, light, truth, and our relation to reality. For today’s observer, every element in the Classicist garden seems to be simple enough for one to capture its size and shape in a glimpse. Arrangements of elements enjoy simplicity in their turn. The garden appears to provide the illusion of simplicity, as if the intention behind it was to convey the impression that those who enjoy the gardens can instantly and doubtlessly understand everything around them. This aspect of Classicism could thereby be considered as an example of a specific way of using scale and symmetry in our understanding of the environment. According to such a reductive and somewhat shallow interpretation of Classicism, the goal was not an actual understanding of the world, but the creation of an artificial world, meant to make us believe that we understand our environment. Can this interpretation teach us something about our current views on our understanding of reality? When we fully immerse ourselves in this cultural current, the number and diversity of elements that claim their place in our investigation are impressive. Overwhelmed by the breadth and richness of the emerging perspective, we may fail to penetrate beyond their shapes and colors, in order to distinguish their common, defining traits. We may even miss the role of scale. Scale enjoys, in fact, a privileged presence in every aspect that we analyzed here. On one hand, the clear, geometrical shapes of individual objects, whether columns, trees, or bushes, offer a decisive power to scale as size. The relative positions of objects, which are constrained by evident features of symmetry, have the same effect. On the other hand, the perspective that emerges on a larger scale enriches one’s perception with gradually shrinking objects and converging lines. These lines materialize aspects of translation symmetry, with their series of entities placed at equal distances in space, which become smaller and smaller until they vanish far at

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the horizon, where the sun sets. From individual objects, to groups of objects, to the spatial arrangements of such groups, scale is a silent but persistent presence, acting everywhere.

8.1.2 Romanticism: The Subtle but Pervasive Presence of Scale Given the numerous symmetry aspects in the Classicist gardens, only some of which were described above, it is easy to overlook the absence of one type of symmetry: scale symmetry. In Romanticism, it is the role of all the other mentioned types of symmetry that drops in importance. As soon as we begin a comparison between these currents, considered from the perspective of gardens and parks, a symmetry-­ based distinction becomes almost unavoidable. Romanticism emerges mainly towards the end of the eighteenth century, and it propagates fast throughout Europe; its “state of existence” extends deep into the nineteenth century. The fact that it is utterly different from Classicism does not mean at all that there is no relation between the two. On the contrary, opposing core features of Classicism and Neo-Classicism (including its “exclusive” focus on reason, its neglect of sensitivity and passion, its negative attitude towards the unusual and the outstanding, etc.), Romanticism boldly offers a quite different attitude towards reality and our relation to it. The Romanticist feels suffocated by the Classicist’s closed shapes, the limitations imposed by the “pure” line, the boundaries set by a form of reason that does not seem to try to fly high, but to linger in forms that must satisfy the conditions of absolute clarity. It is not reason in general, but reason bound only to things that are seen up close, that Romanticism rejects. It does not look kindly upon a view of reason that clings to the simplest aspect of objects, to their (idealized) straight edges and smooth surfaces, to the claim that the simplicity, the order, and the solemnity reflected in objects represent their true value. Therefore, we could say that with the arrows that it shoots into esthetic and intellectual space, Romanticism escapes the gravity of classicistic form, reaching new and at the time unheard-of realms. After exploring dizzying heights, it brings back stories about untamed faces of reality; about foggy territories and strange realms, dominated by mysteries, not by rigid order; about lands where novelty blooms and is not kept in check, even when it looks “wild”; about uncommon or even unique spaces, where imagination mocks the cages of symmetry in which it was held in the past. Concepts such as passion, disorder, and the irrational acquire novel meanings, which, in their turn, reveal unsuspected values. All these traits of Romanticism do not claim to “define” it; they can only illuminate, for the duration of a lightning flash, an environment of broad complexity, much like a moment’s flicker of the sun reflected on the glass surface of an opening window. They point, however, to some premises for the distinctive character of gardens that thrive in the air of Romanticism.

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Scale, in this unusual world, is assigned a different role compared to the ones it played in Classicism – so different a role, in fact, that it may make scale almost imperceptible at first sight (Fig. 8.2). There is a tremendous contrast between the majestic and ample manifestation of scale in Classicism and its subtle presence in Romanticism. Indeed, we can only speak about scale in Romanticism’s gardens because today we have the concepts of scale symmetry and scale-free patterns. Lacking this newer lens, those who imagined and who appreciated these gardens would not have been able to “see” scale at all. From this point of view, descriptions insist on the general impression of mere “disorder,” of “diversity” and “uniqueness” – at a time when all these were considered to compete against the ordering effect of scale. Sometimes, the absence of concepts is equivalent to blindness towards certain aspects of reality. In fact, it is highly probable that no art historian of earlier times ever mentioned scale as a meaningful element in the gardens of Romanticism. It is easy to see why. If we could dive in the environment shown in Fig.  8.2, we would find it difficult to identify any “size” around us. Everything seems to be too “unique” and too “diverse,” spelling “disorder.” If we look at these gardens today, having in mind the abovementioned elements of pattern properties, we see abundant manifestations of scale symmetry. Trees and bushes that are not shaved into cylinders and parallelepipeds, but allowed, instead, to grow “wild,” are typical examples of scale-free patterns. The spatial distribution

Fig. 8.2  Typical elements of a garden that opens a window towards Romanticism. The types of symmetry that are characteristic to Classicism are completely absent, but scale symmetry can be identified in numerous instances, from the structure of individual tress to groups of trees and the wall in ruins. (Photograph by Malcolm Bott)

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of plants of all sizes falls into the realm of scale-free patterns that are characteristic for naturally emerging vegetation. Not surprisingly, garden paths that find their way among wildly grown plant life also tend to have scale-free shapes. Last, but not least, the gardens of Romanticism sometimes include human-built features. However, normally, those are scalebound elements, dominated by straight lines and flat surfaces. The gardens would not accept them unless they change. Although this is hardly a criterion ever mentioned in art theory, we can see that even buildings must fulfill the condition of being scale-free. Therefore, we do not find Euclid’s clean and sharply defined geometrical features in castles, and not even in isolated walls or towers. The garden of Romanticism only accepts, assimilates, and integrates such construction elements, if they surrender to the all-dominating scaling property, by becoming scale-free. Romantic gardens do not tolerate the closing grip of strict symmetry in geometric shapes, whether translation, rotation, or reflection (which characterize Classicism). They have scale symmetry thriving at every step; this is the type of symmetry that feels at home in their world. Not castles, but castle ruins. Not clear-cut nature-separating enclosures, but crumbling walls. Not round and tall observation towers, but rough edges of irregularly weathered tower rims, thriving upwards, as naturally as the plants that surround them. Cracked and crumbled walls (like cracked construction materials in general) are known to be scale invariant (Cherepanov et  al. 1995). They do not stand out in a scale-free environment. How do Romanticism’s gardens perform in terms of information? We are surely in a different situation from the one encountered in gardens of Classicism: a glance at the types of symmetry involved in each of the currents makes this clear. Take any element of the gardens in the realm of Romanticism, and you can see how problematic it would be to capture it in a compact, brief description. Does this means that these gardens are informationally much richer than those of Classicism? Yes and no. It would take, indeed, a large amount of information to describe every branch and every twig in a tree. However, the existence of symmetry – scale symmetry, in this case – changes the situation to a large extent. Once we uncover the invariants, we uncover the pattern. Let us consider a tree. Invariants would include branching angles, branch lengths between bifurcations, shortening ratios between successive bifurcations, the amount of variability associated with the mentioned invariants, etc. Based on this description, we could produce a code based on iterated function systems (Barnsley 1988), i.e., using recursively defined and rescaled elements, and the code would be very brief, considering the richness of the actual real-world structures. Using this code, we could generate, for instance, any number of trees with the same pattern. Would those trees, or at least one of them, be identical to the real one? No. Would we recognize them as being different from the real one? Probably not; not if the invariants are properly identified and implemented in the shape-forming algorithm. Similar approaches can be applied for the spatial distribution of plants, the shapes of crumbling walls, etc. Much like in the case of Classicism, but in a very different way this time, the presence of symmetry can crush the quantity of information needed to describe objects and their spatial distribution.

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8.1.3 Views from the Garden: Romanticism Versus Classicism When we focus on gardens, Classicism and Romanticism can be clearly distinguished by the role played by scale in each of their worlds. On one hand, there is a major difference between them from the point of view of the abundance of certain types of symmetry and the absence of others, as we have seen above. On the other hand, for each of them, there are elements that would normally not be compatible with their patterns and which they modify in order to make them acceptable for their environment, so much so that their very nature and their sense of being are drastically altered. In Classicism’s gardens, these elements are the plants. Bushes and trees lose their natural aspect and are made to look like human-built structures: cubes, cylinders, and spheres. In this way, everything is made to belong to the same category, of rigid, scalebound-based symmetry. In Romanticism’s gardens, these elements are the human-built structures. Walls, towers, and castles lose their original role as shelters and dwellings and are turned into scale-free shapes, integrated in the wild vegetation-dominated environment. Thereby, here too, all the elements are included in the same category, of scale-free figures, in a realm where scale symmetry is omnipresent. This is not to say that scale and its relation to symmetry should, or even could, be used as “the” criterion for characterizing artistic currents. From these brief considerations, we can see, however, that the role played by scale can make a contribution to a necessarily broader approach to “states of existence” expressed in culture. Oscar Wilde’s observation about truth being “rarely pure and never simple” (enunciated in The Importance of Being Earnest) is particularly relevant in this context. No single trait can exhaustively capture the essence of something that is as alive, as rich, and as complex as a cultural current. Even in the described simplified case, where scale appears to operate with remarkable effectiveness, one must resist the temptation of assigning labels based on scale-related features alone. Sometimes there are additional relevant questions that must be taken in consideration. For instance, in the eighteen’s century, in England, two complementary processes were taking place at the same time. While the buildings were more and more dominated by “universal order and mathematical geometry,” their surrounding gardens were increasingly characterized by irregularity (Bassin 1979). In such cases, one must prudently apply scale-based criteria while remaining alert to a wide range of insights into the “states of existence,” which can be acquired from various angles. Our brief comparison might suggest that such highly consistent aspects of scale and symmetry, which appear to drive a crisp boundary between the two currents, were willingly and explicitly pursued by the creators of those times. The unfailing presence of certain symmetry types and the absence of others (Table 8.1) make it hard not to think that this is the result of purposeful action. And yet, the terms “scalebound” and “scale-free,” which were illustrated in this brief inquiry, were not used to describe these currents, neither by Wölfflin or Wellek nor by more recent art historians. This is not surprising. This distinction was made quite recently. Consequently, one might think that the relations of artistic currents to scale and

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Table 8.1  Comparison of symmetry types that are present in gardens of Classicism and of Romanticism Symmetry type Translation symmetry Rotation symmetry Reflection symmetry Scale symmetry

Scale-related property Scalebound Scalebound Scalebound Scale-free

Classicism ● ● ●

Romanticism

●

symmetry were unavoidably discussed in the past, but that this must have occurred in the absence of clearly developed mathematical considerations. Interestingly, however, art history appears to ignore the notions of scale and scaling, as well as the role they have been playing in cultural currents. This suggests that insights on scalebound and scale-free features might not have been reached in those times, but such an assertion should only be made after an in-depth investigation into this matter. If confirmed, it would undermine the idea that scaling concepts have been around for a long time and that they were only formally spelled out over the last several decades. Accordingly, we would have to face the possibility that the extraction and distillation of fractal concepts from our environment, as performed by Mandelbrot and other scholars, have brought scale symmetry to a novel state of visibility, which opens up different ways of looking at our environment. And yet again, the consistent application of scalebound and scale-free patterns, reflected in Table 8.2, cannot be reasonably considered compatible with the lack of purpose and an outcome of mere chance. How can we make sense of this apparent contradiction? We must admit that notions such as scaling, scalebound, and scale-­ free may indeed not have been accessible hundreds of year ago. In that case, they could not have been applied by the architects of those times. Not explicitly, not as such. This would explain the complete silence of art history, older and newer, in this regard. Nevertheless, we feel compelled to recognize that the lack of a theoretical framework and of clearly shaped concepts regarding scaling does not have to imply that people did not deeply perceive the distinction between scalebound and scale-­ free patterns. There may have been strong links in their minds, connecting these different categories of patterns to the multiple interrelated aspects of their state of existence. Without some deep connections between their broader worldviews and scale-related aspects, the numerous and always flawless relations between scale and the elements in their constructed environment within each of the currents would be difficult to explain.

8.2 Insights into Individual Artistic Styles If scale-based perspectives can be helpful in our inquiries into artistic currents, can they also be applied at the level of individual artistic styles? Could they even become reliable instruments used to determine the authenticity of paintings? It is normal to

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Table 8.2  Spatial and temporal scales involved in triggering factors and in their effects for several types of mass movement Spatial scale

Temporal scale Cause or Type of mass movement Cause or trigger Effect trigger Earthflow after prolonged rain in the region Large Medium Large Rockfall due to earthquake tremors Large Large Small Rockslide following accelerated creep Medium Large Large

Effect Small Small Small

be skeptical when faced with such questions. There are, however, researchers (Taylor et al. 1999) who answered the last of these questions with a resounding yes. However, their conclusions were not unanimously embraced by other scholars. A controversy ensued regarding the application of fractal analysis with such objectives in mind. The debate began almost 40 years after the death of the artist. The painter was Jackson Pollock. Taylor et  al. (1999) applied the box-counting method (described above) to Jackson Pollock’s “drip paintings.” They found that Pollock’s paintings enjoyed scaling properties and a fractal dimension could be determined. Moreover, they showed that while the painter gradually changed his style over time, the fractal dimension they determined for his paintings also gradually changed. They could thus conclude that fractal analysis can help us to pinpoint the actual artistic period to which a certain painting belongs – a fascinating result. As long as the analysis is correctly applied and interpreted, these findings might be, in a way, acceptable. What makes them less exciting is the fact that, over time, the surface of Pollock’s paintings also became more and more covered by “paint trajectories,” advancing from 20% to over 90% coverage. In other words, simply measuring the area that is covered by paint, without even taking into consideration any patterns, would still reflect a change in style occurring in time. The fact that the value of the fractal dimension also increased is not surprising, but expected. However, the researchers concluded that “fractal analysis could be used as a quantitative, objective technique both to validate and date Pollock’s drip paintings” (Taylor et al. 1999). These findings stirred a lot of interest. Nevertheless, these statements are problematic in more than one way. On one hand, there is no doubt that the fractal dimension increases for paintings created later and later in time, when the paintings became richer and denser in “paint trajectories.” On the other hand, the major point of controversy referred to the idea that the scaling properties in the painting, captured in one number, the fractal dimension, may offer an instrument of painting authentication. Jones-Smith et  al. (2009) pointed out methodological issues involved in the analysis. Most importantly, however, they showed how little relevance the fractal dimension of a painting can have with respect to artist authentication, by presenting paintings by other artists and even childlike drawings, which led to the same outcomes of box-counting analysis. The issue became particularly relevant when a group of dozens of paintings, possibly created by Pollock, was found, and fractal analysis was considered as a potentially useful method of authentication.

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Taylor et al. (1999) were neither the first nor the last ones to apply fractal analysis to paintings. Such results must be considered within the limits that scaling aspects in works of art can offer. For instance, Alvarez-Ramirez et al. (2019) compare paintings by Jean-Paul Riopelle with those of Jackson Pollock and find that the former are “more complex.” First, the preliminary operations of the analysis applied to a painting, including color separation, meant to make it approachable by fractal analysis, can have a significant influence on the result of the analysis. Second, the term “complexity” used in this context is misleading. A higher fractal dimension, a more densely filled surface of the canvas, etc. have little or nothing to do with the complexity of a work of art. Perhaps the most relevant outcome of such studies is the fact that paintings, drawings, etc. do indeed enjoy scaling aspects. In other words, invariants with respect to scale change exist. Nevertheless, scale invariance revealed by the relation between scale and some property evaluated by the applied method does not imply that we can find similar shapes on different scales. Neither box-counting nor a range of other fractal analysis methods can – or even attempt to – capture actual shapes on different scales. For instance, what box-counting establishes is a relation between scale and the number of boxes needed to cover the object, as a function of the cell size. In other words, a pattern where similar shapes are present on a range of scales does reveal a power law relation and thereby a fractal fingerprint. The opposite is not necessarily true: a power law relation between scale and the number of boxes, for a range of box sizes, does not have to stem from a pattern with similar shapes of all sizes. This is why, in many cases, the “fractal” result regarding an artwork can be surprising, when we can clearly see that there are no similar shapes on different scales. This is one of those occasions when Mandelbrot’s broad definition based on similarity “in some way” becomes particularly relevant. What the scaling relation captures is the spatial distribution of the pattern. It is the way in which scale is implied in such patterns that is interesting. Various brush strokes, sets of lines, etc., on different spatial scales, from tiny details to significant portions of paintings, can be indeed consistently and robustly related to scale. As we have seen, when scaling aspects are identified, the box-counting method reveals strong correlations between scale and the measured parameter (e.g., number of boxes). When a strong correlation is found and when it spans orders of magnitude, this is usually not a meaningless result. The remarkably consistent scale dependence, found so often in natural patterns but also in artwork, must be recognized. What is this property of patterns telling us? We do not have a simple explanation that would cover “all” cases. While even the idea of looking for such an explanation may seem absurd for some, one cannot help asking if we should not let such a question linger, and possibly bear fruit, instead of uprooting it from the beginning. Whether or not scaling properties can be studied in order to learn more about art, especially about individual styles, is a decision to be made with care, after rigorous investigations. Attaching labels to visual art simply based on a numerical value obtained by fractal analysis can be a risky, often misleading endeavor. In the end, no matter how effective and valuable scale-based pattern analysis can be, questions regarding artistic value cannot  – by principle  – be answered by “objective”

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measurement. Establishing value in art involves a different approach and different criteria. There is, however, at least one more merit associated with the work of Taylor et al. (1999) and the controversy it triggered. They made us aware that in contemporary art we might actually need, at least in some cases, a mathematical method of analysis to make the distinction between artworks worth hundreds of millions of dollars and pictures drawn by a child in several hours of fun. What this says about art and about us reaches beyond the realm of scale and scale-related themes.

