Movements of Form (Vision, Illusion and Perception, 6) 3031448200, 9783031448201

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JOACHIIM JOACHIM MH H.. M MOWITZ OWI OW WIT TZ ARNO L G OUDSMIT T ARNO L.. GO GOUDSMIT

MOVEMENTS OF FORM

Vision, Illusion and Perception Volume 6

Series Editor Nicholas Wade, School of Psychology, The University of Dundee, Scotland, UK Editorial Board Benjamin Tatler, University of Aberdeen, Aberdeen, UK Frans Verstraten, School of Psychology, UNSW Sydney, Sydney, NSW, Australia Thomas Ditzinger, Springer-Verlag GmbH, Heidelberg, Baden-Württemberg, Germany

The Vision, Illusion and Perception (VIP) book series publishes new developments and advances in the fields of Vision and Perception research, rapidly and informally and with a high quality. The series publishes fundamental principles as well as stateof-the-art theories, methods and applications in the highly interdisciplinary field of Vision Science, Perception and multisensory processes related to vision. It covers all the technical contents, applications, and multidisciplinary aspects of fields such as Cognitive Science, Computational and Artificial Intelligence, Machine Vision, Psychology, Physics, Eye Research, Ophthalmology, and Neuroscience. In addition, the series will embrace the growing interplay between the art and science of vision. Within the scope of the series are monographs, popular science books, and selected contributions from specialized conferences and workshops.

Joachim H. Mowitz · Arno L. Goudsmit

Movements of Form

Joachim H. Mowitz (Deceased) Amsterdam, The Netherlands

Arno L. Goudsmit EDT Maastricht Psychotherapie Maastricht, The Netherlands

ISSN 2365-7472 ISSN 2365-7480 (electronic) Vision, Illusion and Perception ISBN 978-3-031-44820-1 ISBN 978-3-031-44821-8 (eBook) https://doi.org/10.1007/978-3-031-44821-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 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 Paper in this product is recyclable.

Joachim Mowitz passed away shortly before the publication of this book. He will be remembered by the many who loved him and admired his brilliance.

Foreword by Prof. Koichiro Matsuno

Any descriptive enterprise adopted in practicing the standard physics is sharp in distinguishing between the law of motion and its invariant boundary condition on the methodological ground. In contrast, biology is different in allowing for whatever boundary condition to become variable in time as demonstrated in the empirical phenomena of development and evolution. Still, biological motion is certainly acceptable on the materialistic or physical ground. At issue must be how to descriptively approach dynamic boundary conditions common in the biological realm. The authors of the present monograph try to seek a possibility of dynamic boundary conditions in the network of action and reaction, especially as focusing on its geometrical and topological characteristics. One merit of this proposal is found in cultivating the likelihood of a new type of cohesion not conceivable in the traditional scheme of the standard physics. For instance, the repeated reentrant quantum particle such as a carbon atom into the biochemical reaction network full of loops is cohesive in being fed into the network. Such cohesion serves as an empirical factor making the network more robust and more durable. Once the network happens to be durable, it may turn out that the reentrant particles of the quantum origin are constantly exchanged with those particles belonging to the same class, but different individually. Biology is unique in exploring a new type of cohesion not conceivable in the standard physics. The authors’ focusing on dynamic boundary conditions rests upon the observation stating that the product of the preceding production is going to cohesively set the condition for how the succeeding production would proceed. An essence of dynamic boundary conditions has skillfully been picturized in movements of form. In particular, movements of form as a heuristic descriptive convention of temporal cohesion capable of relating the preceding production effectively to the subsequent one are addressable within the scheme of dynamic boundary conditions. Pivotal to the dynamics internal to the system’s boundary conditions is the activity of their emergence, coalescence, or even sensing and forestalling the unwelcome conflicts likely to come up among themselves in due course of time. What is unique to the authors’ attempt is to shed a new light on extending the notion of cohesive interaction so as to make it applicable even to biology at large vii

viii

Foreword by Prof. Koichiro Matsuno

without offending the established physics so far. To the best of my knowledge, their work has been the first attempt of this kind in explicating the intricate relationship between the geometric topology of the network and the realizable temporal cohesion to be observed widely in biology. Prof. Koichiro Matsuno Nagaoka University of Technology Nagaoka, Japan

Foreword by Dr. Aloisius H. Louie

Geometric tools have been in our repertoire since the very beginning of the Rashevsky-Rosen school of relational biology. Nicolas Rashevsky brought forth relational biology with his paper “Topology and life” (1954), Bulletin of Mathematical Biophysics 16, 317–348. Many papers on the geometry of life followed: e.g., “The geometrization of biology” (1956), Bulletin of Mathematical Biophysics 18, 31–56; “A comparison of set-theoretical and graph-theoretical approaches in topological biology” (1958), Bulletin of Mathematical Biophysics 20, 267–273. I am delighted that the authors Joachim Mowitz and Arno Goudsmit use Robert Rosen’s (M, R)-systems—impredicative networks that are inherently geometrical— to illustrate (see Chap. 4 of this book) their self-referential systems of geometrical expansions. The simplest (M, R)-system, a staple of relational biology, in circular and triangular forms are the following directed graphs (found in, respectively, Louie (2009) and Rosen (1991)). Compare their forms with those in Fig. 3.1 in this book.

Mowitz and Goudsmit conclude that “The entire project was strictly formal; there was no obvious connection to material realizations such as biological phenomena.” Their expanding forms behave “as-if ” they were the clef (closed to efficient causation) network of (M, R)-systems. The authors may self-effacingly describe their movements of form as “shadows,” but often one learns from these incomplete simulations of reality, even when one only has shadows cast on walls inside Plato’s Cave. ix

x

Foreword by Dr. Aloisius H. Louie

I look forward to reading about the “further explorations” on the three issues the authors mention in the culminating paragraphs, as they carry on with unfurling in subsequent publications their Movements of Form. Ottawa, Canada

A. H. Louie Mathematical Biologist https://www.ahlouie.com/

Synopsis of the Book

Movements of Form presents a dynamic geometry of expanding forms so as to unfold self-referential indications and relate them to themes from theoretical biology. Chapter 1 gives some background information on the long term collaboration that gradually evolved into the project of which this book is the result. Chapter 2 departs from a boundary, defined as a self-referential indication, and describes how it can be translated into two types of expanding geometrical forms, with a particular focus on triangles and circles. This chapter presents the idea that paradox can be forestalled if boundaries are allowed to move outwardly. Chapter 3 contains the precise typology of the interaction patterns between the expanding forms, not only how these forms expand and coalesce with others, but also how their interactions lead to closed loops of definitions between processes, where triangles and circles coexist in separate spaces and reciprocally define one another. Chapter 4 connects the relations between geometric properties of expanding contour lines, their expansion rules and the various types of bisectors with the relations that constitute the so called ‘closure to efficient causation’, a core concept from the Rashevsky-Rosen-Louie tradition of relational biology. While the expanding forms themselves change both in continuous and discrete ways, their network of mutually defining relations remains intact. This exemplifies how relations between geometrical properties can be more important than the constituent parts. Chapter 5 deals with issues of computing the expansion processes. A computer simulation (“Zigzag”) has been built in order to visualize the expansions as conceived. Nevertheless, the latter elude exact calculation and cannot be predicted other than by numerical approximation, and only for short intervals of unpredictable duration. Their radical context sensitivity pushes beyond the traditional notions of states and state transitions and shows how boundary conditions are not fully controllable. Unlike cellular automata, these expansion processes defy stepwise progression on a predefined grid, presenting themselves as unprogrammable constructions that proceed with a speed that is necessarily finite. Chapter 6 poses some further research questions on the analogies between the presented geometry and the organization of living beings.

xi

Contents

1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 A Self-referential Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Forestalling the Stalemate of Paradox . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A Translation into Geometrical Relations . . . . . . . . . . . . . . . . . . . . . .

3 3 5

3 Expanding Forms: the Geometrical Representation . . . . . . . . . . . . . . . . 3.1 Expansions in Circles and in Triangles . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Expansion Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Bisectors (Symmetry Axis/Nucleus) . . . . . . . . . . . . . . . . . . . . 3.1.3 Singularity Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Coalescence, the Domains of Validity of Expansion Rules, and the Local Symmetry Axes that Delimit these Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Coalescing Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Gain of Form in Coalescing Circles: The Type 0 ⇒ 1 . . . . . 3.2.2 Loss of Form in Coalescing Circles: The Type 2 ⇒ 1 . . . . . 3.2.3 Enclosure and Loss of Form in Coalescing Circles: The Type 3 ⇒ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Coalescing Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Gain of Form in Coalescing Triangles . . . . . . . . . . . . . . . . . . . 3.3.2 Loss of Form in Coalescing Triangles . . . . . . . . . . . . . . . . . . . 3.3.3 Positive, Negative and Mixed Expansion of Contour Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Enclosure and Loss of Form in Coalescing Triangles: The Type 3 ⇒ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Acute and Obtuse Angles, Internal and External Bisectors of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Mixed Expansion Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Circles and Triangles: Three Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Termination and Inception: The Transmutation into a New Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 9 10

11 12 12 15 16 19 19 21 21 23 27 28 31 34 xiii

xiv

Contents

3.5

3.6 3.7 3.8

3.4.2 The Establishment of New Expansion Rules during Transmutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Examples of Transmutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Generations of Configurations . . . . . . . . . . . . . . . . . . . . . . . . . Bisectors, Singularity Points, Enclosures, Transmutations: Recapitulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Bisectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Singularity Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Transmutations: Transitions between Spaces . . . . . . . . . . . . . On the Geometrical Relations Implicit in the Construction of a New Triangle from Three Coalescing Circles . . . . . . . . . . . . . . . On Increases and Decreases of Irregularity . . . . . . . . . . . . . . . . . . . . . A Process in Three Spaces: Illustration . . . . . . . . . . . . . . . . . . . . . . . .