8.3 Scale and Nonlinearity 8.3.1 Scale and Nonlinear Processes An effect which is not proportional to its cause leads only imperfectly to knowledge of the cause. —Saint Thomas Aquinas (1252/1998)

Many centuries before the emergence of nonlinear science, Saint Thomas Aquinas strikes the right chord regarding nonlinearity with the observation quoted above. Nonlinearity makes it indeed more difficult to understand systems and processes based on recorded “effects.” The field of nonlinear science was mainly developed in physics, and its point is not to focus merely on relations between cause and effect that are not linear. It refers to a series of interesting aspects of the behavior of interacting systems; however, in its simplest form, nonlinearity can be captured by statements such as “the effect is not proportional to the cause” and “the whole is not equal to its parts.” On one hand, this means that since both cause and effect are associated with some forms of scale, relations between cause and effect are unavoidably reflected in relations between scales. For this reason, the study of nonlinear phenomena offers fruitful opportunities for us to see scale from a new perspective. On the other hand, this implies that due to interactions among the system’s components, one cannot study those subsystems in separation, as if they were independent entities. Nonlinear science is mainly characterized by its view on complex systems, consisting either of mathematical objects or of concrete physical objects, and this view has direct implications for its specific approaches and methods developed to support our understanding of reality. While its mathematical framework is often intricate, its main principles can be captured, if adequately presented, without a background in advanced mathematics (Peitgen et  al. 2004). Moreover, James Gleick (1987) has proven that with intelligent writing, key ideas can be transmitted and passion can be conveyed even at the popularization level. The most striking manifestation of a nonlinear system is one in which “small causes” produce “large effects.” For instance, a catastrophic landslide can be triggered by minor additional work in a quarry. A forest fire of a huge size can be caused by one cigarette. However, for many real processes, it is not the case that

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small triggering factors must have large effects. More often, those small causes can trigger effects of a wide range of sizes, from very small to very large, but is the “small cause and large effect” situation that is usually outstanding, since it does not match common expectations based on systems with a linear behavior. In most cases, the discrepancy in terms of cause and effect “size” refers, more or less implicitly, to the energy involved in the initiating process, compared to the triggered process. However, spatial and temporal scales (scale as size, in this case) are always involved too. The relations between triggering factors and triggered effects can therefore also be seen from the point of view of scales in space and in time. Cause-effect characteristics may be, in fact, quite different in relation to space and time scales, compared to the commonly used “event size” perspective. Table  8.2 shows several examples associated with different types of mass movement (landslides). In the first case, rainfall affects a wide area (spatial scale of the cause: large), and it triggers an earthflow event which corresponds to a much smaller surface (spatial scale of the effect: medium). On the other hand, the temporal scale of the cause is large (it rained for a long time), and the temporal scale of the effect is much shorter: tens of seconds. In the second case, earthquake tremors affect a much wider area than the one where the rockfall occurs (spatial scale of the cause: large), but, depending on the topographic context, the rockfall can occur over a long distance, and its width can considerably increase in this process (spatial scale of the cause: large). In contrast, the temporal scales for the triggering earthquake and for the ensuing rockfall are both short. In the third example, accelerated creep starts very slowly, and it can occur for a very long time (it can be closely monitored if the proper sensors are in place): creep can be identified, however, on a spatial scale that is much smaller than the one corresponding to the ensuing rockslide (spatial scale of the cause, medium; spatial scale of the effect, large). In time, scale looks very different. While creep can cover a very long time scale, the rockslide is much more rapid. As we can see, the “scale of the cause”-“scale of the effect” relation, in space and in time, is far from always consisting of “small cause” and “large effect.” A variety of combinations of scale sizes can be encountered, even for a small set of landslide examples. These examples illustrate the fact that we should not confine our overall picture of an event to its “size”: it is useful to include in the analysis cause-effect comparisons based on spatial and temporal scale. Spatial and temporal scales regarding nonlinear cause-effect relations can be as important as the scale of the “event size,” if not more important. A significant example is offered by vulnerability with respect to a natural hazard – a powerful concept. It is not limited to a community’s aptitude to withstand the event’s impact, but also to its capacity to recover from its implications (Hewitt 1997). In the case of a natural disaster, the temporal scale that concerns recovery can stretch far beyond the scale of the actual occurrence. In some cases, domino effects of a disaster can spread spatially, which implies an increase in the spatial scale as well. Mitigation measures

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must carefully consider the spatial and temporal scales both for the causes and for their effects. An interesting category of nonlinear processes is characterized by temporal asymmetry regarding energy accumulation and dissipation. The accumulation phase can be very slow, while the dissipation phase can occur over very short time scales. For instance, trees accumulate energy from the sun, and they gradually build up their biomass (using, of course, also matter absorbed from the atmosphere and from the soil). This accumulation process can continue for many decades. At some point, the stored “sun energy” can be dissipated quickly and violently in a forest fire. There is thus a major difference between the two phases regarding temporal scale. A fast energy release following a slow accumulation also takes place in the case of earthquakes. In this case, both the temporal and the spatial scales stand in stark contrast regarding accumulation and dissipation. The accumulation takes place gradually, over long time scales, as well as over large spatial scales. Dissipation, in its turn, is drastically limited in time – the events take place over seconds to minutes. In this case, however, it is also limited in space: the evolving slip on a fault or on a fault system is concentrated on spatial scales that are much smaller than those involved in accumulation. The often stated idea that we should focus on the “big picture,” implying sometimes that details can be neglected, has an origin rooted in a linear view on reality. Although it is intended to be helpful, making people to avoid losing sight of the forest while analyzing the bark of the trees, this idea can also be impractical, hazardous even. Its dangerous implications become clear when we operate in a context with strong aspects of nonlinearity. In fact, “details,” e.g., processes on the “small scale,” can exert a drastic influence on the large-scale behavior of the system. For example, following an underground pollution event, pollutant propagation can depend on details of the rock structure, including the shape and the relative position of cracks. Working on the acquisition of as much information as possible on these “details,” such as the fracture patterns in that area, can be useful to the modeling of pollutant migration, compared to general premises of isotropic percolation. Similarly, knowledge about “details” concerning a forest – the spatial distribution of tree species, the accumulated biomass due to fallen trees, the position of creeks and rivers, etc.  – can offer valuable information for forest fire modeling regarding a given area. While knowledge about details is, by principle, difficult to achieve and grasping “all” the details is an impossible task in the real-world environment, awareness of the fact that aspects on a small scale in space and/or in time may play an important role can be helpful: it allows us to consider scenarios and research paths that might have otherwise been overlooked.

8.3.2 Scale and Chaos The appropriate notion of scale is one that emerges as a consequence of strong nonlinear dynamics, rather than being imposed a priori from without. —Shaun Lovejoy (2019)

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If we consider nonlinear science, in general, as an important area of scholarship, chaos theory represents one of its principal fields of study. Undoubtedly, the directions of research in chaos theory have projected nonlinear science forward. Not only is chaos a remarkably interesting concept: it has particularly relevant links to scale. Chaos has, of course, many different meanings: here, we only refer to chaos in science. Therefore, we exclude in this chapter any alternative connotations carried by this notion in everyday life, such as those regarding spatial aspects that we perceive as “disorder.” In fact, chaos does not refer to the way things are, but to the way they behave. It is a feature of a system’s dynamics (Fig. 8.3). Chaos theory has been developed in relation to a wealth of research topics, from the dynamics of the atmosphere to the functioning of the human heart, from traffic to the behavior of the geomagnetic field, from economics to natural hazards. However, chaos theory is also considered to be a subfield of mathematics. In fact, regardless of the area of application, the mathematical treatment of chaos plays a prominent role. The theoretical framework of chaos theory focuses on “dynamical systems.” In spite of the numerous applications in the real-world environment, the first meaning of a dynamical system is a mathematical construct, which – in most cases – is centered on differential equations, along with a set of initial conditions. When put in motion, this mathematical apparatus evolves in “time,” and we can follow the system’s trajectory in an adequately defined state space. While this is not the place to bring in details on chaos and the dynamical systems theory, we wish to underline

Fig. 8.3  An illustration of chaotic dynamics: the growth of a volcanic plume in the Halemaʻumaʻu crater, Hawaii. (Courtesy of USGS)

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that every time we talk about initial conditions, trajectories, etc., it is to this mathematical image of the system that we refer. When we explore a real-world problem in the framework of chaos theory, we must first bring it to a mathematical form. The most common defining trait of chaotic dynamics is the fact that it is sensitively dependent on initial conditions. This is precisely where scale comes in, right at the center of the most distinguishing expression of chaos. By sensitive dependence on initial conditions, we mean that if a system is considered in two different scenarios, which start from two different, yet very similar sets of initial conditions, the system’s evolution in time in the two scenarios takes trajectories that diverge fast from each other. In fact, the distance between the two trajectories grows exponentially in time, and the actual exponent quantitatively indicates how sensitive to initial conditions a certain system is. A good example is offered by the experiment that leads to the discovery of this “hallmark of chaos” by Edward Lorenz (1963). Lorenz set up a system of differential equations to model the dynamics of the atmosphere, and he followed the evolution of the dynamic system reflected in a graph (like the one in Fig.  8.4). When trying to rerun the model after specifying again what he thought to be the same initial conditions as in the preceding run, which were printed by the computer, something strange happened. The evolution of the dynamic system was this time very different from the previous one. The cause turned out to be the fact that the printout specified a lower number of decimals than the computer actually used, and therefore there was a very small difference between the initial conditions in the first run and those specified by Lorenz for the second one. Neglecting such tiny “numerical dust” was expected to be insignificant. But, in this case, it was not. After careful studies, Lorenz realized that he had discovered a new type of system behavior. Moreover, this was not only a property of a mathematical model: it reflected an actual feature of processes in the material world. He had discovered chaos.

Fig. 8.4  Sensitive dependence on initial conditions. Two trajectories that start from highly similar initial conditions, reflected in some model parameter on the Y-axis, diverge fast in time. (Courtesy of NOAA)

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As we can see, the unusual dynamics is reflected in scale. First, the difference between two locations, on a small scale, can be decisive for the subsequent behavior of the system. “Small,” in this case, has a particular meaning. It is not given by the size of some distinguishable subsystems. Actually, the distance between the starting points in the two scenarios can be not just small, but vanishingly small. The fact that such a scale as size plays a critical role in the subsequent evolution of the system is surprising, to say the least. Second, the remarkable effect of this small-scale difference is not limited to a brief time interval: its impact lasts throughout the life of the system, which will never be the same after the small change was applied. The effects last over long time scales. Third, the effect of the small-scale difference grows quickly. The distance between the two trajectories with neighboring points of origin covers fast-increasing scale values. When we explore such a system based on a set of experiments, the scale on which we study it can be small initially, but then it must be increased, like the radius of a bubble that should be able to comprise all trajectories.

8.3.3 Scale and Self-Organized Criticality Self-organized criticality (SOC) offers us outstanding opportunities to see scale at work in a multifaceted, yet highly transparent medium, in which its different aspects interact, consistently occupying a prominent position in the studied systems. Moreover, its applications concern numerous processes in the natural and human-­ made environment. For these reasons, self-organized critical systems can offer us a uniquely valuable guide to the role of scale in our understanding of the environment. Despite the surprising simplicity with which it can be introduced, self-organized criticality represents a concept with a high degree of generality, which is applied to a wide range of open systems with a large number of degrees of freedom. SOC captures a remarkable phenomenon: driven from outside, these systems spontaneously reach a stationary state that is characterized by particular properties. This state enjoys interesting attributes in space, in time, as well as in terms of event size; its most obvious feature is the fact that it comprises events with self-similar distributions. At the same time, it is important to note that while SOC has power law distributions as a characteristic fingerprint, the reverse is not always true: not all power law distributions are produced by SOC processes. The concept of SOC was introduced by Per Bak and collaborators (Bak et al. 1987), and it was perceived right away as a groundbreaking approach to reality. This first article became quickly the most cited paper in physics (Bak 1996). The concept and its implications were further explored and developed by Bak and many other scholars and applied in a wide diversity of areas of science. Topics addressed from the perspective of SOC include seismicity, landslides, forest fires, river networks, solar physics, biological evolution, neurobiology, sociology, economics, etc. (Bak 1996; Hergarten 1992; Sornette 2006).

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The relevance of scale in SOC is manifested in distinct, yet interconnected ways. Moreover, it can be explored without involving any theoretical intricacies. As Bak often pointed out, the whole SOC universe is permeated by simplicity: the concept is easy to understand, the models are easy to write, and the outcomes are easy to analyze, with no need for mathematics above high school level (Bak 1996). On the other hand, the outcomes and their implications can be highly interesting. Therefore, we will proceed with the presentation of the simple, yet highly suggestive sandpile model, which opens valuable windows to the core aspects of SOC. We should keep in mind that what we describe is a mental model (even if it was actually implemented in a variety of laboratory experiments: here, however, we are not interested in implementation details). A cellular automata version of the model can be set up without advanced programming skills. Writing and applying such a model can greatly support a better understanding of SOC, of the role of scale, and – last but not least – of certain physical processes in the environment; details can be found in the SOC-launching paper (Bak et al. 1987), as well as in classical monographs (Sornette 2006; Turcotte 1997). More advanced models can then be constructed based on the same principles (Liucci et al. 2017). The sandpile model setup consists of a horizontal disk, on which sand grains are dropped, one by one, from a device located right above the disk’s center (Fig. 8.5a). Over time, a small growing pile of sand grains is formed. With the sandpile growing larger and larger, certain sand grains get to slide off the edges of the disk. Eventually, a conical shape of the pile is reached (Fig. 8.5b). From this point on, in this stationary state, the slope remains unchanged, while new grains are added, and some of the grains on the sandpile fall off. At this point, a landing grain can get stuck somewhere on the sand slope, or it can roll down the slope and fall off the disk; more interestingly, however, it can also hit other grains, which are less stable, which are thus kicked out of their initial position, and begin to descend the slope in their turn, possibly hitting other less stable grains, so that a possibly large number of grains get to fall off the disk. For obvious reasons, such a chain reaction involving rolling grains is called an avalanche. Every time one

Fig. 8.5  Experiment that illustrates the canonical model for self-organized criticality. Grains of sand are dropped, individually, in the center of the disk (the grain dropping device is symbolized by the funnel): (a) initial state – soon after the experiment started; (b) final (stationary) state

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grain or a group of grains fall off, we identify “an event.” The size of the event is given by the number of grains that are involved. We can keep track of event sizes, because the mass of the disk and the sandpile is accurately monitored. Overall, the mass of the sandpile is constant, with occasional fluctuations depending on the avalanches that take place. Avalanches do not interfere with each other: a new grain is only dropped when the previous avalanche has ended. Let us review the experiment and highlight its most important aspects: 1. We can distinguish two stages in this model: one in which the sandpile gradually grows to its maximum size and one in which avalanches of various sizes occur, while the shape of the pile stays unchanged. In the latter stage, the system is in what we call a “critical state.” In other words, the critical state is reached after some time of system functioning. 2. An avalanche is a chain reaction, which can stop, continue as such, or get amplified, depending on the small-scale configuration: the tiniest details concerning grain position and grain stability can have implications for the effect of the oncoming grains. 3. The “history” of the system, the way in which it was built – the positions of the accumulated individual grains – matters for the outcome of the falling grains’ impact and for the subsequent avalanches. System history is important. Besides, the system history keeps changing. 4. The shape of the sandpile – characterized by the “critical slope” angle – is constant. A subcritical sandpile grows until the critical slope is reached. A supercritical sandpile (if we artificially prepared one) experiences avalanches until they bring the slope to its critical value. 5. No event, whatever its size, can remove the system from its critical state. 6. The system is unstable on the smallest spatial scales – the scale of individual sand grains. The grain configuration – given by the actual positions of various grains – is changing permanently. 7. The system is stable on the largest spatial scale – the scale of the whole system. The slope of the sandpile as a whole never changes. Criticality is a property of the entire system. 8. The system involves events on all spatial scales. The processes involved (grains that are rolling, hitting other grains, displacing some of those grains, which may roll and displace other grains in their turn) occur over short distances, as well as on longer distances, all the way to the size of the whole system. 9. The size distribution of avalanches corresponds to a power law. We are thus witnessing a self-similar pattern on a wide range of scales. This event size distribution does not change in time; it characterizes the stationary state of the whole system. 10. Time scales are tightly related to spatial scales. The temporal fluctuations in the mass of the sandpile lead to a pattern that corresponds to 1/f noise. 11. It is impossible to tell the size of the avalanche that will be triggered by a particular grain. Avalanches occur on all scales, and they are influenced by the details at the smallest scale. The effect of a falling grain can be negligible; it can

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be small; it can be very large  – depending on the avalanche that is formed, which, in its turn, depends on the detailed grain positions and grain motion. 12. One of the remarkable features of this process is the fact that the system input consists of a constant, small-scale rate in terms of event size, while the system output consists of events of all sizes, spanning all scales, governed by a power law. As we can see, the name “self-organized criticality” refers to two main features of such systems. First, they are self-organized: the system reaches spontaneously  – without being specifically fine-tuned for this purpose – a certain type of organization, a state characterized by power law-distributed event sizes, as well as by specific spatial and temporal correlations. Second, such systems are in a critical state. Dictionary definitions of “critical” refer to a situation in which a significant change – for the better or for worse – can occur, but the outcome is unknown. The keywords here are “significant change” and “uncertainty.” We can see how both of them can be applied to our systems. On a small scale, a grain can enter a collision with one grain or another, and following the collision, it may stop there, or roll down the slope, and/or displace one or more grains. At every step, both a possible major change and uncertainty are involved. Interestingly, the same factors can be found on larger scales as well. The generation of an avalanche, as well as its temporal development – whether it dissipates on the way, whether it reaches the edge and falls off, whether it grows during its sliding down, and how much it grows – all imply a possible major change, as well as uncertainty. This type of dynamics can only take place in a system that is open, in the presence of a permanent “flow.” In this model, as in certain processes in the environment, the flow refers to mass, small particles (compared to the size of the system), which are subject to gravity. In other cases, there is an energy flow that keeps the self-organized system going. Interactions between subsystems are another essential condition for SOC to emerge. Scale appears here again, front and center. The grains are held in place against the effect of gravity due to the presence of immediate neighbors – small-­ scale interactions are involved here. However, when an avalanche occurs, more and more grains are dislocated from their positions, paths of various lengths are covered by the rolling particles, and successive collisions take place between them. As a result, the scale of interaction comprises a wide range of values, from very small to very large. In summary, static small-scale interactions and dynamic “all scales” interactions occur at the same time, contributing to the self-organized critical stationary state. Event sizes are only limited by the particle size, at the lower end, and by the system size, at the higher end. It is important to keep in mind that the aspects of SOC highlighted above are well captured by the sandpile model, but they are not only a reflection of the properties of a model. They represent general features of many real-world systems characterized by SOC. It is this remarkable generality that created the main premise for the study of SOC in the mentioned variety of fields.

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For instance, earthquakes have been fruitfully studied from this perspective (Sornette 1991; Varotsos et al. 2020): the lithosphere is seen as a system in a critical state, slowly but incessantly driven due to tectonics on the planetary scale, while earthquakes represent events of all sizes, which  – when expressed in terms of energy – are distributed according to a power law (which corresponds to the well-­ known Gutenberg-Richter law). The driving mechanisms of the system span a very wide range of scales: from the dynamics on individual faults to the size of the planet. The events themselves are spatially concentrated in much smaller areas compared to the driving processes. Another example concerns studies regarding forest fires. Research involving models and comparisons to real-world patterns has led to useful insights regarding, for instance, the identification of distinct origins of forest fires (lightning vs. human-­ made) in patterns of events (Krenn and Hergarten 2009) or implications of parameters such as tree density and tree growth probability, even when the studied real-world systems may differ in some ways from the theoretical SOC framework (Rybski et al. 2021). Useful comparisons between patterns in the environment and outcomes of models based on SOC also contributed to the understanding of landslides and the effect of their triggering factors (Liucci et al. 2017): in this case again, scale – both in space and in time – plays a relevant role. The field of SOC is very wide and diverse. However, even based only on the described sandpile model and on above observations (1)–(12), we can draw important conclusions regarding the main characteristics of systems with SOC. As we can see, scale can be recognized as a key factor in the functioning of these systems. Focusing on the multiple, interconnected aspects of scale can reveal essential aspects of SOC, as well as of many real systems in the environment. As we could see above, the dynamics of a system in a critical state involves events of all sizes. Most importantly, it includes many small events and very few large ones (Fig.  8.6). In the light of numerous application areas such as natural hazards, environmental degradation occurrences, social and economic challenging events, etc., it is reassuring to know that most events are small and that very large events are rare. This is the good news. Of course, it already implies that other kinds of news will follow. Although large events are few and rare, they are part of the pattern. In other words, large events are unavoidable – sooner or later they must happen. This is the bad news. Not all events occur on small scales – but fortunately, most of them do. For systems that operate according to this model, the moment of occurrence of large-scale events is not predictable. What we know, however, is that they are weaved in the fabric of our environment. When the pattern is reflected in a distribution like the one in Fig. 8.6, large events are bound to happen. As we could see above, a large-scale event, no matter how large, does not and cannot destroy or even change the pattern: every such event, small or large, confirms and supports it. There is no reason to believe that the sequence of events of all sizes will end if an event of very large magnitude occurs. The idea that a “wiping out” of the pattern may be achieved can still be encountered nowadays. It may rely on people’s personal experience with small, quasi-linear, quasi-isolated systems. In

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Fig. 8.6  The power law distribution of event size: number N of events of size s

contrast, SOC is continuously driven from outside. Its numerous interacting, metastable subsystems and their spatial distribution imply that potential locations of future avalanches are present all throughout the system. Many events of all sizes are ready to be triggered, even after a large event. In our study of natural systems, there are situations in which details may be neglected. In the case of SOC, details are decisive for the way in which every individual event occurs. In fact, whether a group of grains stops, or continues to roll and to push other grains from their positions, always depends on the local situation on the smallest scale: the size of every individual avalanche depends on small-scale details. In contrast, the number of events in every size interval is constant, and it characterizes the system. In other words, the overall distribution of avalanche sizes is independent from small-scale details. Last, but not least, in the case of SOC, all the events, from the smallest to the largest, are produced through the same mechanisms. Sometimes large events with a disastrous impact, which are, unavoidably, rare, are naturally perceived as unusual. Therefore, the fact that when SOC can be recognized the same mechanisms are at work on all scales can be particularly valuable. To conclude, scale – in space, in time, and in relation to other variables as well – can be particularly important, in many ways, to the study of nonlinear processes. Scale can support us in our exploration of the environment, even when – and especially when – we are facing aspects of high complexity.

References Alvarez-Ramirez J, Rodriguez E, Martinez-Martinez F, Echeverria JC (2019) Fractality of Riopelle abstract expressionism paintings (1949–1953): A comparison with Pollock’s paintings. Physica A: Statistical Mechanics and its Applications 526:121131

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Aquinas Th (1252/1998) Commentary on sentences. In: McInerny R (ed) Thomas Aquinas  – Selected writings. Penguin, London, p 50–84 Bak P (1996) How nature works – the science of self-organized criticality. Springer, New York Bak P, Tang C, Wiesenfeld K (1987) Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett. 59:381 Barnsley M (1988) Fractals everywhere. Academic Press, New York Bassin J (1979) The English landscape garden in the eighteenth century: the cultural importance of an English institution. Albion: A Quarterly Journal Concerned with British Studies 11(1):15–32 Cabanne P (1988) L’Art Classique et le Baroque. Larousse, Paris Cherepanov GP, Balankin AS, Ivanova VS (1995) Fractal fracture mechanics—A review. Engineering Fracture Mechanics 51(6):997–1033 Eco U (1986) Art and beauty in the Middle Ages. Yale University Press, New Haven Eliade M (1958) Patterns in comparative religion. Sheed & Ward, New York Gleick J (1987) Chaos: Making a new science. Viking Books, New York Hergarten S (1992) Self-Organized criticality in earth systems. Springer, New York Hewitt K (1997) Regions of risk: A geographical introduction to disasters. Addison Wesley Longman, Harlow Jones-Smith K, Mathur H, Krauss L M (2009) Drip paintings and fractal analysis. Physical Review E 79(4):046111 Krenn R, Hergarten S (2009) Cellular automaton modelling of lightning-induced and man made forest fires. Nat Hazards Earth Syst Sci 9:1743–1748 Liucci L, Melelli L, Suteanu C, Ponziani F (2017) The role of topography in the scaling distribution of landslide areas: A cellular automata modeling approach. Geomorphology 290:236–249 Lorenz EN (1963) Deterministic nonperiodic flow. Journal of the Atmospheric Sciences 20:130–141 Lovejoy S (2019) Weather, macroweather, and the climate. Oxford University Press, Oxford Papu E (1977) Barocul ca tip de existenta (Baroque as a type of existence, in Romanian). Editura Minerva, Bucharest Papu E (1980) Existenta romantica. Schita morfologica a romantismului (Romantic existence: a morphological outline of Romanticism, in Romanian). Editura Minerva, Bucharest Papu E (1985) Apollo sau ontologia clasicismului (Apollo or the ontology of Classicism, in Romanian). Editura Eminescu, Bucharest Peitgen H-O, Jürgens H, Saupe D (2004) Chaos and fractals: New frontiers of science. Springer, New York Rybski D, Butsic V, Kantelhardt JW (2021) Self-organized multistability in the forest fire model. Physical Review E 104(1):L012201 Shannon CE, Weaver W (1949) The mathematical theory of communication. University of Illinois Press, Champaign Sornette D (1991) Self-Organized criticality in plate tectonics. In: Riste T, Sherrington D (eds) Spontaneous formation of space-time structures and criticality. Springer, Dordrecht https://doi. org/10.1007/978-­94-­011-­3508-­5_6 Sornette D (2006) Critical phenomena in natural sciences – chaos, fractals, selforganization and disorder: concepts and tools. Springer, Berlin Taylor RP, Micolich AP, Jonas D (1999) Fractal analysis of Pollock’s drip paintings. Nature 399:422 Turcotte D (1997) Fractals and chaos in geology and geophysics. Cambridge University Press, Cambridge Varotsos PA, Sarlis NV, Skordas ES (2020) Self-organized criticality and earthquake predictability: A long-standing question in the light of natural time analysis. Europhysics Letters 132(2):29001 Wellek R (1964) Concepts of criticism. Yale University Press, New Haven Wölfflin H (1915/1950) Principles of art history: The problem of the development of style in later art. Dover Publications, Mineola

Chapter 9

The Essence of Scale

Abstract  Whereas the first chapter served as a prism that helped us to split a diffuse flow of notions of scale into three crisply defined streams, this last chapter brings the streams back together and focuses on their common core. We illuminate the three types of scale from several angles, based on concepts rooted in logic and mathematics – guided cut, logical field, mapping, morphism, and representation, all of them focusing on the central traits of the scale types. These views are shown to converge towards a general concept of scale. Thereby, the three main scale types can be recognized as branches originating in the same trunk, which can be explicitly outlined. Overall, scale proves to be fruitfully viewed as a process, rather than merely a static medium that is available for other processes to occur. The future of scale is addressed in light of its history as a living concept.

Keywords  Scale · Mapping · Precision · Objectivity · Reversibility · Guided cut · Logical field · Morphism · Representation · Similarity · Memory · Comparison · Quantification · Transformation

In order to make our journey secure, we must look ahead as well as look back. —Søren Kierkegaard (1847/1956)

9.1 Scale Types: An Overall Perspective Rather than introducing the topic of scale by considering a variety of conceptual instruments and a cloud of meanings assigned to them – some appearing to diverge from each other, some seeming to overlap, and some bearing names that can easily create confusion – we have chosen to begin our exploration by introducing a ramified structure (Chap. 1). The strength of a ramified structure resides in its simplicity at every level and in its versatility: only three types of scale are distinguished first, and subtypes are identified afterwards. The adopted grouping is designed to be not © Springer Nature Switzerland AG 2022 C. Suteanu, Scale, https://doi.org/10.1007/978-3-031-15733-2_9

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only theoretically meaningful but also operationally helpful in the framework of practical applications. Distinct as they are, the three groups are branches of the same trunk. The three types of scale are characterized by deep-seated common features, which will be the subject of the next section. Most importantly, they all include some type of transfer between different realms, as we could see in the previous chapters. Those must include, at some point, an abstract space, which is essential to their functioning. Distinctions among groups concern mainly the operations involved in this transfer and the nature of subsequent processes that are involved in the application of scale. A concise overview of some significant differences and similarities between the scale types is presented in Table 9.1: 1. Striking differences start with the use of units. Scale as size often includes physical units, usually units of distance in space and of duration in time. However, it can also refer to units applied to other relevant variables. In many instances, no units are used at all: this makes it possible for scale as size to use the same label (e.g., “large scale”) for very different features, such as the rock surface of outcrops, the number of individuals considered in a social network, the ecosystem in a forest, and the spatial distribution of matter in the universe. In each case, it is the context that supplies the proper size on which we should focus. In contrast, scale as ratio is unitless, being defined as a ratio between two sizes. Therefore, any spatial units can be applied in order to establish the relationship between elements on a map and those elements in the environment. Measurements in both realms – in the real environment and on the model – surely require some kind of units. However, once the scale value is specified, the presence of units vanishes. A similar situation is encountered when scale as ratio is applied to time: a unitless transformation is used, whether we address changes in the speed at which a recorded sound is played or the pace of change in a computer model. Finally, scale as rank involves no physical units at all, since its outcome lies entirely in abstract space. This is the only scale type for which the application process does not involve the returning from abstract space directly to the material environment. It is the abstract outcome itself that we use when we assess certain aspects of the environment.

Table 9.1  Comparative image of key features of the three types of scale # Scale type 1 Use of units 2 Applicability domains 3 Precision 4 Objectivity 5 Reversibility

Scale as size Physical units or no units Space, time, or other variables Low to high Medium to high Medium to high

Scale as ratio Unitless

Scale as rank Conventional units

Space or time High Very high High

Space, time, or space-time, or no space, no time Medium From very low to very high Low or very low

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2. Not surprisingly, the applicability domains of the three types of scale involve a common feature: all are  – or at least can be  – applied to space and/or time. However, their similarity in this regard stops here. Unlike scale as ratio, scale as size is often applied to other variables too. In its turn, scale as rank can be – and often is – applied to situations in which neither space nor time have to be explicitly implied in the use of scale. For example, it can offer a pollution-level index or address the intensity of pain. However, scale as rank can also be very useful when applied to space, or to time, or to space-time. 3. The three scale groups also differ in terms of precision. Although scale as size is widely applied in science, it does not always have to be precise. It can refer indeed to specific, precise sizes, but it can also be used to indicate a domain (whether in space, or in time, or with reference to another variable) over which some property can be identified, certain processes followed, etc. This is why scale as size is often expressed in terms of certain intervals, such as “on a scale of 10 to 100 kilometers,” or even by identifying units only (“on a scale of millimeters”), or without using anything but one of the qualifiers, “small” or “large,” as mentioned above. Scale as ratio, on the other hand, is characterized by mathematical precision. In its case, rigor is essential, and elaborated methods are applied to correctly determine, for instance, the right scale of various portions of maps, or to find out the right sizes in the environment, based on a map. Finally, since scale as rank is constructed from the beginning based on a coarse-graining principle, with “bins” to be assigned to various quantities, precision is not a requirement in its case. The coarse-graining can never offer the precision of scale as ratio. However, it constraints its outcomes to well-defined intervals. From this point of view, its precision can be considered to be “medium.” 4. In the case of scale, objectivity is not always independent from precision. For case of scale as size, objectivity is usually high: in fact, its limits are set by precision. The wider the limits of the size domains that are addressed, the more room for subjective assessment. The outcomes of this limitation are, however, dependent on the actual field where scale is applied. There are areas of scientific research where the use of scale as size is more objective than in others. While scale as size can benefit from strong objectivity, the fact that measurements are sometimes not directly applied – but rather implied – leaves more room for subjective interpretation, depending on the studied topic. Scale as ratio enjoys an unquestioned high degree of objectivity, due to its functioning principle, which is based on strict, mathematical transformations. While maps can both absorb and exude a significant level of subjectivity, this is usually not happening in relation to scale. Even if scale is involved in subjective aspects of a map, it is the choice of scale, or an incorrect application of scale, that may give rise to the manifestation of subjectivity: it is not the scale as ratio as such that produces or encourages it. Scale as rank, on the other hand, includes a broad spectrum of implementations. Depending on the actual process of transfer from reality to the abstract

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categorization framework, its application can range from the strongly subjective all the way to the strongly objective (as shown in Chap. 6, Fig. 6.4). 5. Depending on the research field, the reversibility of a scale transformation can play an important role in the functioning of scale. In this case, the question we ask is: How reliable and how useful is the process of moving from the realm in which the scale has carried the studied system, back to the material environment, given its particular kind of transformation? The answers offered by the three types of scale are quite different from each other. Scale as size can yield a high level of reversibility, since its labels such as “small” or “large scale,” or “a scale of meters,” can easily be projected to the material environment, where its meanings are then unfolded. In these cases, its low precision may, however, influence the outcome of this transformation reversal, reducing reversibility. When precision is high, the reversibility is high too. Scale as ratio is strikingly different: not only is the reverse transformation possible. It is even part of the normal use of this scale. In fact, once a map is produced, transformations take place in both directions: from the material environment to the map, to find there the corresponding positions of locations, features, etc., and from the map to the material environment, to identify there the elements of interest that lie on the map. On the other hand, investigations can be confined to the world of a map, with no explicit transfers between realms taking place. Such endeavors, often dedicated to a better understanding of certain aspects of reality, implicitly and permanently occur with the other, “real,” realm in mind: in the process of interpretation, the material environment is virtually linked with its map representation. In other words, although transfer operations between realms are not explicitly performed, a permanent hence-and-forth flow is occurring for the map user. Scale reversibility is thus an essential condition for the proper functioning of a map. Finally, scale as rank, which relies on a coarse-graining principle, is not designed to be reversible. When a reversal operation is attempted, it can only be modest. Only large chunks can be identified in the environment: it is as if we looked at it through glasses that only let us distinguish several levels of brightness. Once the infinitely rich nuances of reality are poured into a small number of discrete bins, there is no way back to the initial diversity. Irreversibility is built in the process that lies at the core of scale as rank. More details regarding the factors behind reversibility properties are analyzed below. It is not surprising to see that a concept considered to be fundamental to our understanding of reality is expressed in different forms. What we can find to be remarkable is the fact that these forms of scale, in all scale types, share core properties. The next section discusses outstanding features that the three types of scale have in common. When it comes to uncovering profound meanings of a concept, the clearest view may be offered by a mathematical perspective. This is particularly true in the case of a rich, multifaceted, and ramified concept like scale. A perspective that draws on some basic mathematical notions will allow us to leave out misty details and implementation-­specific aspects: as the passionate mathematicians Avner Ash and

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Robert Gross (2009) note, “if we included every bump on every log we would never see the forest for the trees.” After following the path made possible by the clarity of a “perspective from above,” it is easy to descend into the concreteness of actual scale types and to explore their ways of helping us to see the environment. Therefore, we will address some familiar concepts by looking at their mathematical meaning first.