35 38 42 44 51 52 54 56 57 57

4 The Geometrical Expansions as a Relational Network . . . . . . . . . . . . . 4.1 Rosen’s Basic Idea of Closure to Efficient Causation . . . . . . . . . . . . . 4.2 A Geometrical Interpretation of ‘Closure to Efficient Causation’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Structurally Stable Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Regular Singularity Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Special Singularity Points: Transmutations . . . . . . . . . . . . . . . 4.2.4 On the Construction of Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Expansions, as-if Reading and Writing . . . . . . . . . . . . . . . . . . . . . . . .

69 69

5 The Geometrical Representation and Its Computability . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Arbitrary Properties of the Geometrical Representation . . . . . . . . . . 5.3 Limits to the Predictability of Expansions . . . . . . . . . . . . . . . . . . . . . . 5.4 On Simulation: the Program Zigzag . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Unprogrammability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 An Informal Argument about the A Priori Incomputability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Expansions as A Priori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Limited Control of Boundary Conditions . . . . . . . . . . . . . . . . 5.5.4 Constructions and Sensitive Movements . . . . . . . . . . . . . . . . . 5.6 Differences from Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 80 81 83 87

6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Appendix A: The Shrimp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

71 72 73 74 76 77

89 90 91 92 92

Appendix B: The Workgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

About the Authors

Joachim Mowitz (1944–2023) had degrees in mathematics from the universities of Hamburg and Heidelberg. He worked as a mathematical programmer at the University of Amsterdam.

Arno Goudsmit (1955) has degrees in psychology and philosophy from the universities of Utrecht and Groningen. He works as a psychotherapist in Maastricht and Amsterdam.

xv

Chapter 1

Introductory Remarks

Life has always been a miracle that escaped our intellectual capacities. If we want to investigate it, then it may be helpful to study its shadows. The movements of form to be presented here are meant to offer such shadows. This book is about moving geometrical forms and how they define each other in ways that may resemble circular relations between processes in living beings. If their organization is characterized by a kind of closure that withdraws it from the inspection by us outsiders, then can we unfold the complexity of this closure? This is what we will try to do: departing from paradox, to study the closure and to see if we can postpone paradox by adding a dimension of time to it and by representing it in a domain of geometrical forms. We will not deal with material realizations of the presented processes. We will not make efforts to describe empirical phenomena of biology or semiotics, but it cannot be excluded that the geometrical representation may contribute to these fields as well. As is mentioned in the Appendix B, there was a workgroup, once upon a time in the old days, that was concerned with preparing a course on System Dynamics. The models in it did allow the variables involved to be mutually influencing their so called ‘levels’ (values) and ‘rates’ (speeds of value change), but, once defined, there were only limited ways to adapt the formulas and the relative weight of the variables and their properties. The first author of this book missed the options for these variables to interact so as to create new variables or redefine existing ones. His intuition was not deeply appreciated by all participants immediately. After the course had been prepared and taught, the workgroup slowly changed its scopes and interests. Our project has been, first of all, an exploration into changes of boundary conditions of systems. Geometry turned out for us to be helpful as a way to express our themes in terms of moving forms. Once our wonder of bisectors in triangles and polygons had started, it was as if new horizons of ideas and phenomena became visible, inviting for further steps. The moving forms were helpful for understanding how the parameters of a system could come to define each other mutually and in circular ways.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Mowitz and A. L. Goudsmit, Movements of Form, Vision, Illusion and Perception 6, https://doi.org/10.1007/978-3-031-44821-8_1

1

2

1 Introductory Remarks

The present authors have had an intensive and long term collaboration. Joachim Mowitz is to be credited for almost all geometry and for all programming work. Arno Goudsmit did all writing and almost all theory beyond the mathematics. The result of this enterprise is a shared domain of concepts and their elaborations, for which both authors feel ownership and responsibility. If these are the shadows in the cave, then we wish they may prepare us for sunlight.

Chapter 2

A Self-referential Boundary

An indication is an act of pointing at something. Self-reference is an indication that points at itself. Self-reference, therefore, has as its distinctive property a moment of non-distinction between the act of indication itself and the thing indicated by this act. This moment of non-distinction appears for instance in the famous paradox of Epimenides: the Cretan who says that all Cretans are liars. Here we find the moment of non-distinction between the Cretan’s speech act (the act of indication) on the one hand, and on the other hand the Cretan who is said to be lying (the thing indicated). It is their moment of non-distinction that renders the famous expression paradoxical. Would the two logical levels afford to be kept strictly separate, no paradox would arise. The idea of this book is that we can translate the self-referential indication into a process unfolding in time that consists of expanding geometrical forms and their interaction patterns, as a translation and visual representation of the unthinkable non-distinction between an act of indication and its object. We will use the term geometrical representation throughout this book, to denote the outcome of this translation.

2.1 Forestalling the Stalemate of Paradox Let the boundary of a closed geometrical form be defined in terms of a self-referential indication. Then the definition of such boundary will have to contain a reference to itself. This means that such boundary can exist only within the domain of validity of its own definition. Hence, beyond this domain of validity the boundary is not defined. Therefore, this boundary demarcates the domain of validity of its own definition. Let us call this

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Mowitz and A. L. Goudsmit, Movements of Form, Vision, Illusion and Perception 6, https://doi.org/10.1007/978-3-031-44821-8_2

3

4

2 A Self-referential Boundary

boundary a one-sided boundary.1 As is the case with a Möbius strip, any distinction between its two sides is not well-defined. One way to understand the one-sided boundary is that it is demarcating the domain where it is defined from where it is not defined. Hence, its ‘other side’ is where the boundary is not defined. This ‘other side’ cannot be indicated in relation to this boundary, but obviously the idea of ‘other side’ cannot be entirely unrelated to the boundary either. We can think of this ‘other’ non-defined side as something that, indeed by definition, escapes our intellectual capacities. Wittgenstein has put something similar in his Tractatus: “For in order to be able to set a limit to thought, we should have to find both sides of the limit thinkable”.2

If we want to speak of the ‘other side’ of this boundary, we will have to indicate an area adjacent to the boundary. This adjacency relates the boundary to its undefined ‘other side’. That is to say that this ‘other side’ has now come to be situated within the domain in which the boundary is defined. That is: as soon as we indicate this ‘other side’, the definition of the boundary pertains to it and by our act of indication the boundary is displaced so as to include the ‘other side’ into the domain of its definition, so that it is no longer ‘other side’. The reverse is also a valid option: it becomes possible for us to indicate a position that was previously situated on the displacing boundary’s ‘other side’, adjacent to the boundary. Then as soon as the boundary moves towards and across this position, the latter has become situated within the confines of its defined side and can be indicated. Thus, we find two ways to relate the displacements of the boundary (A) to the acts of indication (B): relation 1: the boundary’s displacements (A) are enabled by the acts of indication (B); A (operand) is enabled by B (operator). and: relation 2: the acts of indication (B) are enabled by the boundary’s displacements (A); B (operand) is enabled by A (operator). Thus:

1

cf. Goudsmit (1992, 1998). Wittgenstein (1963, Preface). The reference to this quotation is owed to Heinz von Foerster (personal communication, Bergamo 1990).

2

2.2 A Translation into Geometrical Relations

5

B

A

relation 1:

as operator

as operand

relation 2:

as operand

as operator

In either way of relating, the one-sided boundary displaces in the direction away from its defined side and towards (and beyond) its undefined side. In this way these displacements can be seen as directed outwardly. Unlike Spencer Brown’s (1969) act of distinction, our indications involve a movement. Accordingly, we will use the term ‘expansion’ for these displacements of geometrical forms. Given these expansions we can now use both ways of relating A and B. The self-referentiality of the one-sided boundary can be conceived as curled up in some ‘private’ dimension of its own. Allowing boundaries to have outward movement will add a dimension of time, by means of which the self-referentiality of the onesided boundary can be unfolded. Movement, as the displacements of the boundary, forestalls3 the contradictions that would have occurred otherwise in our attempts to indicate the boundary’s undefined ‘other side’. These displacements can be studied in terms of expanding geometrical forms, so that the self-referentiality no longer remains unthinkable or unaccessible. Let us remark that the expansions are not a feature of the self-referential indications, but they are added to them by us, artificially. Thus the expansions do not occur as properties implied or entailed by any of the geometrical relations that we will encounter. This will be of our interest in Sect. 5.4.

2.2 A Translation into Geometrical Relations The translation goes thus: displacements of the boundary (A): the positions of an expanding geometrical form acts of indication (B): the expansion directions of a geometrical form as operator:

as operand:

means:

type 1:

position (A)

direction (B)

position rules direction

type 2:

direction (B)

position (A)

direction rules position

Type 1 is present in expanding circles. These have a midpoint from which all available expansion directions depart. The expansion directions may decrease in the 3

cf. Matsuno (2002). See also footnote 9 of Chap. 5. cf. also: Matsuno (2006, p. 339), Gunji et al. (1998, p. 182), Gunji (1994, p. 265): “self-contradiction is perpetually generated and is to be removed at the same time”.

6

2 A Self-referential Boundary

course of various interactions between circles, but the midpoints remain available as a rule of the expansions. Type 2 is present in expanding triangles. These have expansion directions that are stable for all sides. The side lengths will increase or decrease in the course of various interactions between triangles (and polygons), but the expansion directions remain available as the expansion rules. Our usage here of these operator-operand relations is only in a loose and unspecific way, meant to distinguish two ways of relating positions and directions. We will see in the next chapter that these two ways specify two kinds of spaces and we will see in chap. 4 that within each of these spaces more complex operator-operand relations occur. Let us explore these spaces now.