9.2 Scale and the Guided Cut When the need arises to develop new theories, the basically new step is generally an act or a series of acts of understanding. —David Bohm (1965/1996)

9.2.1 The Selective Approach to the Environment As we have seen in the preceding chapters, selection repeatedly comes up when we analyze the way in which scale operates. In fact, a selection process is always present when we interact with the world around us. There are several explanations for this. An obvious initial reason is a matter of access. From where you are right now and definitely from where I am sitting while writing this chapter, we may not be able to see the blue shades of the Ruwenzori mountains nor the Plato crater on the Moon. We may, however, be able to make the necessary changes that would allow us to see them. In contrast, there are things that we will never see, whatever we do, such as a galaxy that has become unreachable due to the expansion of the universe. Moreover, even an easily accessible and familiar environment can only be selectively perceived. We usually cut a window from the surrounding environment, a segment of a roughly horizontal stripe around us, which is expected to be useful: it may help with our safe movement, our perception of people, the detection of relevant objects such as doors or various obstacles, etc. This author sometimes asks students if they agree with the messages scribbled on the floor and those that are painted on the ceiling, in the hallway from which they have just entered the auditorium. There are no such messages, of course, but the point is to make students aware that usually we do not look up or down: in a building, we usually don’t need to. This would be different, of course, if we were climbing rocks, for instance. On the other hand, even in one and the same environment, our window selection can vary depending on our interest, our goals, etc. For instance, it is one thing to be hiking and passing through a meadow and another to search for something we have dropped in the grass. In most of these instances, we can notice that our choice is meaningfully linked to scale. When we scan the surroundings to detect possible threats from bears, for example, we apply a different scale of observation than we do if we examine the movement of ants or if we watch the clouds.

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Regardless of what we select, there are always criteria that we apply in the selection process. There is no selection without criteria – not even when they are vague or when we are not aware of them. Clearly, the applied criteria are decisive for the outcome of selection. When Voltaire (1763/1875) decided to leave out the innumerable details usually collected by historians who mainly described battles and to select instead “the most important and best documented facts” that reflect “the extinction, revival, and progress of the human spirit,” he took history to a different level. He shifted the attention of humanity from “unique incidents” to the “spirit of nations” (Cassirer 1951/2009): a major change in scale.

9.2.2 Introducing the Guided Cut There are several reasons why our selective approach to the environment is important to an understanding of scale, on one hand, and an understanding of the role of scale for our understanding of the environment, on the other hand: (i) a selection process is unavoidably present when we direct our attention to the environment, (ii) scale is involved in this selection process, and (iii) in its turn, the selection process is involved in the operating mode of scale. It is time to better define the selection process and to identify some of its most relevant properties. Let us start with an example – a set of photographs we took on a trip in a faraway country. The easiness of taking pictures and the abundance of novelty in the visited land encourage us to take plenty of photos. So many photos, in fact, that it is impossible to show them all when we return home. We must make a selection. Our selection must rely on some criterion, whether or not we choose to explicitly define it. As a matter of fact, we can apply a variety of criteria when we pick the photos. What we choose may depend on our goals, on our previous experience, on the occasion, and on the audience (a family gathering, a party with a certain group of friends, a travel presentation made to colleagues at work, etc.). Each selection criterion may lead to a different set of photos. Still, the various sets of pictures chosen according to distinct criteria may have a number of photos in common. For instance, some photos may be included both in the set called “children” and in the set called “people interacting with nature.” On the other hand, we can apply more than a single criterion to our selection. For instance, “fruit” and “people interacting with nature” might lead to photos in which people are cultivating fruit. A set of photos selected according to a certain criterion, or a set of criteria, will be called a guided cut. Actually, we can use a more precise language to focus on the guided cut (not to be confused with the “Dedekind cut”). Before taking this instrument, the cut, out there in the field, and applying it in our exploration, we will briefly present its main characteristics. Given a set A of n elements and using a set of criteria S, we perform a selection of elements into the initially empty set B by applying the selection operator or guided cut, K:

B  K S  A



(9.1)

9.2  Scale and the Guided Cut

283

Equation 9.1 reads: set B is the result of applying the guided cut operator KS to choose elements of A according to criteria S. The resulting set B, which has m elements, with m ≤ n, is the guided cut of A through criteria S. In all circumstances, a guided cut only makes a selection of elements from a given set (Fig. 9.1); it does not modify any of the elements. The magnitude k of a guided cut applied to a finite set is given by the ratio between the number n of elements in the initial set (i.e., its “cardinality”) and the number m of elements in the resulting set: k  K S  A 



n m

(9.2)

Therefore, the magnitude of a guided cut must always be k ≥ 1. The situation in which k = 1 represents the “null” or “do nothing” guided cut. For example, the initial set A may consist of the natural numbers from 1 to 20, and our selection criterion S can be “numbers that are perfect squares.” In this case, B contains the elements {1, 4, 9, 16}, and the magnitude of the cut is 20/4  =  5. Another cut can be produced on the same set A by applying another criterion S, referring to “prime numbers.” In this case, B = {1, 2, 3, 5, 7, 11, 13, 17, 19}, and the magnitude of the cut is 20/9 = 2.22…. If the original set A is an infinite set, the outcome of the cut, set B, can be either finite or infinite. For instance, A could be the set of natural numbers, and the guided cut B could be the set of natural numbers smaller than 5 or, alternatively, the set of

A

B2

B1

B3

B4

Fig. 9.1  An initial set (A) and four different guided cuts (B1 to B4), illustrating different criteria that can be applied: empty or filled contours, rectangular shapes, or even orientation: in B4, we select all figures having at least one side parallel to the horizontal direction

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prime numbers. As we will see below, the situation in which the cut is applied to an infinite set is particularly relevant. A guided cut applied to a set comprising an infinite number of elements is called a guided supercut. Successive cuts can be produced starting from the same set. The order in which we apply the cuts does not make a difference: given a set A and two cut criteria S1 and S2:

K S1  K S 2  A   K S 2  K S1  A 



(9.3)

In other words, the selection operation is commutative. For instance, starting from our previous example, the set A of natural numbers from 1 to 20, we can pick the prime numbers first and choose among them those that are perfect squares, ending up with B = {1}. Alternatively, we can choose perfect squares first, and select the prime numbers among them, and the result is the same. If we apply a cut to a set, and then apply the same cut to the outcome of the initial cut, and again to the outcome of that cut, etc., nothing changes after the first cut. In other words, the guided cut is an idempotent operator, i.e., an operator for which only the first application counts:

K S  K S  ( K S ( A )   K S  A 



(9.4)

Other examples of idempotent operators consist of taking the absolute value of a number or projecting a line segment on a line with a different orientation than the initial line segment. It goes without saying that the selection criteria set S can include more than one criterion; the order in which criteria are applied does not have to be specified, since the guided cut operation is commutative. In other words, S is indeed a set of criteria: the order of elements in the set (in this case, the elements in the set are selection criteria) does not count. Although the guided cut is explained here with the help of numbers, we are introducing it with the aim of releasing it in the real world. It is there, in the material environment, that it demonstrates its value. In fact, not only does the guided cut play a vital role in our understanding of the environment: the guided cut is one of the core features of all the three types of scale. For these reasons, it helps us to circumscribe the general concept of scale.

9.2.3 The Guided Cut and the Environment When confronted with a new theory or method, it is natural to compare it and perhaps (mistakenly) identify it with something old and well-established. In particular, previous theories or methods can work as powerful magnets that end by attracting anything that comes close to their space of influence, blurring differences. —Luciano Floridi and Jeff Sanders (2004)

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Arriving at the core of scale takes more than defining and applying the guided cut. We are moving forward step by step, building the necessary framework, and advancing towards a general concept of scale. In this process, we analyze the way in which the introduced elements operate, both in a theoretical space and in the real-world space where they engage with the environment. Our introductory example for the cut operator referred to collections of photographs, not to numbers. In fact, a selection can – and does – also take place when we observe and represent parts of the environment. Here again, whether consciously or not, we always apply some criteria in our selection, when we investigate the environment. As shown above, a selection based on a set of criteria defines the guided cut: when we perceive and interact with our environment, we apply guided cuts. Guided cuts applied to the real-world environment enjoy specific characteristics, which distinguish them from those used in the case of mathematical objects like sets of numbers. As discussed in Chap. 6, when we perform our selection in the environment, we do not refer to objectively and uniquely distinguishable “objects.” There are innumerable ways of identifying objects in the natural environment: we can think of a leaf in a tree, the branch, the whole tree, a group of trees, the whole forest, the valley, etc. Moreover, whether limited or unlimited in number, each of the possible real-world “objects” can be characterized from many different points of view: in fact, an unlimited number of perspectives may be applied to the chosen segment of the environment. It is on all of these  – objects and perspectives  – that we must apply the guided cut when we refer to the environment. In the real-world environment, performing a guided cut implies the selection of “what,” as well as the selection of “how.” With his background in mathematics and his focus on logic, Edmund Husserl took groundbreaking steps regarding the endeavor in which we view and understand reality. In his Logical Investigations (1913/2000), he highlights the fact that our attention is directed not only to objects but always to objects in terms of certain aspects. In this sense, Sartre (1943/2001) points out that when we consider an object and we access some of its possible aspects or “profiles,” even one and the same profile can be considered from different points of view. We can notice that these ideas are incorporated in the concept of guided cut applied to the environment. They also resonate with the numerous instances, encountered in the previous chapters, in which we mention the drastic selection from the environment that we are compelled to apply when we explore our world. Our choice is made from an infinity of possible “elements” and an infinity of possible “aspects,” seen from various possible points of view. According to the terms introduced in the preceding subsection, we can state that a guided cut applied to the real-world environment is always a guided supercut. As expected, in the real-world environment, a guided cut is less sharply defined than in the case of mathematical objects. For instance, some elements are more clearly distinguishable than others. It has already become obvious by now that the guided cut is directly related to the system image, which was presented in Chap. 1. We can be now more specific with respect to the system image and state that it is

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always a selection from the accessible amount of information about the studied part of the environment. This selection is performed according to a set of criteria: the system image is a guided cut; since the system image refers to the natural environment, its selection is, actually, a guided supercut. We can also view our discussion on the process of abstraction from Chap. 1 in the light of the concept of guided cut. An important conclusion we can draw is that there is no abstraction without a guided cut. On the other hand, as we will see below, the guided cut has strong links to the operation of representation, without being exhausted by it, and both are intrinsically linked to the heart of the concept of scale.

9.2.4 The Guided Cut and the Logical Field The above sections on the guided cut are useful steps in our analysis of the concept of scale, especially in view of our endeavor of converging on a consistent view that would be applicable to all its forms studied here. It would be useful at this point to highlight the main stages that we will apply to reach this goal. First, we underline the fact that when we interact with reality, we select certain elements that we take in consideration. We never address “the whole” of reality, or of the environment, regardless of how we define them. Second, the selection of elements occurs according to certain criteria. It is not random. Third, the selection criteria are the result of acts of choice, even when choices are not perceived as such. In fact, we often make choices that we perceive as self-understood, as if no serious alternatives existed (Jaspers 1983). One way or another, the selection is not “given.” Nor is it uniformly defined for all individuals and all situations. Selection criteria depend on many factors, ranging from the motivation of the cut and the observer’s previous experience, all the way to moral choices. Most importantly, the selection not refer not only to elements but also to “aspects” of those elements, as clearly established in Husserl’s Logical Investigations (1913/2000). These aspects are approached from certain perspectives, and aspects and perspectives are often correlated with each other. There are perspectives that reveal certain aspects, while there are aspects that are uncovered, which are conducive to the application and enhancing of certain perspectives. In this context, we can already see scale as size moving closer to our focus and offering criteria for guided cuts. Also, based on our experience with the three types of scale, we can already tell that guided cuts are involved in all three of these types. While approaching the key idea of representation, we can benefit from the valuable support of the concept of logical field, introduced and applied by Constantin Noica (1986/2011). In the realm of logic, where it was introduced by the Romanian scholar, the logical field is defined by the situation in which the whole can be found in the part, and not only the part in the whole (the concept must not be confused with the logical field used in data science). Noica illustrates it with contrasting examples.

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For instance, in the case of a clockwork, a part is incorporated in the whole, but it never “is” the whole. In contrast, one can see from a set of books from someone’s library what that library is about. Each set of books sampled from the whole is an interpretation of the whole. The part is a reflection of the whole. This would not be the case with a pile of books arbitrarily heaped together. In logic, this concept plays a crucial role in what we perceive as “logical” situations. It is also successfully applied to the knowledge framework developed in science, where, as Noica showed, it holds together and conveys coherence to parts that would otherwise be limited to mere statistical collections. At the same time, however, the concept is fruitful in relation to the operation of selection, which we find at work in the processes involved in scale. Considered in the light of the guided cut, the logical field helps us on our way forward. On one hand, it has a positive side: it reveals a vital feature of the selection process to be applied to our investigations on the environment. On the other hand, it points to a negative side, by raising a question that may seem to expose a major problem in our exploration of the environment. The positive side consists of the insight it offers regarding the selection process, the guided cut. Not every selection is, or can be, fruitful. We know this very well, of course, but the logical field applied to a guided cut can better circumscribe the conditions that must be considered when the cut is applied. A special value is assigned to the capability of the guided cut to extract a particular “part”: one in which the “whole” can be found, from the perspective chosen by the observer. It is the extent to which this can happen that makes the result of a guided cut more or less valuable. This has significant implications for the way in which we approach our research on the environment, starting with the selection and definition of the system image. The logical field emphasizes the value of the “guiding” act that is involved in the guided cut. The concept of representation, which will be addressed below, will benefit from this observation. The negative side refers to an alarming question. How could a guided cut be performed in order to capture the “whole” in an infinitesimal part of it? How could a logical field even exist under these circumstances? Can we apply a guided cut, informed by the concept of logical field, in the real-world environment? Can we ever capture the “whole” regardless of what we want the whole to be? The answer to these questions is “yes.” Not only can we access “the part” using a guided cut, in order to bring over and reveal “the whole.” Looking at the part is the only way for us to learn about the whole. All we can ever capture from the investigated system identified in reality is a part of it: some of its aspects, from some perspectives. The question is thus not whether or not we must rely on the part to study the whole. What we ask concerns the “what” and the “how” that we should involve in our selection from the investigated system, in order to understand it better. While not every guided cut can be successful, deciding upon it from the logical field perspective can support a useful and responsible selection.