Chapter 3

Expanding Forms: the Geometrical Representation

In order to investigate systems that consist of expansion processes, we will restrict our discussions to spatial relations in flat geometry. We are interested in closed configurations, especially those that can expand from a minimal extension (a single point). We will investigate two types of expanding configurations and their geometric relations: those based upon circles and those based upon triangles, for reasons mentioned in the previous chapter. We will first present expansion processes in circles and in triangles, introducing the notion of symmetry axes and the various types of singularity points that occur at the moments of inception and/or termination of these expansion processes. Then we will present the various spaces in which these expansion processes take place and discuss the interactions between the processes of these spaces.

3.1 Expansions in Circles and in Triangles Let each circle and triangle expand and start at a single point. For circles this inception point is the circle’s own centerpoint; for triangles it is the incircle’s midpoint, the point from which the triangle’s angular bisectors incept. Let circles and triangles expand as continuous processes that have a constant velocity. When we represent expansion processes by means of consecutive steps or stages, such representations are not meant to suggest or imply that the processes themselves consist of discrete steps or stages, nor that these steps or stages exist simultaneously. Figure 3.1 presents a few stages of the ongoing developments in a circle and in a triangle. The shape of an expanding boundary at a particular moment will be called the contour line.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Mowitz and A. L. Goudsmit, Movements of Form, Vision, Illusion and Perception 6, https://doi.org/10.1007/978-3-031-44821-8_3

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3 Expanding Forms: the Geometrical Representation

Circles and circle configurations will be shown in red. Triangles and polygon configurations will be shown in green and blue, or in cyan, as a color between blue and green, whenever a distinction between green and blue is not relevant. In circle expansions the contour line is related to the position of the circle’s midpoint; the contour is the set of points that are of equal distance from the midpoint. The expansion of the circle’s contour line can be visualized as the ripples in a pond caused by a pebble. In triangles the contour lines expand in the directions perpendicular to the sides of the triangles. There is a constant expansion velocity of the triangle’s sides,1 so that for each triangle the sides remain at equal distance from the triangle’s inception point. Notice that the vertices of an expanding triangle always move faster than the sides, and the more so if the vertex’ angle is more acute.

3.1.1 Expansion Rules An expansion rule is the prescription according to which an expansion process takes place. The displacements that have already been performed in accordance to an expansion rule can be called the rule’s domain of realization; the displacements to be expected in the future can be called the rule’s domain of possibility. The rule’s domain of validity at a particular moment is the extension to which the rule applies at that

Fig. 3.1 Examples of outward expansions in a circle and in a triangle. Each expands with equal and constant velocity from its own single point of inception. The circle expands in all directions (four of which have been drawn), the triangle in three directions. The bisectors shown are symmetry axes between the triangle’s expansion directions 1

Also called ‘prairie-fire’ displacements, cf. Duda and Hart (1973).

3.1 Expansions in Circles and in Triangles

9

moment. We will use the term ‘substrate’ as an alternative to ‘domain of validity’. An expansion rule can be said to exist (qua operator) as long as its substrate exists (i.e. having a non-zero extension qua operand). For circles the rule’s domain of validity is the angular extension of the circle’s arc. The rule according to which a circle’s contour line expands is the position of the circle’s midpoint.2 For triangles the rule’s domain of validity is the linear extension of a contour line segment. The expansion rule of a triangle’s contour line is the direction of its expansion movement.

3.1.2 Bisectors (Symmetry Axis/Nucleus) We will use the term ‘bisector’ for a symmetry axis between expansion rules. A bisector is a straight line or a curve that describes this symmetry between expansion rules. Bisectors are specified by contour lines, more specifically: each point of a bisector is a position of encounter between contour lines. We will also use the term ‘nucleus’ as a generic term for ‘bisector’ and ‘symmetry axis’. A bisector occurs between circles as a separation trajectory between two expanding circle sectors; the latter are the domains of realization of two circle midpoints (that are the expansion rules). The expanding contour lines of circles specify bisectors that are related to the circle midpoints. A bisector of two circles is a curved line, each point of which has a constant difference of distances to the two circle midpoints that are the expansion rules of the two circles.3 The bisector is a straight line if this constant difference equals zero. Likewise, in triangles and between triangles a bisector occurs as a separation trajectory between two expanding line segments; the latter are the domains of realization of two expansion directions (that are the expansion rules). The expanding contour line segments of triangles and polygons specify angular bisectors.4 A bisector is here a straight line that equally divides the angle between 2

In a different context (Dobrin 2005) these points have been called ‘generators’. See Sect. 3.2.1. 4 These are called ‘medial axes’ by Duda and Hart (1973). Furthermore, Aichholzer et al. (1995) offer a formalized presentation of these axes, which they call ‘straight skeleton’, ‘a new internal structure for polygons’. The idea that these authors presented was that a given polygon is shrunk inwardly. However, an entirely new internal structure it wasn’t, for in a less formal style the present authors already introduced and specified these axes, as well as their ‘roof model’ (Mowitz and Goudsmit 1988). (We tried to contact these authors from Graz about our findings, in vain, but only once.) We used there the term ‘folding’ (instead of ‘roof’) and we are reminded of the late Gordon Pask’s typical encouragement when in 1986 we showed him our tentative ideas: “there is no reason not to fold it up”, (ibid., p. 174). We also started, as of 1985, with the idea of an internal nucleus of a polygon, later added by the idea of an external nucleus of the same polygon (see Figs. A.1–A.3). Then we added the idea of both an inward and an outward expansion of the same polygon. Then we decided that outward expansion was more elegant as a concept if the expansion incepted from a set 3

10

3 Expanding Forms: the Geometrical Representation

the two expansion directions that are the expansion rules of the two contour line segments. It is obvious that the displacement of an expanding triangle contour line is larger when measured along its bisectors than along its displacement direction itself. More generally, all vertices of a polygon will expand faster than its sides, and the more so if the vertex has a more acute angle. Indeed, displacement speeds going to extreme values can be conceived as the angular extensions of the vertices approximate zero. It may look surprising that we do not define a nucleus as a symmetry axis between substrate parts. For simple triangles (as in Fig. 3.1) this would not be untrue, but for circles and for more complicated types of encounter between triangles or polygons (as in Sect. 3.3.5) this would not fit. In Sect. 4.2 this symmetry will be described as a function of the expansion rules, in terms of the circularity of relations in which it partakes.

3.1.3 Singularity Points The parallel expansion of displacing contour lines leads to the expansion of configurations in such a way that (a) trajectories occur during which a particular expansion rule is valid. As long as the rule is not changed, the expansion trajectory is called structurally stable. The rule’s domain of validity changes here in a continuous (parametric) way. (b) at certain points structural changes may take place, that is, points where the expansion rules are changed in a discrete (topological) way, with ensuing new domains of validity. These points are characterized by the discrete change of the number of bisectors that are involved. The singularity points at which radical changes occur between structurally stable trajectories are visible as special points in the developments of bisectors. There are five types of singularity points that we encounter in the geometry that we are presenting here. These types are named according to the number of the involved bisectors. For instance, a singularity point of the type 2 ⇒ 1 has two incoming bisectors and one outgoing bisector. The singularity points can be grouped as ‘loss of form’ and ‘gain of form’, in accordance with a decrease or an increase of the number of bisectors involved. Accordingly, in the following sections we will encounter the various types: 2 ⇒ 1, 3 ⇒ 0, 0 ⇒ 1, 0 ⇒ 3 and 1 ⇒ 2. Thus, the singularity points can be seen as the switch points between the regimes of expansion rules; themselves they are beyond these regimes and they delimit these regimes! This holds for the singularity points of all types mentioned here. This will be elaborated in the following sections. of initial points, whereas internal expansions were to occur only within enclosures (cf. Sect. 3.3.4). This yielded the triangle expansions as described in this book.

3.1 Expansions in Circles and in Triangles

11

3.1.4 Coalescence, the Domains of Validity of Expansion Rules, and the Local Symmetry Axes that Delimit these Domains A major class of events of structural changes is the confrontation of expanding configurations (triangles/polygons or circles). At points of encounter between expanding configurations the domains of validity of several (at least two, at most three5 ) expansion rules are constrained. That is, a point of encounter between two expanding configurations is a point situated at the expanding boundaries of these configurations. Let the domain of validity of any expansion rule be such that it pertains only to points of an expanding contour line that are not a point of encounter between expanding configurations. Thus, no conflict can arise between expansion rules, as the latter stop being valid for any point of encounter. Accordingly, a bisector can be understood as the course of all points at which two expansion rules stop being valid. A bisector demarcates the domains of validity of the expansion rules.6 Both for expansions of circles and for expansions of triangles a bisector is a vector composed of the expansion movements of two coalescing forms. The extensions of contour lines in triangle spaces are understood in terms of their length (linear extension of contour line); the extensions of contour lines in circle configurations are understood in terms of their arc (angular extension of contour line). Thus, in the triangle spaces the reduction of a domain of validity to zero means a reduction to zero linear extension (a line segment of zero length7 ), whereas in the circle space it means a reduction to zero angular extension (a circle sector of zero central angle8 ). We use the terms ‘coalesce’ and ‘coalescence’ for the process of encounter between two or more expanding configurations, as they meet and develop their symmetry axes (bisectors). Notice that the contour of a resulting configuration (either consisting of circles or of polygons/triangles) remains closed during transformation, either in gain of form or in loss of form, at singularity points of all types. We will discuss in Sect. 3.2 the types various types of singularity in circle configurations and in Sect. 3.3 we will discuss those in triangle configurations.

5

Special situations of overdetermination, as in regularly shaped forms (equilateral triangles, squares etc.) are left beyond our consideration. 6 The coalescence of expanding forms can also be seen as (mutual) internal measurements (cf. Matsuno 1989, e.g. pp. 38ff.) being taken with mutual impact, in real time and with finite constant velocity. Bisectors that emerge as a result of these coalescences can be understood as relatively stable emergent properties of these measurements. Notice that such internal measurements are not expressed in (rational) values, as would be the case in ‘external measurement’. The internal measurement is the occurrence of the coalescence of expanding forms. It develops without a numerical value being imposed upon it from outside. 7 As for example in Fig. 3.16. 8 As for example in Fig. 3.6.