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9.3 Scale Involves Mapping 9.3.1 Key Elements of Mapping The statement in Sect. 9.3 might seem more natural if we reverse the order of the words. However, while mapping surely involves a certain type of scale, all three types of scale imply a mapping operation. We can think about mapping in a strictly confined intellectual space, as it is used in one of the many fields in which it operates: cartography, logic, mathematics, etc. We could also relate the concept of mapping to one of its applications, such as brain mapping, gene mapping, or data mapping, to list just a few. Since all of these instances of mapping are loaded with many domain-specific aspects of their definition and application, it is tempting to conclude that there is no mapping “in a general sense.” Nevertheless, we can indeed talk about mapping in a general sense, and it is worth doing so, especially when we focus on the core of the three types of scale, although each of them has its distinct forms of manifestation. In a general sense, when applied to sets, mapping is defined as a transformation, which establishes a correspondence between the elements of one set and those of another set. Given two sets A and B, a mapping M can be written as: M



M : A → B or A → B

(9.5)

An element of set A is called an object, while its corresponding element in set B is its image. The ensemble of all elements in set A is the domain of M, while the elements in B are its codomain. The arrow symbolizing the mapping can be read in two ways: either A onto B or A into B. This sounds like a small distinction, but it is a meaningful one, and it can be useful when mapping is addressed in relation to practical problems. If all the elements of B are an image of at least an object in A, this is an “onto” mapping. If there are elements in B that are not an image of any object, the relationship is an “into” mapping (Rosen 1983). A defining condition of mapping is that any object – any element of A – can have a single image in B, and not more. This key rule of mapping has major consequences in the case of scale: it allows us to unambiguously establish the outcome of a scale-based transformation. We do not operate in a room full of mirrors. The fact that certain geographical maps – world maps, for instance – show the same geographical areas in two places (at both ends of the map, for instance) does not contradict this rule: the representation of those areas is still unambiguous, and their relation to other areas on the map does not change, even if an area is shown twice on the same map. Although the concept of mapping is often used to denote a function, its meaning can be much broader, and we are using it indeed in its more general sense here. For instance, we do not constrain the domain and the codomain of M to sets of numbers, but rather leave them open to a wider diversity of “things” that are handled by the mapping. In fact, mapping has vast areas of application, many of which are extremely

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different from each other. For instance, we can think about aspects of language or about various forms of reasoning. Or we can consider the mapping of geometry into algebra, as it was used by David Hilbert to check the consistency of the Euclidian postulates (Nagel and Newman 2001). Mapping can be both very widely applicable and very precise.

9.3.2 Relations in Mapping and the Three Types of Scale The property of interest in this case concerns the type of correspondence determined by the mapping. Several situations are of particular relevance to our discussion, since they provide insights into the scale types considered in this book. A graphical illustration of some of these situations is presented in Fig. 9.2. A mapping is one-to-one or injective mapping of A into B if no element of B is the image of more than one object. This kind of mapping implies that there may be elements of B that are not an image (Fig. 9.2b). In other words, an injective mapping is “into.” On the other hand, if we change an injective mapping so that no element in B is left without a corresponding object, i.e., if the mapping is “onto,” then every object has exactly one image, and every image corresponds to exactly one object: the mapping is called bijective. A bijective mapping must thus fulfill two conditions: it must be injective and “onto.” For instance, in Fig.  9.2b, the mapping becomes bijective if we remove the element “r” from set B.

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Fig. 9.2  Four significant situations regarding mapping: (a) not a mapping, since an object has more than one image; (b) injective mapping, where it is not allowed for two or more objects to have the same image; (c) many-to-one mapping, where multiple objects can have the same image, and images without an object are available; (d) many-to-one mapping in which no image is object-less (surjective mapping)

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If one or more elements in B can be the image of more than one object, the mapping is many-to-one. We should note again that in this case certain elements in B may not be an image of any object at all (Fig. 9.2c). If, on the other hand, all elements in B must be the image of at least one object, the mapping is an A onto B, and it is called a surjective mapping (Fig. 9.2d). Since scale starts from a domain that is part of the environment, the “elements” in the domain of mapping are not automatically defined. It is the observer who must establish what they are: not to recognize them, since elements are not “given,” but to decide upon them. One must make informed choices, in the same way in which the system of interest was defined (Chap. 1) and in direct relation to the actual outlining of the system. The implicit role of scale can already be perceived at this level: the choice of elements depends on the choice of scale. Thereby, scale as size is present from the very beginning of a process of investigation. Even when the scale is chosen, the mapping elements are not crystallized yet: one must first select specific aspects of the environment to be subject to the transformations involved by scale. Only after making a clear choice regarding scale and the elements to be involved can the scale-based transformation begin to operate. For these reasons, in the case of scale, the mapping that is actually involved is a partial mapping. A partial mapping from set A to set B is a mapping for which only a subset of the elements in A is made to correspond to elements in B. This is precisely what happens when we apply scale to the real-world environment. As we know, a drastic selection of aspects of the environment (out of its infinitely many) takes place when we represent the environment, i.e., when we transfer it into abstract space, in any way and for any purpose that might be, whether scientifically or artistically, for instance. This highly selective approach is applied in the case of scale too. We can never involve in the domain of mapping “all” the elements in the environment, to establish their corresponding image in the codomain. Therefore, rigorously speaking, it is partial mapping that we apply within the scale process. We can, however, still use the word “mapping” if we keep in mind that every time we apply scale, regardless of its type, only certain elements are picked from the environment. It will be thus implied that it is to this, “preselected” set of the domain, that we apply the transformation. We will see in the next sections that the selection process can be specified in a more precise and operatively more useful way. When we consider the three types of scale, each of them branching out into subtypes, we are struck by the overarching nature of the concept of mapping. In fact, all the three types involve mapping as a primary operation to be performed. In some cases, mapping can be easily recognized at the core stage in the application of scale. In other cases, the presence of mapping is less palpable. In spite of their common roots in the mapping concept, the scale types seem to point in quite different directions. We will begin with the scale type that offers the most straightforward identification of mapping, scale as ratio. If we start from the preselected set of elements as discussed above, scale as ratio can be recognized as a one-to-one mapping. Moreover, it is an A onto B mapping. Therefore, scale as ratio corresponds to a bijection. In fact, this is a key condition for scale as ratio, as well as for maps and

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scale models in general: every (preselected) element in the environment has its counterpart on the map, and every element on the map has a meaning that relates it to an element in the environment. Scale as ratio plays a major role in the transfer, which is, or should be, a thoroughly bijective transfer. We must also remember that the nature of the mapping codomain is radically different from the one of the mapping domain: the image of elements has a symbolic form. Scale as rank is different. Its codomain consists of a set of sharply delimited boxes, and each of these boxes is an image designed to collect a range of objects. Therefore, regardless of the element preselection, scale as rank involves many-to-­ one mapping. The allocation of objects to their corresponding image boxes is performed in a static or in a dynamic manner. In the case of static assigning, a set of elements (objects) is given, and the transformation is applied to all of them. The result is a coarse-grained image, which shows how the objects are distributed among the scale’s categories. Dynamic assigning is applied when, from time to time, an individual element must be projected into the image framework, and a category is assigned to it. For example, wind speed data are available, and the result of scale as rank is the category of an individual hurricane at some point in time. In the first case, there is no guarantee that all the boxes will contain one or more elements; in the second case, all the boxes except one (the image of the assessed object) are empty. Not all the elements in the codomain are an image of an object. Therefore, scale as rank corresponds to a many-to-one mapping of the type “into.” Finally, scale as size has its own special flavor too. When scale as size does not work towards the goal of obtaining sharply defined numerical values, the elements – both in the domain and in the codomain – have fuzzy boundaries. This being said, the condition for scale as size to function is that a reliable correspondence exists between the studied processes and structures in the environment and those in the abstract space where scale as size presents its outcomes. This condition implies that scale as size involves bijective mapping, but both its objects and its images are very different from those of scale as ratio.

9.3.3 Reversibility in Mapping and in Scale Transformations The mentioned relationships between objects and images which characterize mapping have implications regarding possible subsequent transformations. If we reverse the direction of the transformation “A to B,” we produce an inverse mapping. The inverse mapping of M: A → B is denoted by M −1, and it is written as: M 1

B  A 

(9.6)

From the description of the possible relations between the domain and the codomain, which characterize a certain mapping operation, we can see that not all can sustain inverse mapping. First, objects and images must stand in a one-to-one relation, and second, the operation must be an onto mapping. In brief, a mapping has an inverse if it is bijective. What are the implications for scale?

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We have seen that scale as ratio involves bijective mapping. The mentioned consistent relation between objects and their images assigns to the operation of scale the important role of a safe transfer in both directions: from the environment to the map and from the map to the environment. We must keep in mind that mapping has indeed an inverse in this case, but only within the boundaries of the mentioned element preselection: it is the latter that imposes reversibility on an operation that attempts to capture real-world features in an abstract model. If we were to consider any part of the environment without including the stage of preselection, without brutally discarding all but an infinitely small fraction of aspects of the environment, i.e., without a guided supercut, an inverse mapping would be unthinkable. However, within the limits of the preselected objects and their images, which are set aside and “frozen” for the operation of transformation governed by scale, the mapping is lossless. Consequently, it can be produced in both directions, repeatedly, without the danger of a growing distortion. In its turn, scale as size involves bijective mapping too. However, as shown in Sect. 9.3.2, if its precision is low, i.e., if there is no sharp delimitation of its element boundaries, inverse mapping is less reliable than in the case of scale as ratio. Therefore, it is not recommended to move repeatedly between its realms and to reverse its domain and codomain: while an inverse of an inverse of an inverse, etc. would not affect the outcome of a reversible mapping using like scale as ratio, this is not the case with scale as size when its precision is low. For instance, it is often the case that approximations intervene when we decide what we call small scale, and what we call medium scale, and where the limit between those would be. Therefore, repeated inverse operations may easily lead to a drift that could be difficult to control. Finally, scale as rank does not involve bijective mapping, and reverse mapping is out of the question. The massive coarse-graining, which is a key factor for its functioning, is the main reason for this unavoidable irreversibility.

9.4 Scale Involves Representation 9.4.1 Morphism and the Choice in Representation Although mapping is one of its main defining traits, scale does not involve just any kind of mapping. If we go deeper, we notice the presence of a particular kind of transformation: representation. It is the concept of morphism that will pave the way for us in this direction. Like many other mathematical concepts, morphism takes different forms in various domains of mathematics, whether linear algebra, topology, category theory, etc. Here, we are interested in its most general meaning. A morphism represents a mapping that makes the relations among elements in one realm correspond to those connecting the elements in the other realm of the mapping. Morphism is a structure-­ preserving mapping.

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However, there is a specific aspect of morphism that is of utmost importance in our case. According to its mathematical meaning, morphism is mapping from a set A into a set B that “captures at least part of the essential nature” of the mapped set A in its image B (Ash and Gross 2009). A representation is a morphism from a “source object” to a “target object,” which is supposed to be well understood. Consequently, we expect to learn about the source object based on our understanding of the target object, on one hand, and on our knowledge about the representation that is involved, on the other hand. This may seem to pose the learning problem in the wrong order. Is morphism, seen in this light, really telling us that we actually learn about an object by studying its projection, like its shadow on the wall, and not the object itself? Don’t we actually learn more from the object than from a representation? Before addressing this question, we should carefully analyze some key aspects of the process of representation. While representation is a mathematical concept that reveals its power in our inquiry, we will address it here in a general sense, without entering the ramified world of representation theory (the latter is a broad field of mathematics applied to abstract algebraic structures, focusing mainly – as representation theory of groups – on symmetries of vector spaces). First, it is worth paying attention to the defining trait of representation: its capability of grasping something of the essential nature of the object. We can already see why this is important here. When we discuss properties of representation, in light of the previous chapters, we can recognize properties of the way in which the three types of scale operate. One might suspect that the mentioned representation’s aptitude of absorbing some highly significant aspects of the object, and transferring them to the object’s image, is only valid in the abstract realms of mathematics. This is not the case. This property of representation has been an important issue in various approaches regarding the way in which we understand the environment. For instance, a major figure of the enlightenment, J. G. Herder (1772/1986), not only recognizes this property as a basic condition of representation: he is even capable of zooming into it, establishing a way of “checking” when this condition is fulfilled. In fact, stated outside of the domain of mathematics, the idea of seizing essential properties of an object is rather vague. How do we know what is essential and what is not? Herder has an answer to this question. He states that in spite of picking very little from the object, an image that represents an object should pass a key test: the representation must occur in such a way that the object is recognized in the image. The act of recognition should therefore play a fundamental part in the process of representation. This observation is highly pertinent in relation to all three types of scale. While it can be more easily identified in the case of scale as size and scale as ratio, it is also valid in applications of scale as rank. Let us consider a simple example. When we analyze various kinds of buildings that were damaged in different ways by a tornado and we use them in order to establish the proper bin to assess the tornado intensity, there is, in theory, an endless number of aspects we must choose from, which makes the task seem impossible. However, the aspects we capture, among all those that would be available, are consistently those that we would be able to recognize. In fact, by applying the defining traits for structure damage, distinct observers usually converge on the same

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aspects to be retained, which contribute to the consistency of the assessment process. This conclusion stands in flagrant contrast to the next observation. Second, we should be aware of the fact that the process of representation is highly selective. After all, only a small fraction of aspects of the object can be taken in. On the other hand, there is no single way of choosing those particular aspects. Kant (1787/2007) makes it clear that an infinity of possible representations may be available and it is under these circumstances that we must begin our endeavor of understanding. We are thus not handed over an object along with a method to precisely determine what properties of the object must be selected. Whether we realize it or not, there are always choices to make. In his highly influential work on symbolic forms, Cassirer (1929/2020) highlights the distinction between the objects of mathematics and those studied in the natural sciences from the point of view of this necessary choice. He points out that the all-powerful structure-preserving mapping found in mathematics is nowhere to be seen in the real world. This makes the act of representation particularly challenging. We admit today that even if we broadly distinguish different ways of producing a representation (the so-called representation styles), rather than different actual representations, the possible list of styles is still endless (Frigg and Nguyen 2021). If the above statements are true, we might land into the trap of a paradox. Given that an object allows us to represent it in an infinity of ways, the probability of having the same object being represented in similar ways in different representations must be vanishingly low. In other words, there are premises that push reasoning towards an uncomfortable conclusion. If every single representation is different from all others, this means that we learn different things about the object, depending on the representation that we happen to use. With all of us having different representations in mind, it would be hardly possible to even communicate about the same object. And yet, this is not what happens. What saves us from such a situation? Briefly put, there is a good reason for the above premises to be correct and yet the conclusion to be wrong. The existence of an infinity of possible representations does not mean that all of them are equally probable, nor that they are strikingly different from each other. Not all the possible representations fulfill their representative role. Callender and Cohen (2006) argue that there are “fundamental representations,” a privileged set of core representations, which have the power of representing the object. On the other hand, even if those privileged representations are different from each other, many of them are highly similar to each other and distinct from the rest of possible representations. For example, if various individuals are asked to represent a tree, although innumerable representations are possible for a tree, the majority of them will draw a trunk with branches or the contours of a crown. Very few, if any, will show the tree as a shape seen from above or provide a description of the taste of its bark. This also explains Herder’s accurate observation from over 200  years ago, concerning the condition for a representation to be recognizable. Let us add right away that similarity between all the representations that are comprised in a set of representations of higher relevance, as mentioned above, is conceptually not the same with similarity between the representation and the represented

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object. The latter has long been considered a characteristic feature of representation, from Plato’s Republic all the way to these days – when Giere (2004), for instance, who focuses on this similarity in the fine arts, as well as in science. We will steer away from the tendency to impose such a condition of similarity to representation, since it is a condition that would be fulfilled in some forms of scale, but not in others. The fact that certain representations have a much higher relevance than others, along with the property of those more relevant representations to be similar to each other, greatly reduces the variability of actual representations that are produced for a given object. And yet again, when we refer to the material environment, the number and the diversity of aspects that could be selected in a representation are often very large. In fact, we are rarely given tasks as straightforward as the one of representing a tree. In many real situations, we cannot simply distinguish a unique set of fundamental representations. How can our representations still converge towards results that are useful in our communication and to our understanding of the environment? This is where another trait of representations steps in: intentionality. Representations are highly dependent on the intentions, on the objectives of the observer (Frigg and Nguyen 2021). It is intentionality that helps us to distinguish indeed fundamental representations. Different objectives lead to different such sets of representations. Our choice among possible sets is based on our intentions, on the aspects of the environment to which we dedicate our inquiry. We can now recognize that this loop is closed by connecting us back to the concept of guided cut introduced in the beginning. It is only now that we can finally leave behind the basic questions and tensions raised by ambiguities, which are predictable, as the consequence of confrontations between the world of mathematics and the material environment.