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3 Expanding Forms: the Geometrical Representation

3.2 Coalescing Circles In the coalescence of circles we encounter the types 0 ⇒ 1, 2 ⇒ 1 and 3 ⇒ 0.

3.2.1 Gain of Form in Coalescing Circles: The Type 0 ⇒ 1 Gain of form in coalescing circles only occurs in singularity points of the type 0 ⇒ 1, i.e., at the moments of first encounter between expanding circles. Two expanding circles will inevitably meet at one moment. At their moment of encounter, there is one single point that they have in common, and it is at this point that their expansions stop. Figures 3.2, 3.3, 3.4 and 3.5 show a few stages of several ongoing processes of coalescence. Their first points of encounter are singularity points of the type 0 ⇒ 1. The points where the expansion processes subsequently stop together constitute a line. As the circles coalesce, their bisector is specified by the consecutive points of their encounter. In the special case that the circles simultaneously start to expand, this line is the bisector of points as known from Euclidean geometry; the configurations developed thus are known as Voronoi diagrams,9 as in Fig. 3.3. In the general case where the circles do not start simultaneously, this line becomes curved. We may call them ‘generalized Voronoi diagrams’,10 but they are also called ‘Johnson-Mehl models’,11 ‘additively weighted Voronoi diagrams’,12 or ‘hyperbolic Dirichlet tessellations’,13 as in Fig. 3.5. We will use the term ‘bisector of circles’ or ‘medial axis’ for these lines or curves that are constituted by the encounter points of expanding circles. For each point on a bisector the difference between the distances to the midpoints of either circle is a constant value. Accordingly, when the circles start at the same moment, this constant difference is zero and their bisector is a straight line, as in Figs. 3.2 and 3.3. When they start at unequal moments, as in Figs. 3.4 and 3.5, the developing bisector is a branch of a hyperbola. At the points of encounter between two coalescing forms the expanding contour lines can be seen to connect pairwise at their points of encounter. Hence the newly composed forms are themselves closed configurations again. Notice that for every pair of coalescing circles their bisector delimits the angular extension of each of the circles’ expanding sectors, irrespective of the circle’s moment of inception. In all cases the angular extension decreases and tends to an asymptotic value. 9

Cf. Ahuja and Schachter (1983, p. 17); Ash and Bolker (1985) call them ‘plane Dirichlet tesselations’. 10 Cf. Ahuja and Schachter (1983, p. 51). 11 E.g. Aurenhammer (1991, p. 351). 12 Anton et al. (2009). 13 Ash and Bolker (1986).

3.2 Coalescing Circles Fig. 3.2 Two circles start to expand at the same moment. Their bisector (symmetry axis) is a straight line. Their point of first encounter is a singularity point of the type 0 ⇒ 1. As the circles coalesce, their bisector is specified by the consecutive points of their encounter

Fig. 3.3 Coalescing circles that start expanding at the same moment specify straight lines or line segments that are bisectors (symmetry axes) between the circle midpoints, yielding a regular Voronoi diagram

13

14 Fig. 3.4 Two circles start to expand at two unequal moments. Their bisector is a curved (hyperbolic) line

Fig. 3.5 Coalescing circles that start expanding at unequal moments specify bisectors (symmetry axes) between the circle midpoints, yielding a generalized Voronoi diagram, or hyperbolic Dirichlet tesselation

3 Expanding Forms: the Geometrical Representation

3.2 Coalescing Circles

15

Fig. 3.6 Loss of form (of the type 2 ⇒ 1) in a circle configuration. The circle with midpoint B has one sector of which the angular extension goes to zero (at the singularity point Z) and one sector of which the angular extension will tend to a stable nonzero value

3.2.2 Loss of Form in Coalescing Circles: The Type 2 ⇒ 1 When three circles coalesce pairwise, loss of form (of the type 2 ⇒ 1) will occur. We will present in Sect. 3.2.3 the singularity points of the type 3 ⇒ 0. At this place we will first focus on the situation in which the expansion of only one of the three circles reduces to zero, while the other two circles continue to coalesce. Various instances of loss of form can be found in Figs. 3.3 and 3.5. Figure 3.6 shows this in more detail. Here we find three circles, with midpoints A, B and C. The pairwise coalescence occurs between the pairs of circles (A, B), (B, C) as well as the pair (A, C). The middle circle’s (midpoint B) expansions can be seen to be delimited by two bisectors (between A and B and between B and C respectively) that, upwardly, diverge, and that, downwardly, converge and meet at the singularity point Z. The two bisectors that meet at Z delimit the angular extension of the expanding circle’s sector. The line BZ indicates the final (zero) extension of this circle’s expanding sector. The expansion rule (the circle’s midpoint B) terminates here. The new bisector between A and C starts at Z, which accordingly is a singularity point of the type 2 ⇒ 1. Notice that the expansion rule of midpoint B does not terminate for its upper circle sector, where the bisectors between A and B and between B and C also delimit this sector of the expanding circle. However, here the bisectors can be seen to diverge. Although the length of this circle’s arc grows as the distance between the two delimiting bisectors grows, its angular extension decreases steadily, approximating a nonzero value as the curvatures of the two (hyperbolic) bisectors tend to zero.

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3 Expanding Forms: the Geometrical Representation

3.2.3 Enclosure and Loss of Form in Coalescing Circles: The Type 3 ⇒ 0 As a configuration of circles is coalescing with sufficiently many (at least two) other configurations, it may also come to coalesce with itself and enclose some of its external space. Figure 3.7 shows a configuration of circles, in which not only three circles coalesce pairwise (such as at points C and D) but also in a way that creates enclosures around the points A and B. The enclosure occurs after the coalescing circle pairs (P, R) and (P, Q) have become a configuration that meets itself where the circle pair (Q, R) meets and coalesces. This is shown in Fig. 3.8. We see the deltoid (in fat lines) that has become an enclosure as soon as the circles P and R meet. The line segment QR is obviously longer than both PR and PQ, so the enclosure occurs only after the last pair of circles has met. Notice that this enclosed and shrinking deltoid is external to the three constituting circles, as we will see in the enclosed triangle of Fig. 3.22. The deltoid of Fig. 3.8 is going to shrink and terminate in point A, as the three contributing circles continue to expand. Meanwhile, the angular extensions of the shrinking arcs of the three circles (midpoints P, Q and R) are reduced to zero as the deltoid converges to point A. In Fig. 3.6 we saw at point Z the reduction of angular Fig. 3.7 Expanding circles coalesce and eventually enclose deltoids that converge at singularity points A (see Fig. 3.8) and B (see Fig. 3.10), both of the type 3 ⇒ 0. The singularity points C and D are of the type 2 ⇒ 1. The points shown in Fig. 3.7 will be referred to in the Figs. 3.8, 3.9, 3.10, 3.33, 3.34 and 3.35

3.2 Coalescing Circles

17

extension down to zero and the termination of a single expansion rule. Here (Fig. 3.9) the same reduction happens to the three arcs of the deltoid, simultaneously. The sectors of these circles are mutually delimiting, thus defining their hyperbolic bisector lines and enclosing the shrinking deltoid that ends in the termination point A. The sectors and their arc lengths grow smaller as the circles’ expansions continue up to the termination point A. At that point the sectors and their arc lengths have become of zero extension. The same type of circle convergence can be described with respect to the deltoid enclosure that shrinks around point B of Fig. 3.7 and terminates in it. This is shown in Fig. 3.10. Notice that the points to which the enclosed configuration converges by its inward expansion are the termination points of a tree, as in Fig. 3.11. These termination points are always the points towards which three expanding circles make an enclosure and converge. This enclosure has the shape of a deltoid, as in Figs. 3.9 and 3.10. The position of such a termination point and the time at which it is attained both depend on the configuration only. The termination points are the final singularity points where the three final sectors of zero extension meet (3 ⇒ 0). Fig. 3.8 Three circles (see Fig. 3.7) come to enclose a deltoid that shrinks as the circles expand

18 Fig. 3.9 The convergence towards termination at point A. The enclosed deltoid shrinks and the angular extensions of the three converging sectors decrease accordingly as the three circles continue to expand. At the termination point A the sectors end at zero angular extension. The line segment PA represents the final moment of the shrinking sector of circle P. Here it has zero angular extension, as have the line segments QA and RA. The three directions of these terminating sectors will inform the three expansion directions of the triangles that incept from A (as in Fig. 3.33). (The points are those of Fig. 3.7.)

Fig. 3.10 Circle configuration (detail from Fig. 3.7): the deltoid enclosure terminates at the singularity point B, thus determining the line segments RB, SB and TB, which are the circle sectors that have decreased to a zero angular extension. (The points are those of Fig. 3.7.)

3 Expanding Forms: the Geometrical Representation

3.3 Coalescing Triangles

19

Fig. 3.11 An enclosure of circles with multiple points of termination

3.3 Coalescing Triangles In the coalescence of triangles we encounter the types 0 ⇒ 3, 1 ⇒ 2, 2 ⇒ 1 and 3 ⇒ 0.

3.3.1 Gain of Form in Coalescing Triangles The most elementary type of gain of form in triangles is the type 0 ⇒ 3. This occurs when a triangle originates at a particular point. The points A and B in Fig. 3.12 illustrate this type. They are the points of inception of the outward expansion of the triangles. Two expanding triangles will meet in a way that their expansion processes stop at their subsequent points of encounter. The most generic case is that of a triangle’s angle meeting one of the other triangle’s sides. This type of coalescence is shown in Figs. 3.12 and 3.13. Notice that the point C of the initial encounter between the two triangles is a singularity point of one incoming (terminating) and two outgoing (incepting) bisectors. Especially at the points of encounter between two coalescing forms, as in the point C of Figs. 3.12 and 3.13, and point R in Fig. 3.14, the expanding contour lines can

20

3 Expanding Forms: the Geometrical Representation

Fig. 3.12 Gain of form in expanding triangles: contours and bisectors. The triangles incept at A and B at singularity points of the type 0 ⇒ 3. The point C is a singularity point of the type 1⇒2

Fig. 3.13 (detail from Fig. 3.12) One bisector splits into two bisectors at the encounter point C: gain of form. The point C is a singularity point of the type 1⇒2

be seen to connect pairwise at their points of encounter. Hence as was the case for circles, the newly composed forms are themselves also closed configurations again, i.c. polygons. Finally, we mention here a special type of gain of form in triangles that we will call ‘coalescence from within’, see Sect. 3.4.4.2 and see Figs. 3.24, 3.44 and 3.46.