9.4.2 Representation and Our Understanding of the Environment Representations are much more difficult to treat rigorously when they apply to objects in the real-world environment, than to mathematical objects. Clearly, it is in the former that we are interested here. Even if this is the case, it is important to keep in mind the abstract nature of morphism: we are talking about “relations down here” matching “relations up there” and, more importantly, about the structure of relations being the same in the two domains (Nagel and Newman 2001). Beyond the abstract character of relations, when it comes to scale, there are other dimensions of abstractness too. Indeed, abstract relations can refer, and often do refer, in their turn to abstract properties of the elements involved. The extent to which this is the case varies from one scale type to another, but the abstract essence of morphism is deeply embedded in the substance of each type of scale. Regardless of these differences, the main goal of the process involved in the various types of scale is to produce a shift from the objects of interest, which exist in the environment, to another realm, after selecting only some of their features and

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leaving everything else behind. The shift does thus not consist of a point-by-point change of the original objects: it includes an essential phase of choosing only a tiny, infinitely small fraction of what the objects are. This almost inconceivably drastic selection is one of the reasons for the differences between the types of scale, which apply specific rules to the selection processes. By taking into consideration the steps taken up to this point, we can see that the shift that must be performed is, in fact, a guided cut – to be more precise, a guided supercut, as defined in Sect. 9.2. In other words, the guided cut is a particular type of representation. Moreover, the guided cut must fulfill the conditions of a logical field, by selecting the part that can “stand for the whole”, and it must do that in a way that would make the whole recognizable in the cut. All scale types operate in such a way that the shift to another realm enables us to obtain a specific kind of outcome, based on the initial object, i.e., the initially considered part of the environment. This new outcome is meant to help us to refer to (and learn about) the original object. In other words, the transformations involved in scale are aspects of a representation. A representation can be symbolized as:

  A

(9.7)

which is read as follows: A represents Γ (gamma). The arrow is called a fact of representation. It is different both from Γ, the source of representation, and from A, the target of representation. Equation 9.7 uses letters from different alphabets for the source and the target in order to highlight their radically different nature in the case we are interested in: the one that is involved in scale and in all the forms of scale. The arrow always points from the represented object, or the source, to the target. The situation in which the source and the target are one and the same object is not of immediate interest here. As we know, in contrast to the world of mathematics, where the source and the target can be of the same “kind” of mathematical objects, the real-world context in which we apply scale typically involves two very different types of realm and different kinds of objects. The arrow highlights the irreversible character of the representation: A represents Γ, and not the other way around. This is the so-called property of directionality, a required feature of representation (Frigg and Nguyen 2021) and, in particular, of the guided cut. It is now time to return to the question asked above: How can it make sense to study what seemed to be the shadow of the object, rather than the object itself, in order to learn about the real-world environment? The representation process is a transformation performed with the goal of understanding the object. It goes without saying that the representation plays a critical role in our understanding of reality. It may seem surprising that according to the meaning of representation developed in mathematics, it is the target that we know better: we use A in order to better understand Γ. One may expect any representation to start from a thorough knowledge of the object to be represented: surely such firm knowledge should be a necessary condition for a representation to be generated in the first place. If we produce a

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painting of an object, we are expected to learn about the object first, to get to know it, before representing it. At first sight, we would assume that we learn more about a tree by observing it ourselves, by touching it, by smelling its leaves, etc. than by looking at a painting. After all, it is a carefully observed object that we normally try to represent in some way, with – unavoidably – imperfect results. We are acutely aware of the fact that a representation, no matter how carefully it is designed and how successfully it is completed, can never capture the actual object. How can we rely on the representation then in order to better understand the object itself? The fact is, however, that we don’t know anything and we know our representation. The representation is our construct. It is understandable by design. Most importantly, the representation is exhaustible, while the real object is not. Multiple, innumerably many representations are possible for real objects, for the real environment, but no number of representations, no matter how “good” they are, can ever be enough to capture the real-world object. Therefore, a representation is meant to offer us a safe access – albeit a limited access – to the real-world object. There is, however, a more powerful argument regarding the idea that we should learn from the representation rather than from the object itself: representations are the only way of learning about the object. Whether or not we call them as such, representations are everything we can obtain from the object: not surprisingly, we reach the same conclusion that we found in the case of the guided cut. It is from representations that we start, and we rely on them when we construct novel representations. Since we are interested here in representation from the point of view of scale, rather than in the processes of cognition in general, we will not move our discussion into the latter’s realm. We know now that the functioning of scale does not just incorporate representations: transformations involved by scale use a specific type of representations, the one corresponding to a guided cut.

9.4.3 Scale and the Modalities of Representation We mentioned above that a representation of an object – a part of the environment – leads to an outcome that depends on the source of representation (the object, symbolized in Eq.  9.7 by Γ) and on the fact of representation, the transformation (symbolized by the arrow). When we consider a transformation, we must distinguish the “what” (the selection of aspects to be represented) from the “how” (the way in which these aspects are represented). We focused above on the role of the choice regarding the aspects that are subject to the transformation. The previous sections have prepared us for the fact that specifying this choice does not completely describe the representation: there is not just a single way of applying the representation. The arrow in Eq. 9.7 is not as neutral as it seems: it carries an algorithmic load, which may have a decisive effect on the outcome. While this is relevant for representation in general, it is particularly significant in the case of scale.

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The two operations, aspect selection and transformation, are not always seen as distinct phases in the process of representation. They can be, and often are, lumped together, in one and the same step. Although this may make their identities less distinguishable, both features of representation, the “what” and the “how,” must still be explicitly identified and characterized. We can notice that the “how” is related to the so-called style in representation mentioned in Sect. 9.4.1. Style, however, refers to the selection process, the “what,” as well. Therefore, instead of only looking at the amalgamated operations as presented in style, we will consider the “how” in separation from the selection task. We will call modalities of representation the representation-specific ways of operating on the selected elements. We must underline that the concept of modality is only applied here in the abovementioned sense, which does not overlap with other meanings, like the one in semiotics, or with the notion’s semantic aura in other fields. We can now look at the three scale types from the point of view of representation, including its modalities. Scale as size applies the most exigent selection operation of all scale types. The selection is so brief and clean that it becomes almost unnoticeable. For instance, one can estimate the size of time intervals on which certain social processes are followed. Similarly, the observer can make a rough estimation of the spatial extent over which the physical processes of interest occur or the range for which another kind of variable should be studied. The selection is converted by scale, applying a modality that depends on the scale subtype. In the case of scale as size without units, the selected size is compared to reference sizes, such as the spatial extent or the time interval corresponding to other processes, the spatial size of certain subsystems, etc. The outcome of the comparison decides on the resulting scale label: for example, “large scale.” For scale as size with units, the estimation is more precise, and it may rely on measurement. The comparison occurs between the size in question and the spatial units, the temporal units, etc. used in the study. The outcome is of the type “on a scale of millimeters, etc.” or “a size of n millimeters.” Scale as ratio performs a different kind of selection. It is directed to elements like those selected for the geographical map. The selection of elements leads to the selection of distances (in time or in space). The modality of representation is uniquely characteristic to scale as ratio. It involves a mathematical operation that can be as simple as the application of a ratio, or it can be more elaborated, as is the case of various map projections. In all cases, the goal is one and the same: converting one size to another based on an algorithm. Thereby, scale as ratio performs more than a comparison between distances in the model and distances in the modeled environment: it produces a quantitative transformation in a precisely defined manner. Scale as rank is applied in a very wide diversity of situations, and its modalities of representation reflect this diversity. The selection operation is guided by the objectives of the transformation, as it is typically the case for representation. It can refer, for instance, to YES/NO answers to clear questions asked about observed features in the environment. For example, this is the case of the EF tornado intensity scale, where the observer compares the information from the environment to the situations provided in the scale setup: “Are there uprooted hardwood trees?”, “Are

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parts of the roof of this house missing?”, etc. The answers are further processed following an algorithm, which eventually leads to a tornado intensity category. Similarly, questions can refer to the subjects’ perception regarding a certain situation, a change in the neighborhood, etc.: a succession of YES/NO answers helps the subject to draw a picture based on an ensemble of perceptions. The representation modality can comprise a set of different stages, as in the case of the EF tornado intensity scale, or a one-stage allocation of a set of answers to a certain category, or even simply a coarse-graining operation applied to numerical values, as in the case of the Saffir-Simpson hurricane scale. In each situation, regardless of the modality, the outcome is a place in a hierarchy, a rung on the ladder. Thereby, the results are the expression of a comparison. When an event is subject to scale as rank, the comparison concerns the object (the event) and the existing categories (possibly some on a higher and some on a lower position than the one of the object), including the end-of-scale categories. The object is thereby assessed in relation to these scale bins. We can see, repeatedly, that scale is directly and pervasively implied in the endeavor of our understanding reality. It is a component of the way in which we assess our environment, in which we make sense of it: it is an informational process par excellence. The role of representation in the case of scale – in fact, the role of representation based on a guided cut – underlines its informational nature.

9.4.4 Mapping, Representation, and Scale Although the representation of an object is only a fraction of what we could represent, we learn about the object, the source, from the fact of representation, i.e., the process, and from its outcome, the target. Representation is more than just one way among many others of learning about the environment: as we could see, it is a quite peculiar way of accomplishing the endeavor of understanding. In summary, we found that representation is a particular type of mapping. Therefore, it enjoys the general properties of mapping: we have seen how the properties of mapping are linked to the different types of scale. On the other hand, scale proves to have a highly representative character. This trait of scale confirms – but also goes beyond – the scope of the concept of “partial mapping” discussed above. In fact, not only do we leave out some elements from the source. We leave out virtually all elements of the source, except a few, but those few are of great importance to us, compared to all others. Since the way in which the selection process is performed is of vital importance, what we see at work in the case of scale is not just representation in general. It is a representation of a special kind: a guided supercut. In its turn, scale uses a particular type of representation. Beyond the features addressed above, this type of representation involves one more defining feature, which is at least as fundamental to understanding as mapping and representation. This final property is the topic of the next section.

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9.5 Scale Involves Quantitative Comparison 9.5.1 Comparison and Memory In all its forms and according to all the meanings considered in this book, scale involves comparison. We have seen so far the central role of comparison for each scale type. Comparison is, of course, not limited to the functioning of scale. Comparison is fundamental to human understanding. The role of comparison in thinking – and, in particular, in thinking directed to understanding something new – is so massive that it wouldn’t be possible to open up a substantial discussion on this issue here. It has been amply presented throughout our intellectual history. This began with the founder of logic – Aristotle. Almost two millennia later, Nicolaus Cusanus, another star, one of the brightest, in humanity’s skies, showed that for Aristotle’s logic, all concepts are, in fact, “concepts of comparison” (Cassirer 1929/2020). About four centuries later, Friedrich Schleiermacher – who is widely recognized as the founder of general hermeneutics – found comparative knowledge to be a fundamental approach to knowledge (Schleiermacher 1838/1998). The role of comparison in processes of thinking was addressed, from a variety of angles, by later generations of scholars. It would be pointless, however, to search for the roots of the use of comparison in our history. Comparison is too deep-seated for that. It did not even start with humans. It is present in animals as well. In fact, one may argue that it is impossible to act in the environment, to move, to produce change, without comparison. There is one particular aspect of comparison that must be mentioned here, given its relevance for scale: comparison involves memory. Every act of comparison must entail two (or more) distinct elements to be compared. With very few exceptions, it is always the case that at least one of them has its source in memory. We often compare an object that we have before our eyes with one that we saw earlier (farther or closer in time). Only seldom do we compare, for instance, images of two objects that we perceive simultaneously, next to each other in space. Even then, what we take in is not a flat, static picture as a whole. We rather perform a complex set of quick, jerk-like eye movements, or saccades. Saccades involve complex mechanisms of the eye-brain system. While it is not straightforward to discern memory-related processes on different time scales, they prove to optimize our learning in and about the natural environment (Gibaldi and Banks 2019). Even these situations of apparent simultaneous viewing are, however, exceptions. As a rule, we rely on memory in more obvious ways when we compare. When a leopard is chasing an antelope that is zigzagging to escape, it keeps assessing positions and their change in time, the speed of the prey and its own speed, etc. It compares those with references it already detains about what it is able to do, about patterns learned from experience, etc. An example of comparison – well-known for its fatal outcome due to an alleged inability to make a timely decision – is offered by Buridan’s donkey, which is usually presented in a story about symmetry. The symmetry involved by the two piles of hay, which lie at the same distance to the left and to the right of the donkey, is

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supposed to prevent the animal from choosing, forcing it to starve to death. The story cannot be found in the actual writings of Jean Buridan, but a similar situation had already been brought up, sarcastically, by Aristotle (fourth century BC/1984), who refers to a man having food on one side and drink on the other. Regardless of who is choosing, the relevant issue for us here is the subject’s comparison of the two distances. This comparison is made repeatedly, with the subject looking alternatively to the left and to the right. Every time, one distance can be visually observed, while the other distance is present in memory. The memory function is particularly important in the case of scale, as we have seen in the previous chapters. In brief, in scale as size, the reference term for the comparison – the virtual term, which we hold in our mind – is the extent (the spatial extent or the temporal extent) associated with the studied processes or structures. In scale as ratio, the distance measured in the material environment must be preserved and used as an input in the algorithm that produces, eventually, the scaled version of the environment. In scale as rank, the element to be assessed (which can consist of a single number but also of a set of quantified aspects of the environment, for instance) is memorized, after which it is compared to the reference values of the boundaries characterizing the bins. Only then can a category be assigned to the element in question.

9.5.2 Comparison and Quantity: Looping Back to Mapping It is not only comparison in general that is acting inside the process of scale. After all, comparison may refer to two or more shades of the same color, or to different spatial arrangements of a set of elements, or to different ideas. In the case of scale, however, a specific form of comparison is at work: one that involves quantification. This surely does not come as a surprise. In every type of scale, comparison typically concerns elements of reality, and in each case, there is a point when those aspects of the environment are subject to a transformation that includes quantification. The role of quantitative comparison has been recognized for a long time. In the fifteenth century, when in the climate of the Renaissance the importance of measurement was growing rapidly, Cusanus saw quantification as a cornerstone of comparison. In his De Docta Ignorantia, he maintains that number “is a necessary condition of comparative relation” (Cusanus 1440/1990). Quantitative comparison meant much more than a comparison of an object with other objects. It involved a contact with a whole new universe, which promptly invaded its space: the universe of measurement. The development of a theory of measurement started with Euclid and evolved like a multistaged rocket: successive rocket stages were provided over time by a series of scholars, from Galileo, Newton, and Leibniz to Maxwell, Helmholtz, and Poincaré, all the way to Patrick Suppes and others. What we understand today by “measurement theory” refers, in fact, to mathematical theories of measurement. These are particularly important, because they concern the mathematical

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foundations of measurement as a concept and as a process. The representational theory of measurement (RTM) is currently “the most influential mathematical theory of measurement” (Tal 2020). According to RTM, measurement represents a many-to-one mapping: from a real-world structure, i.e., a set of objects and their relations to each other, to an abstract, numerical structure, consisting of a set of numbers interconnected by mathematical relations. The mapping, which is governed by precise rules, is of the type many-to-one, because multiple objects can end up on the same number when the objects are “of the same size.” If we connect these elements with the key ideas from Sect. 9.3, we notice that (i) mapping can be seen as a central feature that characterizes measurement; (ii) in its turn, measurement plays an important role in most instances of quantitative comparison; and (iii) quantitative comparison is ingrained in the core of scale. Thereby, we can retrieve the presence of mapping, which is critically important for the concept of scale, in a completely different way, right in the heart of scale. It is necessary to specify the quantitative character of comparison in the case of scale, since comparison can operate in many other ways (we can think, for instance, about comparing works of art). Quantitative comparison is at work in every scale type. In scale as size, assigning the label “large scale” to a certain process, for instance, one unavoidably implies a quantitative comparison with some reference quantity; in scale as size with units, the reference is more precisely specified by the units used. Scale as ratio involves the most precise quantitative comparison of all scale types. The way in which scale as ratio operates involves a conversion of sizes, based on a unitless transformation: map distance stays in the same ratio to ground distance regardless of the physical unit that could be applied in the measurement. This physical unit could thus be the distance travelled by light during a certain time interval or the length of a leaf found in the grass on a rainy day. Finally, scale as rank transforms aspects of the environment of many different kinds into an abstract framework where they are not only comparable but also quantitatively comparable. We are able to compare yesterday’s level of pain to today’s and turn pictures of buildings destroyed by a tornado into a number that can be compared to other numbers, assigned to consequences of the tornado produced elsewhere. Aspects of the environment that are complex, vague, and difficult to grasp are assigned their position on a scale where they become quantitatively comparable. At this stage, we can conclude that scale is characterized by an unbreakable link to mapping, as well as to quantity and quantitative reasoning. We can now bring together the outcomes reached in Sects. 9.2, 9.3, 9.4 and 9.5 and look at scale in a new light.

9.6 Converging on One Essence of Scale: A General Definition Starting from the basic properties shared by the three scale types, we can now converge towards a unified view of these three and thus focus on a general concept of scale. The resulting characteristic feature is based on all three types of scale and

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should be designed to capture their essence. It should support the understanding of the concept, as it can be found at work in its various forms. With these requirements in mind, we advance the following general definition of scale: Scale is a conceptual tool used to support the generation of abstract representations of reality that involve quantitative comparison.

According to the meanings of scale addressed in this book, all the scale types and their branching subtypes should be captured by this definition. Four main statements are integrated here: • First, scale is an instrument, a conceptual tool, which can be applied for a certain purpose. • Second, this instrument is meant to be applied to the environment, to reality. • Third, it is applied with the purpose of supporting a specific process: generating abstract representations. • Fourth, these are not merely abstract representations in general: they involve quantitative comparison. The described aspects of scale are represented in Fig. 9.3, which shows them in relation to each other as part of the general outline enunciated above. The four main phases involved in scale shaping are shown to support the conversion process P. The process P has part of the environment as input and a scale-type-specific output in the abstract realm. The output can consist of a unitless label (“large scale”) or a numerical evaluation with units (in the case of scale as size); a map or a higherdimensional representation in its abstract form, which can be subsequently materialized (in the case of scale as ratio); or a set of bins, each with well-defined

Selection process is applied to a set, based on certain criteria.

Guided Cut Logical Field

It is applied to the guided cut in such a way that the selected part “stands for the whole”.

Mapping

It applies additional constraints to the logical field.

Representation

It makes sure that key criteria are met (capturing essence; intentionality). Offers modalities.

Environment

P

Abstract Realm

Quantitative comparison

Fig. 9.3  A simplified diagram symbolizing the described phases involved in the functioning of scale. While the diagram is equally valid for all three types of scale, the scale-specific process is denoted by the gray disk denoted “P”

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boundaries (for scale as rank). The diagram is designed to illustrate all the discussed scale types; however, its components differ from one scale type to another. The quantitative comparison is shown to act on the transfer process P; together with the described phases, it has a significant impact on the outcome of the process in the abstract realm. If the scale concept circumscribed above includes a variety of manifestations of scale, including all scale types, the question of whether or not it may be too inclusive arises. Does it include a concept beyond scale in all its forms? The process of generation of abstract representations does indeed match a wide diversity of processes. The decisive factor that sets a firm limit to the potential aura of meanings is quantitative comparison. It would be possible to find instances when quantitative comparison is applicable to some form of abstract representations without explicitly making use of the concept of scale. One may ask, however, if in such instances, as soon as quantitative comparison is performed on representations, scale is not implicitly present as well. Answers to such questions might have to be considered in confrontation with concrete circumstances. If it would turn out that the scope of the concept would have to be narrowed down, novel constraints would have to be added while carefully preserving the all-­scale-­type-­encompassing property.

9.7 Scale as a Process In the preceding chapters, we have repeatedly addressed scale in terms of a transformation – from one realm to another. Each time, the realm of origin had been a part of our concrete, material environment. The destination realm took various forms, depending on the type of scale and on the ways in which scale was applied. In Sect. 9.4, we showed the transformation corresponding to a representation using letter symbols for the origin and the destination of the operation and an arrow to denote the actual operation (see Eq. 9.7). We could use the same equation with a slightly different meaning for the transformation involved by scale: this time, Γ stands for the selected aspects of the section of environment to be processed, and A represents the result of the operation of applying scale, regardless of the scale type. A could thus be a choice of size (such as “large sale” or “on a scale of kilometers”), a map (or a two- or three-dimensional model), or a certain category among the available ones in a category framework. In this case too, the arrow is a symbol for the transformation. Sometimes, the implementation of mapping is not presented as a process but rather as a set of rules specifying the relationships they stand for. Defining the action of mapping indeed can take the form of a set of rules. If we define such a concept by specifying what it can do, we can thus obtain a useful description that is neither wrong nor formally incomplete. On the other hand, we should not forget that no mapping operation can produce a transformation if it is not applied. By themselves, a set of ingredients and a list of instructions written in a cooking recipe will never turn into soup. To fulfill their role, these transformations must be put to work.

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While we can admittedly bring forward mappings by providing them with a complete “instruction sheet,” we know that they are more than that: they are active processes. When fed with a certain type of input, they produce something new – an output that is specific to their mode of operation. For Kant (1787/2007), this is a pivotal aspect of information handling: one that he uses in order to highlight the distinction between analytic and synthetic propositions. He points out that a proposition like “7 + 5 = 12” may look as if the result were already implied when addition is specified. “The concept of sum of 7 and 5 contains nothing beyond the union of the two numbers into one.” However, “we may analyze the concept of such a possible sum as long as we please, still we shall never discover in it the concept of twelve.” “We must go beyond these concepts” and perform the actual work, using our fingers, or our “intuition,” in order to see, eventually, “the number 12 arise.” The need for such a process is more evident, he shows, when we have to handle large numbers. It is thus useful to notice that when we talk about “mapping,” we should make the distinction between the actual process of mapping and its result. In this sense, distinct interpretations are possible for the arrow in Eq. 9.7. One can consider it isolated, suspended outside the equation – like a graphical sign ready to be picked up from a list of symbols. It can be plugged in, right between the two realms, like Γ and A above, which are then ready to be submitted to the transformation. It is only then and there that the arrow (and thus the scale) would perform their role. This is one way of viewing scale – a tool waiting on the shelf. One can also perceive scale as making more sense when it is seen in action, operating between the two realms, dynamically fulfilling the role assigned to it. We would thus see the arrow as making sense only as playing a part in the equation, rather than as a symbol to be picked from a list and pasted among other symbols. In fact, it is only when we perform the “pasting” of the transformation symbol between the two realms that the actual transformation can begin. It is in this sense that we have addressed scale in this book. This stands in contrast with the view supported by some of the scholarly literature, whether in an explicit or an implicit manner. That is a view that could be likened with a triptych – the kind of static work of art made of three hinged panels, with images painted on each of them; the triptych would show, in this case, the input on the left, the output on the right, and the symbol of the transformation in the middle. However, it is not unusual to see a transformation – such as the one produced by scale – as a process. Such a dynamic view can be even encountered in mathematics, a realm so abstract and so devoid of “time.” The latter is a significant fact, since the flow of time is commonly considered a condition for a process to occur. Language used in mathematics sometimes boldly supports the dynamic view; for instance, in representation theory, a structure-preserving mapping between algebraic structures is called an action. After all, one of the brightest stars in the field of mathematics, Felix Klein, considered transformation to be the actual subject of geometry itself (Gowers 2008). As we could see in the previous chapters, a process can be identified in each case  – of scale as size, scale as ratio, and scale as rank. The actual process that

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characterizes every type of scale, its distinct phases, with the various origins and landing spaces that each phase involves, can effectively help us to distinguish the types of scale from each other. And yet, at the same time, the outstanding presence of a process is a unifying feature of the three categories of scale. As shown above, scale as size relies on the engine of a process of comparison. When a statement is made about “large scale,” it incorporates a comparative appraisal of the size that is meant and a reference value. The comparative evaluation involves the environment, on one hand, and an abstractly conceived reference, on the other hand. Scale as size is informationally operating on the studied environment. Scale as ratio includes several distinct phases. One of them focuses on the real-­ world environment. The other occurs in another material environment, which represents the studied real-world system: a model, a map. In between, there is a transient operation that takes place in abstract space. Both when the model is created and when the model is used, the above phases can be identified. Selected properties from the environment are extracted and processed in an abstract configuration, from where they are transferred, according to the scale, into a map. Conversely, elements can be chosen on a map, processed in abstract space with the help of scale and then projected into the real-world environment. In each case, an informational process is at work. Scale governs the process of change, even if its numerical value is constant. Scale is involved in the process of transition from one realm to the other. One can encounter situations in which scale is seen as a mere number, like the one written close to the edge of the map. Similarly, the scale could be considered to be merely the answer to the question: “What is the scale of this map?” There is no process to be seen there. According to the approach chosen for this book, this view would be misleading. The numerical value of the representative fraction, taken simply as a number, is less than the actual scale. Scale regains its nature when, for instance, we measure a distance between two points on that map and ask ourselves about the distance in the field. It is at that point that the true meaning of the scale can emerge. As long as we only talk about a number, its meaning could be associated with anything. The same number might be found in a myriad of other places, carrying other meanings. When, on the other hand, we take the number along with its meaning, e.g., one unit on the map represents that many units in the field, the transformation is already occurring, and so is the process corresponding to scale. Scale as rank, in its turn, fulfills the phase of severe selection of properties from the environment, and the resulting abstract results are projected into the framework erected in abstract space and consisting of abstract pigeonholes or categories. Both the static and the dynamic views of scale could be correct, equally supported by the same theoretical framework, and equally capable of offering us a useful instrument for our inquiries into reality. The difference between them consists mainly in the attitude, in the intellectual orientation with which the problem of scale is approached: one could metaphorically talk about the “energy” that the concept of scale triggers when the explorer puts it to work. From this point of view, we find the dynamic concept of scale to be more helpful than the static one. We also find that the dynamic view offers the advantage of better highlighting common threads throughout the landscape of scales  – a landscape that is

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remarkably rich. An in-depth approach to certain aspects of scale, based on a dynamic view, seems to be better able to grasp the threads of scale that span various fields. This creates premises for a smoother and more reliable understanding of the concept. However, viewing scale as a process, as an occurring operation, should not be associated in any way with operationalism; the latter considers the meaning of a measured quantity to emerge from the operations that are applied in the measurement process (Tal 2020). In contrast, the choice of the dynamic view resonates with the founder of experimental logic, John Dewey, for whom knowledge itself, our very criteria of meaning, must be related to activity (Dewey 1929/1960). Seeing scale as a process is a consequential decision. It does not directly affect the presented theoretical framework or the resulting meanings – those would not change much if we saw scale as a static device. It is significant, however, for the paths taken to arrive there, the evoked imagery and the connections it generates to other networks of thought. It is a different approach to understanding.

9.8 Scale Is a Living Concept We have no reasons to doubt that scale will continue to develop: its changing theoretical and applicative aspects, in full interaction, may take us to unanticipated places, where new facets of the world will become accessible to our understanding. The further we get and the more we look for deeper insights, the clearer it becomes that our comprehension of where we are and of the path we have covered so far is a crucial premise for moving forward. In fact, the “place to stand,” which Archimedes was asking for in order to move the world, could be our current understanding – unavoidably limited, partly inadequate, but valuable, indispensable. It is based on the work of the past. In contrast, the “lever,” which was also needed in Archimedes’ vision, is very different: it already bears the taste of the future. Whatever the lever is, however, it can only work because the place to stand is here. Our current understanding of scale could be more relevant to the future than we think. The three types of scale analyzed here have been around for a relatively long time. Admittedly, while this speaks to their theoretical significance and their applicative value, it is not a guarantee for their persistence in time. There is, nevertheless, an indication that they will keep existing  – both being and becoming  – in some form, in the future. It is the fact that they are dynamic instruments. They have been growing and changing – reinventing themselves when circumstances changed; taking on new shapes allowing them to thrive in a new world, like the digital environment; and developing additional capabilities, for instance, to effectively handle networks and big data. Tracing the historical development of the various types of scale was not among the goals of this book (another book may have to be written for this purpose). However, even without a dedicated exploration of changes in time, we can see that scale has been evolving. We can find scale as size actively working in the remote past and its branching applications growing in time and in knowledge space. We can

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uncover the emergence of scale as ratio and its continuous metamorphosis, along with the development of maps, including the sparks that are lighting up digital space. We can clearly distinguish the incessant development of scale as rank and the emergence of its various novel forms. Scale is a living concept. Scale is not a helpful remedy to assimilate once and for all, like a pill, and not a powerful but rigid tool, like a hammer, either: it is a trustworthy aid on the journey of understanding, and it is in the space where this interaction occurs that one always has the opportunity to learn. In more ways than one, one stops being a mere “observer.” Considering the typically active role that is assigned to those who apply scale, labeling them as “observers” is even misleading – perhaps words like “investigators” or “explorers” would better characterize them. Professionals from the most different fields, with a wide diversity of backgrounds, at various levels of expertise (Jones and Taylor 2009), found that their understanding of scale has mainly relied on their interaction with reality through various activities and physical movement through the environment. The development of a sense of scale for them was essentially supported by walking, driving, flying, and boating; paying attention to the changes in the landscape as they were getting closer or farther from its various elements; measuring distances in the environment; and finally, interacting with the environment through modeling – which also contributed to the deepening of the meanings of scale, along with studying and creating maps. Our attitude towards scale should not be merely passive. While it is important and useful to know how to make the best use of what the scale can offer us when it is already produced and applied, an active attitude is usually more productive. Mastering the way in which we should decide upon scale, scale range, and their uses, when engaged in a specific research project, can be particularly rewarding. Studies concerning our insights on scale and their change over time, from childhood onwards, found a consistent relationship between the understanding of scale and the understanding of the environment (Jones and Taylor 2009; Taylor and Jones 2009). The very fact of engaging with scale implies incessant learning: this is particularly noticeable when we get involved in some active process, rather than watching a static image. An unavoidable transformation of the learner is taking place – which is the very essence of learning. By engaging with scale, especially by interacting, together, with our environment, we get a better grasp of the ways in which scale helps us to better understand the environment. And by doing so, we implicitly get a deeper understanding of the environment.

9.9 The Open-Ended Scale The surface meaning lies open before us and charms beginners. Yet the depth is amazing, my God, the depth is amazing. —St. Augustine (fourth century/1998)

Scale, in all its forms, implies a comparison, a confrontation with a reference. Scale has a firm connection to the abstract realm, to thought. Scale lives where spirit is alive,

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where search is underway, where understanding grows, sometimes with boundless force. This is not about scientists only. There is the baby who realizes the mystery of a rubber ball, something that is different from herself and reveals distances by its size. There is the musician who, from the very first touch of the strings with the bow, keeps track of ratios among scales in time. There is the thinker, the artist, the writer. They lay secure paths for others to follow, foreseen with safety rails of scale, of perpetual balancing, estimation, and interaction with a reference. There is no scale in lifeless spaces. There is no scale where no one is on an ascending path to comprehend reality, in permanent dialogue with a scale-defining reference. What should our reference be? Should it lie behind us and make us proud when we surpass it? Will it make us feel humbled when we sink for a while and find ourselves lower than we were sometime in the past? Should it always be built anew based on our latest values? Should our reference rely on the loudest voices? Should it follow the largest growth in power, seen as a measure of success? Should it steer towards the brightest and highest orbits that have ever been drawn by human spirit? Should it be instilled by a boundless aspiration to become better human beings? Could “becoming better” mean not just running faster, staying longer without breathing, or handling quickly enormous amounts of information? Scale is open-endedly capable of evolving, transforming itself, and adapting to new circumstances – this is what we have reason to think. Humans, on the other hand, are an open-ended concept – this is what we know. Experiencing incessant change and endlessly following novel paths of transformation are among our defining traits. Our “history of the world compressed in 1  year” metaphor (Chap. 5) might suggest that this view stems from our belief that what started several minutes ago will last forever. And yet, our tendency to look into the future is more than naïve thinking. The truth is that humans can only properly function in a perspective of infinite time. Only in this way can they face and perform countless tasks, big and small, and act in the best way they can while moving through highs and lows, while improving themselves and helping others. When the perspective of infinity becomes clouded, it gets harder to even think of such endeavors. But clouds come and go, as clouds should do, and the perspective of infinite time is always there, somewhere, at the horizon. And yet, the nature of infinity is challenging to grasp. It is one thing to face the infinite in an equation and another to get anywhere near what we would call “understanding” it. We appear to sail between the Scylla of a finitude of the future, which we cannot bear, and the Charybdis of infinity, which we cannot truly assimilate. In these circumstances, our choice is clear. Without hesitation, we prefer the uncertainty of a limitless, albeit mind-boggling, open horizon over the darkening of a closed capsule. How we will actually navigate with this choice being made is hard to tell. It would be unwise to predict what will happen with scale in the future, especially at a point in our history when ample transformations are occurring around us. If, however, we look back, look around, and look out again, from where we stand, there is one thing we can see. For us, right now, scale is open-ended. Karl Jaspers (2003), one of the great companions we have had on our journey, is telling us this: “When I don’t know, I am entitled to hope.”

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References Aristotle (fourth century BC/1984) The complete works of Aristotle, Barnes J (ed.). Princeton University Press, Princeton Ash A, Gross R (2009) Fearless symmetry – exposing the hidden patterns of numbers. Princeton University Press, Princeton Augustine, Saint (fourth century/1998) Confessions. Oxford University Press, New York Bohm D (1965/1996) The special theory of relativity. Routledge, Milton Park Callender C, Cohen J (2006) There is no special problem about scientific representation. Theoria 21(1):67–84 Cassirer E (1929/2020) The philosophy of symbolic forms. Routledge, Milton Park Cassirer E (1951/2009) The philosophy of the Enlightenment. Princeton University Press, Princeton Cusanus N (1440/1990) On learned ignorance (trans: Hopkins J). Banning, Minneapolis Dewey J (1929/1960) The quest for certainty. Capricorn Books, New York Floridi L, Sanders JW (2004) Levellism and the method of abstraction. IEG Research Report 22 Nov 04, digital ed: Greco GM, Information Ethics Group, Oxford University – University of Bari. http://web.comlab.ox.ac.uk/oucl/research/areas/ieg. Frigg R, Nguyen J (2021) Scientific representation. In: Edward N.  Zalta (ed) The Stanford Encyclopedia of Philosophy 2021 Edition, Stanford University, Stanford https://plato.stanford. edu/entries/scientific-­representation. Accessed 30 November 2021 Gibaldi A, Banks MS (2019) Binocular eye movements are adapted to the natural environment. The Journal of Neuroscience 39(15):2877–2888 Giere RN (2004) How models are used to represent reality. Philosophy of Science 71(5):742–52. https://doi.org/10.1086/425063 Gowers T (2008) The Princeton companion of mathematics. Princeton University Press, Princeton Herder JG (1772/1986) On the origin of language. In: On the origin of language – essays by Jean-­ Jacques Rousseau and Johann Gottfried Herder. University of Chicago Press, Chicago Husserl E (1913/2000) Logical investigations. Routledge, London Jaspers K (1983) Philosophische Logik, erster Band: Von der Wahrheit. Wissenschaftliche Buchgesellschaft, Darmstadt Jaspers K (2003) Way to wisdom: An introduction to philosophy. Yale University Press, New Haven Jones MG, Taylor AR (2009) Developing a sense of scale: Looking backward. Journal of Research in Science Teaching 46(4):460–475 Kant I (1787/2007) Critique of pure reason (B). Penguin Books, London Kierkegaard S (1847/1956) Purity of heart is to will one thing (trans: Steere DV). Harper and Row, New York Nagel E, Newman JR (2001) Gödel’s proof. New York University Press, New York Noica C (1986/2011) Briefe zur Logik des Hermes. Traugott Bautz, Nordhausen Rosen J (1983) A symmetry primer for scientists. John Wiley, Chichester Sartre J-P (1943/2001) Being and nothingness. Citadel Press, New York Schleiermacher FDE (1838/1998) Hermeneutics and criticism and other writings. Cambridge University Press, Cambridge Tal E (2020) Measurement in science. In: Zalta EN (ed) The Stanford Encyclopedia of Philosophy (Fall 2020 Edition). https://plato.stanford.edu/archives/fall2020/entries/measurement-­science. Accessed 17 November 2021 Taylor A, Jones MG (2009) Proportional reasoning ability and concepts of scale: Surface area to volume relationships in science. International Journal of Science Education 31(9):1231–1247 Voltaire (Arouet F-M) (1763/1875) Remarques pour servir de supplément à l’essai sur les moeurs. In: Oevres complètes de Voltaire. Moland, Paris