3.3 Coalescing Triangles

21

Fig. 3.14 Gain of form, another example of a singularity point of the type 1 ⇒ 2 (detail from Fig. 3.21)

3.3.2 Loss of Form in Coalescing Triangles The events at those singularity points that have two incoming bisectors and one outgoing bisector are called ‘loss of form’. These events occur both in configurations of triangles (Fig. 3.16) and in configurations of circles (Fig. 3.6). For singularity points of the type ‘loss of form’, the length of one contour line is reduced to zero. In Fig. 3.15 we find the development of the expansion process shown in Figs. 3.12 and 3.13. Here the singularity point D is of the type 2 ⇒ 1: two bisectors terminate and one bisector incepts, as was the case for circles in Fig. 3.6. A special variety of this type of loss of form (2 ⇒ 1) has one internal bisector meet one external bisector, giving rise to one new external bisector, as will be shown in Fig. 3.46. Finally, loss of form of the type 3 ⇒ 0 will be discussed in Sect. 3.3.4.

3.3.3 Positive, Negative and Mixed Expansion of Contour Lines A variety of cases can be distinguished for triangle expansions. First, a contour line’s growth can be fully positive, that is positive at its both bisectors. This is the condition of double positive expansion, as in the triangle of Fig. 3.1. Second, a contour line’s growth can be fully negative, i.e. negative at its both bisector ends. This is the condition of double negative expansion. Figure 3.17 shows this case.

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3 Expanding Forms: the Geometrical Representation

Fig. 3.15 The process of Fig. 3.12 continued: at singularity point D (of the type 2 ⇒ 1) two bisectors terminate and a new bisector incepts

Fig. 3.16 Loss of form at the singularity point D: detail from Fig. 3.15. Two bisectors meet and join as two expanding triangles coalesce. One contour line segment is reduced to zero linear extension at singularity point D

Third, a contour line’s growth may be positive at one bisector, and negative at the other. We may call this the condition of mixed expansion. Figure 3.18 shows this condition.

3.3 Coalescing Triangles

23

Fig. 3.17 Loss of form (2 ⇒ 1) in an expanding polygon configuration (detail from Fig. 3.21). Two obtuse angles between contour lines converge onto one obtuse angle between contour lines. The middle of three expanding contour lines decrease on both sides and terminates at the singularity point P

Fig. 3.18 Loss of form (2 ⇒ 1) in an expanding polygon configuration (detail from Fig. 3.21). Two angles between contour lines, one obtuse and one acute, converge onto one obtuse angle between contour lines. The middle of the three expanding contour lines decreases on one side and increases at the other side (mixed expansion); it terminates at the singularity point

3.3.4 Enclosure and Loss of Form in Coalescing Triangles: The Type 3 ⇒ 0 Figures 3.19 and 3.20 show several expanding triangles, before and after three of which come to enclose a space that is external to these expanding triangles. As soon as the enclosure occurs, the resulting triangle has become a new external contour of the expanding polygon, an enclave of the external space, one that is shrinking as the polygon continues to expand. Figure 3.21 shows the wider context of this event and Fig. 3.22 presents it in more detail.

24 Fig. 3.19 Three expanding triangles shortly before they come to enclose part of their external space, which becomes a shrinking triangle, expanding inwardly. More about this configuration is in Fig. 3.21

Fig. 3.20 The situation of Fig. 3.19, after the enclosure has occurred: the enclosed triangle shrinks. More about this configuration is in Fig. 3.21

3 Expanding Forms: the Geometrical Representation

3.3 Coalescing Triangles Fig. 3.21 The development of the coalescing triangles of Figs. 3.19 and 3.20, before and after the enclosure of the triangle (indicated by the three fat arrows). Details of the enclosure and its termination point M are shown in Figs. 3.22 and 3.37. (See also Figs. 3.17 and 3.18 for more details of the points P and Q; see Fig. 3.14 for a more detailed view of the point R and its environment.)

Fig. 3.22 The onset of a new contour that expands internally (see Fig. 3.21) is marked by the singularity point L, preceded by J and K, each of the type 1 ⇒ 2. The three bisectors JM, KM and LM of the shrinking triangle indicate its termination point M, which is of the type 3 ⇒ 0. The continuation of the process at that point M is shown in Fig. 3.37

25

26

3 Expanding Forms: the Geometrical Representation

The enclosed triangle, or polygon, as it is shrinking, is eventually to vanish. That means that at least one position exists, as a point of termination, towards which this shrinking triangle or polygon converges. Figures 3.19 and 3.20 give a generic example of this. A polygon enclosure does not necessarily start as a triangle, but eventually it will be split and reduced into one or more final triangles. An enclosed space may take the shape of any polygon, as in Fig. 3.23. There the shrinking polygon splits into two shrinking polygons, each of which converges onto one or more points of termination. The number of terminal points is equal to the number of splits that a closed boundary undergoes (including the first one that we called ‘enclosure’). In this way the points to which the polygon converges by its inward expansion are the termination points of a tree, as was the case for inward expansion in circle configurations (e.g. in Fig. 3.11). Eventually all termination points are situated within a shrinking triangle that is converging upon it, comparable to the shrinking deltoids in the circle configurations (cf. Figure 3.8). The position of such a termination point and the time at which it is attained both depend on the configuration only. The termination point is the final singularity point where the triangle’s angular bisectors meet (3 ⇒ 0). Fig. 3.23 An enclosed shrinking polygon is split up (1 ⇒ 2) at A into two shrinking polygons, which leads to B (2 ⇒ 1) and two termination points C and D (3 ⇒ 0)

3.3 Coalescing Triangles

27

Fig. 3.24 Acute vertices of expanding polygons approach and touch ‘from within’ upon expanding contour lines. Acute vertices are faster in their outward movements than the contour lines upon which they bump. As a result, the produced bisectors are symmetry axes (internal bisectors) to the expansion directions, but they are external bisectors to the contour line segments that they delimit.

3.3.5 Acute and Obtuse Angles, Internal and External Bisectors of Polygons For triangles there are three possibilities with respect to the development of contour line lengths. First, if the expanding contour has an acute angle at which a bisector is realized, then the contour lines grow on both sides in the vicinity of the bisector. An example of this case is the increase of the contour lines of the expanding triangle of Fig. 3.1. Second, if two neighboring contour lines have an obtuse angle, then they decrease in the vicinity of the bisector. Examples of such decrease can be found in Figs. 3.17 and 3.18. A bisector is primarily a symmetry axis between expansion rules. For triangles and polygons these rules are the expansion directions of the various contour line segments. Bisectors are the symmetry axes between these directions. In many instances this also means that a bisector is the angular bisector of two adjacent contour line segments. Whenever that is the case, the lengths (extension) of the adjacent contour line segments change equally: in acute angles the contour line segments grow equally on both sides of the bisector; in obtuse angles they shrink equally. When two polygons coalesce such that the subsequent points of their encounter make up the external angular bisector of the contour lines, then their bisector is still a symmetry axis (internal bisector) of their expansion rules (expansion directions), but not a symmetry axis of the coalescing contour lines. For both contour line segments are now on the same side of the angular bisector, which is their external angular bisector. There is increase of length in one contour line segment, and decrease of length in the other, as in Fig. 3.24. These external bisectors can only occur within

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3 Expanding Forms: the Geometrical Representation

triangle enclosures, as described in Sect. 3.3.4. More details about these coalescences ‘from within’ will be discussed in Sect. 3.4.4.2 (Figs. 3.44 and 3.46). Finally, as mentioned in Sect. 3.3.2, there is also a variety of ‘loss of form’ in which a singularity point has one outgoing external bisector and two incoming bisectors, one internal and one external (Fig. 3.46).

3.3.6 Mixed Expansion Triangles Mixed expansion triangles or mixed expansion polygons have contour lines of both inward and outward expansion directions. Either one or two sides of a mixed expansion triangle are expanding inwardly, and either two or one sides are expanding outwardly. These expansion types can only occur when triangles newly incept within enclosures, as described in Sect. 3.3.4.

3.3.6.1

Mixed Expansion Triangles: Two Sides Inward and One Outward

Figure 3.25 shows the type of mixed expansion in which one side expands outwardly and two sides expand inwardly. It is obvious that its two external bisectors and its internal bisector converge at the triangle’s termination point. Figures 3.26 and 3.27 show how this particular instance originates and develops. Notice that this type of mixed triangle is capable of displacing over a relatively long

Fig. 3.25 A triangle that has two sides expanding inwardly and one side expanding outwardly has two external bisectors and one internal bisector. The triangle can be seen to displace and eventually converge upon a single termination point which may be situated far beyond the earlier positions of this triangle. Figures 3.26 and 3.27 show its origination and its subsequent stages.

3.3 Coalescing Triangles

29

distance. In Sect. 3.4.4.1 we will relate this to the circles that incept at positions beyond the circle configurations of preceding generations.

Fig. 3.26 Origination and development of the mixed expansion triangle (see Fig. 3.25)

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3 Expanding Forms: the Geometrical Representation

Fig. 3.27 Fig. 3.26 continued: origination and development of the mixed expansion triangle (see Fig. 3.25)

3.3.6.2

Mixed Expansion Triangles: Two Sides Outward and One Inward

Figure 3.28 shows the type of mixed expansion in which one side expands inwardly and two sides expands outwardly.

3.4 Circles and Triangles: Three Spaces

31

Fig. 3.28 Mixed expansion triangle: like the type of mixed expansion shown in Fig. 3.25, there are two external bisectors and one internal bisector. However, unlike the previous type, this type of triangle expansion does not shrink; it has two sides that expand outwardly and only one that expands inwardly. Figure 3.29 shows its origination

The three bisectors of Fig. 3.28 do not actually spring from a single inception point (as would have been an analogy to the real termination point of the other type of mixed expansion triangles, as in Fig. 3.25). The origination and development of this particular triangle is shown in the series of Fig. 3.29.

3.4 Circles and Triangles: Three Spaces The expansion processes of circles and of triangles are conceived to take place in three separate isometric real coordinate spaces in R2 , one space of circles and two spaces of triangles. Each point in each space uniquely corresponds to a point in either of the two other spaces. We assume one space in which circles have displacing contour lines, appearing as expanding circles. We will use the term ‘circle space’. We also assume two separate spaces in which triangles have parallel expanding contour lines. As these triangles meet and coalesce they constitute polygons. We will use the terms ‘triangle space’ as well as ‘polygon space’. We will assign the color red to the circle space and the two colors green and blue to the two triangle spaces, instead of the ‘neutral’ cyan. We may visualize these spaces by imagining three parallel sheets of paper: the middle one represents the circle space and the two outer sheets represent the triangle spaces. Figure 3.30 shows this idea as a trolley with three layers. As a metaphor this trolley immediately falls short due to the fact that its layers are situated in a third dimension (height), some 30 cm away from each other, whereas between our three geometric spaces no third dimension (height) is assumed, and hence no distance between spaces is defined.

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3 Expanding Forms: the Geometrical Representation

Fig. 3.29 The origination and development of the mixed expansion triangle of Fig. 3.28

Another helpful metaphor for this type of relatedness can be found in electronic charts or atlases: the various layers or slices of the map (streets, accidentation, vegetation, etc.) are considered relevant for each point of the map, although usually not shown simultaneously. All metaphors have their flaws and none of them would be helpful with respect to a particular feature of the three isometric spaces, one that plays a major role in the kind of processes that we describe here and that we will call a transient identification. This is to say that at a particular single moment in time a point in one space will be identified with a corresponding point in another space. We will describe

3.4 Circles and Triangles: Three Spaces

33

Fig. 3.30 Three spaces, depicted as parallel layers on a room service trolley

the processes that specify these transient identifications in Sects. 3.4.1 and 3.5.3 in terms of ‘transmutation’ and ‘termination/inception’. The expansion processes can be represented in various ways: the situation that is the case either at one particular moment in time, as in Figs. 3.19 and 3.20, or at several consecutive moments, as in Fig. 3.1; either one single space, as the three images of Fig. 3.31, or more spaces on top of each other, as in Fig. 3.32. Let us have a closer look at the images that are laying on the three shelves of this trolley. In Fig. 3.31 the three spaces are shown separately and in Fig. 3.32 they are represented on top of each other. Let the three spaces be fully separate, that is, let them exist as spaces that are unconnected for most points at most of the time. This means: we will encounter and pay special attention to those points and those moments at which these spaces will not be disjoint. These particular points and moments of identification will be introduced in Sect. 3.4.1.

34

3 Expanding Forms: the Geometrical Representation

Fig. 3.31 The images at the shelves of the trolley of Fig. 3.30. On the middle shelf we find an image containing circles and on the other two shelves we find images that contain polygons

3.4.1 Termination and Inception: The Transmutation into a New Context Above we have introduced the idea of transient identification between points in the various geometric spaces. We will now describe how the expansion processes of circles and triangles in these spaces are interrelated. The central idea is here that at a point of termination, in the spaces of triangles as well as in the space of circles, there is always a triple of expansion rules that terminate, whereas, at the very moments of termination, the domains of validity of the terminating expansion rules have arrived at zero extension (linear for polygon contour lines, angular for circle arcs). At these moments the final zero extensions specify the new expansion rules, i.e. for a new form within a new space. The points of termination thus immediately serve as new points of inception, from which a new expansion processes develop, such that they make use of contextual properties of the termination/inception point. The transmutation from termination to

3.4 Circles and Triangles: Three Spaces

35

Fig. 3.32 The same three spaces as in Figs. 3.30 and 3.31, shown together. Notice that four pairs of green/blue inverse triangles can be recognized

inception is not a logical consequence14 of the preceding expansions in the preceding spaces of triangles or circles. This transition is a relation between the various spaces, and as such is it additional to the expansion processes themselves. The present section deals with how these transmutations takes place. termination point in

yields inception point of

circle space (enclosed deltoid)

2 expanding triangles

polygon space (enclosed triangle)

1 expanding circle

We will use the term ‘transmutation’ for the conversion from one type of space into a different type of space. We will use the term ‘transition point’ to indicate the points of identification between the spaces, where the transmutations take place.

3.4.2 The Establishment of New Expansion Rules during Transmutations Singularity points of the 3 ⇒ 0 type are always at the inside of a self-enclosed configuration (see Sects. 3.2.3 and 3.3.4). The enclosed form is expected to decrease up to the final convergence points. It is at these termination points that the expansion process can be conceived to transmute into a different context. Within the new context 14

There has been an element of aesthetic judgment in our conceptualization of the transmutations. See also Sect. 5.1.

36

3 Expanding Forms: the Geometrical Representation

Fig. 3.33 The three final directions of the terminating circle sectors PA, QA and RA (see Figs. 3.7 and 3.9) define the three expansion directions of each of the two inverse triangles, shown in green and blue respectively. Both triangles start expanding from the point A. (The points are those of Fig. 3.7.)

Fig. 3.34 The three final directions of the terminating circle sectors RB, SB and TB (see Figs. 3.7 and 3.10) define the three expansion directions of each of the two inverse triangles, shown in green and blue. Both triangles start expanding from the point B. (The points are those of Fig. 3.7.)

the expansion process is to continue. That is, for expanding circles the new context is the two spaces of triangles, and for each of the latter the new context is the circle space. This change of context can be visualized as the process’ passage through a single point of zero extension, a point of transient identification. Thus the process obtains some ‘transfinite’15 properties. Figure 3.33 shows the environment of point A, as in Figs. 3.8 and 3.9, within a new context. As we saw in those figures, the expansions of three circles terminate at A. The line segment PA has a direction that has now become of importance, together with the directions of QA and RA, for the specification of triangles that newly incept at A. Notice that we use to present triangles in green when their expansion rules 15

Cf. footnote [6] of Chap. 5.

3.4 Circles and Triangles: Three Spaces

37

Fig. 3.35 The first few subsequent stages of triangle expansions from points A and B (see the corresponding configurations of Figs. 3.33 and 3.34). It is obvious that at point A triangles start to expand earlier than at point B. This corresponds to the distance of RA being smaller than that of RB, that is, the circle enclosure at B terminates later than the one at A. (The points are those of Fig. 3.7.)

are directed towards the midpoints of the preceding circles, and in blue when the expansion rules point in the opposite directions, away from these circle midpoints. The transmutation in the other direction, from either space of triangles to the space of circles, goes in a similar way. Figure 3.37 shows the triangle configuration of Figs. 3.19, 3.20 and 3.22. The shrinking enclosed triangle has a termination point, at which a circle incepts and expands.

Fig. 3.36 Continuation of the processes of Fig. 3.35: subsequent stages of the polygon expansions. The blue triangles and the green triangles coalesce each in their own separate space of polygon expansions

38

3 Expanding Forms: the Geometrical Representation

Fig. 3.37 The shrinking enclosed triangle of Figs. 3.19, 3.20 and 3.22: the shrinking enclosed triangle’s termination point M is the newly emerging circle’s inception point. (Points K and M as in Fig. 3.22)

3.4.3 Examples of Transmutations This section provides some illustrations of the transmutation processes described above. The transmutations between the spaces of triangles and circles are presented in either direction.

3.4.3.1

From Triangles to Circles

See Fig. 3.38

3.4.3.2

From Circles to Triangles

See Figs. 3.39, 3.40 and 3.41

3.4 Circles and Triangles: Three Spaces

39

Fig. 3.38 Inception of circles after termination of triangles: the first circle stems from a shrinking green triangle, the second circle from a blue triangle, etc.

40 Fig. 3.39 Transmutation from the space of circles to the two triangle spaces. The enclosure between four circles gives rise to two pairs of blue and green triangles. The blue triangle with the very acute angle is the one also shown in Figs. 3.44 and 3.45, as well as in Figs. 3.48, 3.49, 3.50 and 3.51. The details of this transmutation are shown in Fig. 3.41

Fig. 3.40 Detail from the lower left image of Fig. 3.39. The further development of the enclosed form is shown in Fig. 3.41

3 Expanding Forms: the Geometrical Representation

3.4 Circles and Triangles: Three Spaces

41

Fig. 3.41 Details of the circle enclosure and transmutation process of Fig. 3.39. It is obvious that the four coalescing circles lead to an enclosure that has two termination points, so that two pairs of green and blue triangles incept. Blue triangles and green triangles respectively meet and coalesce

42

3 Expanding Forms: the Geometrical Representation

3.4.4 Generations of Configurations As the transmutations between spaces of circles and of triangles proceed, transmutations from a space of triangles to circles is followed by a transmutation from the space of circles to those of triangles, and so on. Thus, two consecutive transmutations give rise to the occurrence of generations of forms of the same kind, circles or triangles.