Index

A Abstract, 178 Abstraction, 4, 7–12, 14–16, 97, 114, 165, 286 Abstractness, 184 Abstract space, 5, 9, 13–15, 31 Accelerated creep, 143, 266 Accelerating rhythm of change, 156 Acceleration, 158 Accessibility, 151 Accessible learning, 151 Accuracy, 105 Accurate, 107 Affine transformation, 95, 98, 130 African, 220 Agatharchus from Samos, 37 Air pollution, 188 Air Quality Health Index (AQHI), 188, 189 Alarm fatigue, 150 Alberti, L.B., 38 Amount of information, 257 Amplification, 131 Amplification factor, 131 Amplitude, 134 Anaxagoras, 37 Ancient China, 37 Ancient Greece, 37 Ancient Greeks, 219 Animal communication, 151 Anisotropic scaling, 96 Applicability domains, 279 Aquinas Thomas, St., 265 Archimedes, 307 Areal local scale adjustment factor, 113 Aristotle, 13, 41, 126, 153, 174, 181, 300, 301 Art history, 253, 262 Artifacts, 97 Assimilation effect, 36

Attention, 154 Auditory, 142 Augustine, St., 1, 152, 153, 160, 308 Automated map generalization, 115 Automated similarity assessment, 116 B Bach, J.S., 158 Bak, P., 74, 270, 271 Barabasi, A.-L., 157 Barbour, J., 153 Bar scale, 101 Basurto, X., 49 Bejan, A., 213 Bergson, H., 58, 59, 153 Bernini, 254 Bijection, 95, 290 Bijective, 289 Bijective mapping, 291 Biomes, 44 Blenkinsop, T.G., 250 Bohm, D., 59, 78, 177, 220, 281 Boisot, M., 10 Bonaventure, St., 76 Box-counting analysis, 236 Box-counting method, 233, 234 Brain monitoring, 150 Brunelleschi, F., 36, 38 Bryson, N., 87, 89 Bunde, A., 231 Buridan, J., 301 C Carathéodory, C., 74 Carroll, L., 82, 107

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312 Cartographic communication, 100 Cartographic scale, 16, 21 Cassirer, E., 294 Castells, M., 157 Categories, 23, 173–175 Categories of meanings, 180 Categories of objects, 180 Categorization, 117, 173, 175, 181 Categorization process, 167 Category dependence of the histogram, 196 Central dilation, 96 Central Italy, 147 Chaos theory, 268 Chaotic dynamics, 269 Charles, A., 49 China, 220 Chlorofluorocarbons (CFCs), 29 Class amalgamation, 203 Classes, 175 Classicism, 254, 256, 257, 261, 262 Classification, 173, 175 Climate change, 48, 70 Coarse-grained histogram, 197 Coarse-graining, 185 Coastal processes, 45 Codomain, 288 Cognitive load, 150 Collinearity, 95 Comparing sizes, 32 Comparison, 300, 308 Compressing time, 126 Compression, 131, 157 Compression of time, 157 Compromise projections, 112 Concepts, 13 Concrete, 178 Conformal projections, 111 Connected, 120 Constructal theory, 213 Context, 86 Context of interpretation, 87 Context of production, 87, 88 Contraction, 106 Crack propagation, 143 Criteria, 282 Criticality, 272 Cusanus, N., 62, 300, 301 D d’Alembert, J., 60 Data storage, 115 Data transmission, 115

Index Degree of damage (DOD), 169 Democritus, 37 De Saussure, F., 76 Descartes, R., 61, 160, 174, 208 d’Espagnat, B., 3 Dewey, J., 307 Diachronic, 151, 159 Diachronic perspective, 76–78 Diegesis, 159 Diffusion-limited aggregation (DLA), 225 Digital geometry, 120 Digital map, 115 Dilation, 96 Dimension, 82 Directedness, 183 Directionality, 296 Discrete data, 63 Discrete sampling, 127 Distance, 116 Distortion, 40 Distributed hierarchy, 50, 51 Domain, 288 Downscaling, 84 Dragomir, V.C., 109 Drip paintings, 263 Duration, 57 Dynamical downscaling, 84, 85 E Early maps, 98 Earthquake, 71, 108, 147–149, 266 intensity, 190 magnitude, 190 Earth tides, 108 Ebbinghaus illusion, 36 Echo, 132 Ecological fallacy, 121 Ecology, 50 Eco, U., 62, 116, 257 Einstein, A., 58, 60 Electroencephalography, 150 Eliade, M., 257 Ellipsoid, 110 Empirical reality, 3 Enhanced Fujita (EF) scale, 168, 169 Environment, 3, 14 Equal-area projections, 111 Equidistant projections, 111 Equilibrium, 73–75 Euclid, 5, 260 Euclidian dimension, 242, 243 Expanding time, 126

Index Expansion, 96, 106, 131, 154 External environment, 155 Extrapolation, 122 F Fact of representation, 296, 299 Falconer, K., 231 False scale as rank, 187–190 Feature-guided exploration, 117, 118 Feder, J., 231 Feynman, R., 219, 220 Film, 138, 140 Filmmaking, 142 Fisheries, 48 Fish stock, 73 Flat ontology, 52 Floridi, L., 284 Flows, 157 Foreshortening, 36 Fourier, J., 138 Fourier spectrum, 138 Fractal, 223, 224, 226–230, 264 analysis, 264 dimension, 233, 242, 243, 263 object, 242, 245 Fracture propagation, 143 Fragmentation, 78 Frequency, 134–136 F-type transformation, 185 G Gadamer, H.-G., 219 Galileo, 6, 7, 41, 82, 301 Gardens, 254 Gauss, 109 General description of scale, 303 Generalized scale invariance, 247 General systems theory, 50 Geodesy, 109 Geographic scale, 17, 21 Geography, 229 Geoid, 109, 110 Geometric contraction, 96, 97 Geometric expansion, 96 Geometric transformations, 94–98 Gleick, J., 156, 265 Globalization, 52 Global knowledge exchange, 47 Global level of attention, 154 Global scale, 47 Goodchild, M.F., 122, 229 Granularity, 64

313 Graphic scale, 101 Ground-penetrating radar, 150 Groundwater, 71 Guided cut, 282–285, 296, 297 Guided supercut, 284, 296, 299 Gullies, 44 H Hamming distance, 116 Harvey, D., 156, 157 Havlin, S., 231 Hawaii, 236 Helical symmetry, 221 Heraclitus, 218–220 Herder, J.G., 293 Hergarten, S., 232 Hierarchy, 50 Hierarchy theory, 50 High-density sampling, 143 High-speed movements, 145 Histograms, 191–196 Hofstadter, D.R., 10 Homogeneity, 122 Human geography, 51 Hume, D., 152 Husserl, E., 178, 180, 285, 286 Hypertension, 198 I Idempotent operator, 284 IF-type transformation, 185 Image, 288 Incorrect size estimation, 36 Independent reality, 3 Index set, 128 India, 37, 220 Informational process, 299 Information overload, 151 Information processing rate, 152 Intentionality, 295 Intermittent pain, 182 Invariance, 218, 219 Invariants, 212–214 Inverse mapping, 291 Irreversibility, 180, 186 Isostatic adjustment, 108 Isotropic scaling, 7, 96 J James, W., 115 Jaspers, K., 309

314 Jones, K., 52 Joyce, J., 160 K Kant, I., 13, 152, 174, 294, 305 Keates, J.S., 100 Kierkegaard, S., 277 Klein, F., 305 Koch curve, 225 Korvin, G., 232 Kruhl, J.H., 232, 250 L Ladder, 184 Lagrange, J.-L., 60 Landslides, 143, 266 Large scale, 21, 46 Lava flows, 198 Leibniz, G.W., 58, 218, 301 Lessing G.E., 76 Le Vau, L., 254, 255 Level of analysis, 49 Levenshtein distance, 116 Levine, R., 158 Lightning, 144 Li, J., 116 Linear perspective, 36 Literature, 159, 160 Li, W., 122 Local communities, 47 Local scale, 47 Local scale adjustment factor, 112 Local to global, 48 Locke, J., 152, 153, 174 Logarithm, 42 Logical field, 286, 296 Lorenz, E., 269 Loudness, 135 Lovejoy, S., 247, 248 Lying with maps, 203 M Magnitude, 147 Mandelbrot, B.B., 223, 224, 226, 229, 230, 244, 247 Many-to-one mapping, 291 Map, 8, 15, 38, 197, 231 manipulation, 201–204 projections, 109–113 scale, 16 Mapmaking, 7 Mapping, 288, 289, 299, 305

Index Marginalization, 52 Mars, 199 Marston, S., 52 Mathematical fractals, 224 Mathematics, 10 Maxwell, J.C., 301 Medicine, 150 Medieval sea charts, 100 Memory, 300, 301 Mercator projection, 111 Microsaccades, 155 Microscopes, 5 Mimesis, 159 Minkowski, H., 60 Modalities of representation, 298 Modifiable areal unit problem (MAUP), 121 Modified Mercalli (MM) scale, 190 Morphism, 292 Morrison, J.L., 100 Movement, 155 Movies, 158 Moving across scales, 216 Multifractals, 248 Multiscale analysis, 103 Multiscale approach, 54 Multiscale spatial databases, 115 Music, 140 N Narration, 159, 160 Narration time, 159, 160 Natural disaster, 266 Natural environment, 215 Natural hazards, 69 Nega, M., 250 Neighborhood, 48 Nested hierarchy, 50 Network, 45, 47, 157 Neural activity, 32 Newton, I., 108, 218, 301 Noica, C., 286 Nonlinearity, 265 Nonlinear processes, 267 Nonlinear science, 50, 265 Non-nested, 50 Nonrenewable, 71 Number of classes, 194 Nyquist criterion, 64 O Object classification, 33 Objectivity, 186, 187, 279 Objects, 285

Index Occurrence time, 159, 160 Ontological condition of scale, 52 Operational scale, 17 Opisometer, 230 Optical instruments, 7 Optical transformations, 5 Orders of Magnitude, 41, 42 P Pain intensity, 182 Paleolithic, 97 Paper maps, 101 Papu, E., 253 Parallelism, 95 Parmenides, 218, 219 Partial mapping, 290 Particulate matter, 188 Pattern analysis, 213, 216, 249 Patterns, 208–214, 216 Peitgen, H.-O., 231 Penrose, R., 10 Perception of time, 154 Period, 134 Periodic function, 134 Perspective, 96 Perspective triangles, 96, 97 Physical time, 59 Pitch, 133, 135, 139 Plane-filling, 242 Plato, 62, 173, 177, 295 Plotinus, 153, 160 Poincaré, H., 301 Pollock, J., 263 Pompeii, 37 Popper, K., 10, 174 Power, 140 Power law, 245 dependence, 231 relations, 120 Power spectrum, 140 Precipitation rates, 198 Precise, 107 Precision, 105, 279 Proportional change, 94 Proportional reduction, 9, 98 Ptolemy, 5, 6 Q Quadtree, 118 Quantification, 301

315 R Raster cells, 114–115 Ratajsky, L., 100 Ratio of magnification, 96 Ratios of distances, 95 Recognizable, 296 Reconstructed, 136 Records, 81 Redshift, 133 Reference ellipsoid, 109 Reflection symmetry, 221 Relations, 10, 12, 13 Relationships, 13, 20 Relationships between elements, 99 Relative motion, 155 Remote sensing, 115 Renaissance, 37 Renewable, 71 Representation, 293, 295, 297, 299, 304 Representational theory of measurement (RTM), 302 Representation theory, 305 Representative fraction, 19, 101, 102, 107, 108 Resample, 136, 137 Resolution, 17, 114, 147 Resources, 71 Reversibility, 186, 280 Richardson, L.R., 231 Rigid hierarchy, 50, 51 Romans, 100 Roman surveyors, 100 Romanticism, 258, 260–262 Root representation, 119 Rotation symmetry, 221 Roughness, 231 Russell, B., 174 S Saint-Exupéry de, A., 49 Samples, 64 Sampling, 64, 67 Sampling rate, 64, 66, 69, 130, 143 Sanders, J., 284 Sandpile model, 271 Sartre, J.-P., 285 Scale as a process, 304–307 Scale as rank, 17, 22, 23 Scale as ratio, 17, 19, 21, 22 in space, 93–122 in time, 125–161 Scale as size, 17–19

316 Scale bar, 21 Scalebound, 244, 246, 255, 261 Scale change, 130, 217, 218 Scale factor, 96, 105, 106, 131, 147, 149, 159 Scale-free, 245, 246, 255, 261 Scale invariance, 218, 219, 222 Scale invariant, 218, 233 Scale in virtual space, 49 Scale of analysis, 49 Scale range, 69, 237 Scale symmetry, 221, 259 Scale types, 277–281 Scaling, 96–98 down, 81 regimes, 238 up, 81 Schertzer, D., 248 Schleiermacher, F., 215, 300 Schopenhauer, A., 122 Scott, W., 160 Seismic moment, 190 Selection, 114, 282, 285 Selection process, 106 Selective attention, 154 Self-affine, 227, 246 Self-organized criticality (SOC), 74, 270, 273, 275 Self-similar, 227 Self-similarity, 229, 241, 256 Semivariogram, 130 Senses, 142 Set, 120 Shakespeare, W., 156 Shannon, C.E., 257 Shape of the Earth, 108 Shot length, 158 Signals, 127 Similarity, 9, 116, 294 Similarity transformation, 96 Simply connected, 120 Sine wave, 134 Size, 4, 5, 18, 32 Small scale, 21, 46 Smith, H., 49 Smith, N., 52 Snow avalanches, 143 Socrates, 62 Sonification, 149 Sound, 138, 151 Soundscape, 150 Space debris, 143 Spatial context, 86 Spatial informational backbone (SIB), 119, 120

Index Spatialization of time, 79 Spatializing time, 62 Spatial orientation, 33, 155 Spatial positions, 108 Specificity, 122 Spectrum, 140 Speech, 140 Spinoza, B., 220 States of existence, 253 Statistical downscaling, 84 Stommel diagram, 44, 45, 59, 71 Stommel, H., 44 Stream of consciousness, 160 Structure-preserving mapping, 292 Subjective perception of time, 154 Subjective time compression, 154 Subjectivity, 186 Subsystems, 50 Suppes, P., 301 Suppressed perspective, 39, 40 Suppression of self-generated behavior, 155 Surjective mapping, 290 Sustainability, 83 Symbolization, 117 Symmetry, 219–221, 255, 262, 300 Synchronic, 151, 159 perspective, 76–78 view, 78 System, 28, 29, 50 image, 28, 106, 181 representation, 106 T Tactile stimuli, 155 Takayasu, H., 231 Target, 299 Target of representation, 296 Tectonic processes, 108 Telescopes, 5 Telescoping bias, 58 Temporal acceleration, 158 Temporal compression, 159 Temporal event density, 133 Temporal informational backbone (TIB), 146, 148, 149 Teresa of Ávila, St., 160 Thales, 36 Theory of categories, 173 Thrift, N., 52 Time compression, 132, 155–158 Time density overload, 151 Time domain harmonic scaling, 137 Time expansion, 132, 154, 155, 159

Index Time perception, 154 Time scale, 69, 108, 154, 155, 158 bias, 79–81 change, 155, 159 factor, 159 perception, 154 Time series, 63, 127, 146 Time-space compression, 157 Time stretching, 137 Tissot indicatrices, 111 Topographic surface, 108 Topography, 199 Topological dimension, 242, 243 Tornado intensity, 167, 171, 183 Tornadoes, 168 Transformation, 305 Translation, 96, 131, 132 Translation symmetry, 221 Trees, 33 Tropical storm intensity, 179 Turcotte, D., 232 U Udden-Wentworth scale, 44 Unequal bin sizes, 197 Uniform sampling, 127 Units, 278 Urban environment, 48

317 Vasari, G., 38 Verbal scale, 101 Viewing angle, 33 Views on time, 58 Viruses, 144 Visual, 142 arts, 142 estimation of sizes, 33 von Helmholtz, H., 155 Vredeman de Vries, H., 39 W Wavelength, 134 Waves, 133–135 Weather hazards, 166 Wellek, R., 261 West, G., 82 Weyl, H., 220 Wheeler, J., 10 The Whole, 287 Wigner, E., 10 Wilde, O., 261 Wölfflin, H., 174, 261 Wolfram, S., 11 Woodward, K., 52 Woolf, V., 160 X X-ray, 140, 141

V Value in art, 265 Vanishing point, 36 van Leeuwenhoek, A., 7 Variability, 128

Y Yan, H., 116