3.4.4.1

Generations in the Space of Circles: Permanent Levels of Nesting

A distinction between generations of circles is established permanently for those new circles that incept within the domain of realization of an already existing circle. The circularity between the spaces of triangles and circles entails that circles can only interact with circles of the same depth of nesting, as is shown in Fig. 3.43. The depth of nesting of a circle is determined by the position of its inception point. No interaction is possible between circles of unequal depth of nesting. Generally, a newly incepting circle will be nested at one level deeper than circles of a generation previous to it. However, due to circumstances in the spaces of triangles (among which the kind of mixed triangles as described in Sect. 3.3.6.1), a polygon configuration may create a point of termination/inception such that a newly incepting circle is situated beyond the boundaries of circles of previous generations. (For instance, a mixed triangle will have moved away from its original position and may have terminated at a point, at which a new circle incepts, situated beyond the confines of an older circle configuration.) In those cases the new circle will expand, meet and coalesce with its previous generation, i.e., it will not be nested at a deeper level where it would remain within the contours of an older circle generation. Figure 3.42 shows the occurrence of such external inception. It can also be recognized in Fig. 3.43. Notice that these generations of coalescing circles bear some optical resemblance to the patterns of the Belousov-Zhabotinsky chemical reaction.

3.4.4.2

Generations in the Spaces of Triangles: Coalescence from Within

On the other hand, for triangles no strict level of nesting can be established. This is due to the elementary property that the displacement of an expanding contour line is larger when measured along its bisectors than along its displacement direction itself (as was mentioned in Sect. 3.1.2). Therefore, a distinction between generations of polygons can exist only temporarily; generations of expanding polygons will meet and coalesce whenever the vertex of a later generation meets the contour line of an earlier generation. This brings us to the issue of ‘coalescence from within’. We already encountered this

3.4 Circles and Triangles: Three Spaces

Fig. 3.42 A circle incepts beyond the boundaries of circles of previous generations

43

44

3 Expanding Forms: the Geometrical Representation

Fig. 3.43 Two stages of a configuration of expanding circles (the corresponding triangle spaces are not shown). The various generations of circles (6 left, 14 right) remain distinct (unlike generations of polygons as in Fig. 3.52). Red and black are used as colors for odd and even generations respectively

type in Fig. 3.24. Figures 3.44 and 3.46 show two examples of this type. Each of these examples will be explained and elaborated, first by an image that shows the subsequent steps (Figs. 3.45 and 3.47 respectively); then by showing these expansion processes as simultaneous events (Figs. 3.48, 3.49, 3.50 and 3.51). Due to the various and multiple occurrences of coalescence from within, triangle generations do not remain distinct, as do circle generations. Figure 3.52 shows the two triangle configurations that correspond to the 14 generations circle configuration of Fig. 3.43 (right).

3.5 Bisectors, Singularity Points, Enclosures, Transmutations: Recapitulation We resume here the variety of bisectors, singularity points and transmutations.

3.5 Bisectors, Singularity Points, Enclosures, Transmutations: Recapitulation

45

Fig. 3.44 A coalescence from within: the quickly moving vertex with acute angle of a newer triangle/polygon coalesces with a contour line segment of an older (previous generation) polygon, creating a singularity point of the type 1 ⇒ 2 (gain of form, cf. Sect. 3.3.5), yielding two external bisectors, as in Fig. 3.24. The series of images in Figs. 3.48, 3.49, 3.50 and 3.51 shows how this coalescence originates. Figure 3.45 shows the various steps in more detail. Figure 3.39 shows how this particular triangle was created

Fig. 3.45 Coalescence from within: various subsequent steps of the process (see Fig. 3.44)

46

3 Expanding Forms: the Geometrical Representation

Fig. 3.46 Coalescence from within: a new triangle’s quickly moving vertex with acute angle meets the inside of a contour line of an already existing polygon at singularity point A (gain of form, the type 1 ⇒ 2, one ingoing internal bisector and two outgoing external bisectors, as in Figs. 3.24 and 3.44, cf. Sect. 3.3.5). Furthermore, singularity points B and C show subsequent losses of form (the type 2 ⇒ 1), but here in a new variety: one internal bisector and one external bisector meet and a new external bisector starts. (Notice that the new external bisector at point C only seems to be in line with the internal bisector that terminates in C.) The series of images shown in Figs. 3.48, 3.49, 3.50 and 3.51 shows how this coalescence originates. Figure 3.47 shows the various steps in more detail.

Fig. 3.47 Coalescence from within: various subsequent steps of the process (see Fig. 3.46)

3.5 Bisectors, Singularity Points, Enclosures, Transmutations: Recapitulation

47

Fig. 3.48 Coalescence from within: the processes shown in Figs. 3.45 and 3.47 are shown here in the simultaneous developments in three spaces.‘Coalescence from within’ occurs in both triangle spaces, where a quickly expanding polygon vertex (acute angle) coalesces with an older contour line. To be continued in Figs. 3.49, 3.50 and 3.51.

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3 Expanding Forms: the Geometrical Representation

Fig. 3.49 Continuation from Fig. 3.48: coalescence from within

3.5 Bisectors, Singularity Points, Enclosures, Transmutations: Recapitulation

Fig. 3.50 Continuation from Fig. 3.49: coalescence from within

49

50

3 Expanding Forms: the Geometrical Representation

Fig. 3.51 Continuation from Fig. 3.50: coalescence from within

3.5 Bisectors, Singularity Points, Enclosures, Transmutations: Recapitulation

51

Fig. 3.52 The two triangle configurations that correspond to the circle configuration of Fig. 3.43 (right). No distinctions can be made between generations of polygons, as newly incepting triangles quickly coalesce with older configurations

3.5.1 Bisectors The expansion processes that are described by the expansion rules are mutually delimiting the domains of validity of the expansion rules involved. Whenever two expanding configurations meet, the expansion rules are no longer valid at the first point of encounter and at all other subsequent points that are found in the course of the ongoing expansion processes. It is at these points of mutual constraint that the bisectors can be understood as local symmetry axes. These symmetry axes (bisectors) can be understood as emergent properties of the expansion processes, so that a reciprocally descriptive relation takes place: • expansion rules describe contour line expansions • expanding contour lines describe bisectors • bisectors describe the limits of the domains of validity of expansion rules. 3.5.1.1

Circle Bisectors

Expansion rules are here the positions of the circle midpoints. For each point on a bisector the difference between the distances to the midpoints of either circle is a constant value. Accordingly, when the circles start at the same moment, this constant difference is zero and their bisector is a straight line. When they start at unequal moments, the developing bisector is a branch of a hyperbola.

52

3.5.1.2

3 Expanding Forms: the Geometrical Representation

Triangle Bisectors

Expansion rules are here the displacement directions of the sides of triangles and polygons. A bisector is primarily a symmetry axis between expansion rules. For triangles and polygons these rules are the expansion directions of the various contour line segments. Bisectors are the symmetry axes between these expansion directions. This holds also in those cases of ‘coalescence from within’, i.e. when triangles from originally diverse generations meet.

3.5.2 Singularity Points Whereas the contour lines (either of circles or of triangles) expand as a continuous process, changes of the number of expansion rules are discrete. Singularity points are the points at which such a change occurs. We encountered various types of singularity points in circle configurations and in triangle configurations: Type of event:

occurs:

Loss of form:

in circles:

in triangles:

2 ⇒ 1

x

x

3 ⇒ 0

x

x

Gain of form: 0 ⇒ 3

x

1 ⇒ 2 0 ⇒ 1

3.5.2.1

x x

Loss of Form

The type 2 ⇒ 1: Two bisectors converge. For triangles see e.g. Figure 3.16 (in Sect. 3.3.2); for circles see e.g. Figure 3.6 (in Sect. 3.2.2). In ‘loss of form’ of this type there are three expansion rules, of which one extinguishes. Figure 3.46 (in Sect. 3.4.4.2) shows two instances of a special case, in which one internal bisector and one external bisector converge and end at the singularity point, from which one new external bisector departs. The type 3 ⇒ 0:

3.5 Bisectors, Singularity Points, Enclosures, Transmutations: Recapitulation

53

Three bisectors converge simultaneously and three expansion rules stop simultaneously. This was called ‘termination’. See Fig. 3.21 (in Sect. 3.3.4) for converging triangle bisectors and see Fig. 3.7 (in Sect. 3.2.3) for converging circle bisectors.

3.5.2.2

Gain of Form

The type 0 ⇒ 3: Three bisectors bifurcate simultaneously out of one point; this was called ‘inception’ (see Fig. 3.1 in Sect. 3.1). This type of singularity only exists for triangles. Here a triple of new expansion rules is established. The type 1 ⇒ 2: One bisector bifurcates into two. This type only occurs in polygon configurations. It was called ‘gain of form’ (cf. Sect. 3.3.1; see point C in Fig. 3.12). Figure 3.24 (cf. Sect. 3.3.5) showed a special case, ‘coalescence from within’ (elaborated in Sect. 3.4.4.2) in which one internal bisector bifurcates into two external bisectors. The type 0 ⇒ 1: This type only occurs in circle configurations. A bisector arises out of one point, as shown in Figs. 3.2 and 3.4. This was also called ‘gain of form’ (cf. Sect. 3.2.1).

3.5.2.3

Enclosure: The Split into Inner and Outer Boundaries

As a configuration of polygons or circles is coalescing with sufficiently many (at least two) other configurations, it may also come to coalesce with itself and enclose some of its external space. This has been illustrated for circles in Fig. 3.7 and for polygons in Fig. 3.19. When this enclosure happens, the configuration’s contour line splits into an outer contour line and an inner contour line, both of which are closed. Thus, the new inner contour line will inevitably be displaced inwardly, due to the outward expansion of the original expansion processes. Accordingly, we may loosely use the terms ‘growing configuration’ and ‘shrinking configuration’ for the expansions of the outer contour lines and for the expansions of the inner contour lines respectively. Thus, we may characterize the enclosures, both in configurations of circles and in configurations of polygons: • two coalescing configurations develop one boundary • at the occurrence of an enclosure a configuration’s boundary splits into an inner boundary and an outer boundary. Given the enclosure of space by a particular expanding configuration, the inner contour line is expanding outwardly, when seen from the perspective of the expanding configuration. On the other hand, when seen from the perspective of the enclosure

54

3 Expanding Forms: the Geometrical Representation

itself, it is expanding inwardly, and it will converge upon one or more final points, to be called points of termination. Losses of form and splits into less complicated separate configurations may happen in the course of these inward expansions. We have discussed this for circles in Sect. 3.2.3 and for triangles in 3.3.4. Both in triangles and in circles the domains of validity of the expansion rules are delimited as the configurations coalesce. In singularity points where loss of form occurs (2 ⇒ 1, 3 ⇒ 0) it is the linear extension that goes to zero for triangles and polygons; for circles it is the angular extension that goes to zero (cf. Sect. 3.1.4).

3.5.3 Transmutations: Transitions between Spaces Transmutations between spaces are the events of a process continued in a different type of space: a shrinking triangle that terminates in a triangle space is continued as an expanding circle that incepts in the space of circles; conversely, a shrinking deltoid that terminates in the circle space is continued as a pair of inverse triangles that incept and expand in their respective triangle spaces. The term ‘transfinite’ was an appropriate qualification. Thus, termination points become inception points between the spaces of circles and of triangles, in both directions. Thus, generations of configurations occur within all spaces. However, within the space of circles these generations remain distinct, whereas in the spaces of triangles the various generations will meet and coalesce, giving rise to the occurrence of external bisectors. Let us discuss in some more detail what happens at these points of transition between the domains of circles and of triangles. For the transition from triangles to circles (Fig. 3.37) we notice that the final position of the shrinking triangle’s contour lines, where they finish in one single point, is now preparing the ground for the midpoint of a new expanding circle. We notice that the final point of the shrinking triangle is the last instance of the three contour lines of the enclosed triangle, which have attained zero extension. Furthermore, this final position, as it becomes the midpoint of a new expanding circle, becomes the rule by which the new circle expansion can be described. Conversely, for the transition from circles to triangles (Figs. 3.9 and 3.33) we noticed that the triple of final orientations of the shrinking circle sectors - which specify the final expansion directions of the three expanding circles at their point of convergence - specify the expansion directions of the pair of inverse triangles (shown in blue and green) that incept at the point of termination of the circle convergence. At the transition from termination to inception there is a rearrangement of relations: • old (final) domain of validity specifies new (initial) expansion rule for circle to triangle transmutations: the final three sectors (of zero angular extension) specify the expansion directions of the new pair of triangles; for triangle to circle transmutations: the final position of the enclosed triangle specifies the midpoint of the new expanding circle

3.5 Bisectors, Singularity Points, Enclosures, Transmutations: Recapitulation

55

• old expansion rule is lost for circle to triangle transmutations: the original circle midpoints do no longer play any role; for triangle to circle transmutations: the shrinking triangle’s (inward) expansion directions do no longer play any role • domain of validity of the new expansion rule is delimited by the newly emerging bisectors for triangle expansions: the new triangle’s bisectors delimit the domains of validity of each of its three expansion directions; for circle expansions: where two circles meet, the points of their encounter do no longer contribute to the circle expansions; thus, for a pair of coalescing circles their bisector delimits the domain of validity of each circle’s midpoint qua expansion rule. Thus, the expansion processes bear a kind of continuity that is beyond the range of expansion rules as described thus far. In this respect the termination/inception points differ from the other types of singularity points (cf. Sect. 3.1.3). The points of termination/inception are the points of transient identification between the spaces of triangles and circles. They make up the connection between the spaces and this connection means that any termination yields the inception of a new expansion process; likewise it means that any inception is based on a preceding termination of an expansion process in the other space.

3.5.3.1

From Triangles to Circles

Enclosures occur in the spaces of triangles as soon as an expanding polygon touches upon itself and coalesces with itself (as in Fig. 3.19). Such enclosures are shrinking polygon configurations and they end as one or more shrinking triangles, each of which terminates in a single point. Such point is the locus where the triangle has become of zero linear extension. A new circle incepts at the point in the circle space that corresponds to the point in the triangle space at which an enclosed triangle has terminated. We consider such momentary correspondence an event of transient identification, namely between the shrinking triangle’s point of termination and the newly expanding circle’s point of inception.

3.5.3.2

From Circles to Triangles

Enclosures occur in the space of circles as an expanding circle configuration touches upon itself and coalesces with itself (as in Fig. 3.7). Such enclosed configurations consist of contour lines that are the sectors of contributing expanding circles. Such enclosure shrinks and ends as one or more shrinking deltoids, each of which terminates in a single point. Such point is the locus where the deltoid has become of zero angular extension. A pair of new triangles incepts, each in its own triangle space, that corresponds to the point in the circle space at which a shrinking deltoid has terminated. As in the transmutation from triangles to circles, we consider such momentary correspondence an event of transient identification between the shrinking deltoid’s point of termination and the pair of inception points of the two newly expanding triangles.

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3 Expanding Forms: the Geometrical Representation

Fig. 3.53 The relations between three circles and their shared tangent circle, the circle of the so called tenth problem of Apollonius. A, B and C denote the points where each of the three circles (midpoints X, Y and Z) is tangent to this circle of the tenth Apollonius problem. The triangle PQR has A, B and C as its side midpoints and D, E and F as its perpendicular feet. All six points are situated on this circle. Hence, this circle is a Feuerbach circle to the triangle PQR. The sides of the pink triangle DEF can be seen to be perpendicular to the three blue line segments XH, YG and ZI

3.6 On the Geometrical Relations Implicit in the Construction of a New Triangle from Three Coalescing Circles The termination points in shrinking circle enclosures are the centerpoint of a circle that is generally known as the “tenth problem of Apollonius”. This problem pertains to a circle that is defined as external and tangent to three other circles. The latter three can be recognized as the circles that make up the shrinking enclosure. Thus, it can be seen in Fig. 3.53 that the two triangles ABC, the red triangle of the tangent points of the convergent circles, and DEF, the green triangle of perpendicular feet, are situated on the Apollonius circle, which is the Feuerbach circle of triangle PQR. The directions of the pink triangle DEF in this figure are those that incept after a transition from circle space to one of the triangle spaces, viz. that in which the expansion directions point away from the circle midpoints (as we use to do here for blue triangles, cf. Sect. 3.4.2). If we imagine the three circles with midpoints X, Y and Z as expanding circles, then the point O can be understood as the termination point of the shrinking deltoid that they would come to enclose. Finding the point O in that way is an alternative to the various constructions of O that are known in literature.16

16

Cf. Kim et al. (2006).

3.8 A Process in Three Spaces: Illustration

57

3.7 On Increases and Decreases of Irregularity The geometric processes that we have presented in the previous chapter are not limited to a particular time interval. However, though they could continue indefinitely, there is always a finite moment in time at which an entire process will have become structurally stable, so that no more losses or gains of form will occur. We may call this the moment of final stability. The interval before, from the first inceptions until this moment of final stability is of our major interest, and we may call this the system’s interval of structural developments. Obviously, when new inception points are fed into the system externally, after its onset, then this interval will be prolonged. Generally, the expansion processes tend to use the irregularity provided by the configurations for the construction of bisectors. These bisectors are always constructed as the symmetry axes of the constituents (the positions of circles in the space of circles, the expansion directions in the spaces of triangles). Hence, the irregular properties of a configuration are being averaged repeatedly, and this yields a more regular configuration, at least in general. With the reduction of irregularity in the geometric configurations the expansion processes tend to have more losses of form. As a result, the system’s opportunities for new gains of form decrease in the course of its development. On the other hand, if the irregularity of the system’s configuration could be increased in the course of its expansions, then this would prolong the system’s period of structural developments. In fact, such increases of irregularity do occur at the transmutations that take place from the space of circles to the spaces of triangles. The red triangle ABC of Fig. 3.53, as any other triangle, has angles that can be expressed in terms of their deviations from the angles of the equilateral triangle (π/ 3): π/3 − α, π/3 − β and π/3 − γ (where α + β + γ = 0). Then the green triangle DEF of Fig. 3.53, orthogonal to the three finally selected radii (XH, YG and ZI), has angles equal to π/3 + 2α, π/3 + 2β and π/3 + 2γ.17 Thus, the shape of the newly incepting triangles is twice as irregular as the triangle between the three coalescing circles. Hence, the transmutations from circle space to triangle spaces contribute to an increase of the system’s irregularity and accordingly prolong its interval of structural developments.

3.8 A Process in Three Spaces: Illustration The following Figs. 3.54, 3.55, 3.56, 3.57, 3.58, 3.59, 3.60 and 3.61 show a development of expansion processes in three spaces. Here we depart from a situation where circles already have developed and pairs of inverse green and blue triangles newly incept at the points of termination of enclosed shrinking deltoids. For instance, let ∠CAB (the angle CAB) = θ, and α = π/3 − θ, then ∠HOG = ∠COB = 2θ and hence ∠GDH = π − 2θ = π/3 + 2α.

17

58

3 Expanding Forms: the Geometrical Representation

Fig. 3.54 Developing expansion processes in three spaces

3.8 A Process in Three Spaces: Illustration

Fig. 3.55 Continuation from Fig. 3.54: developing expansion processes in three spaces

59

60

3 Expanding Forms: the Geometrical Representation

Fig. 3.56 Continuation from Fig. 3.55: developing expansion processes in three spaces

3.8 A Process in Three Spaces: Illustration

Fig. 3.57 Continuation from Fig. 3.56: developing expansion processes in three spaces

61

62

3 Expanding Forms: the Geometrical Representation

Fig. 3.58 Continuation from Fig. 3.57: developing expansion processes in three spaces

3.8 A Process in Three Spaces: Illustration

Fig. 3.59 Continuation from Fig. 3.58: developing expansion processes in three spaces

63

64

3 Expanding Forms: the Geometrical Representation

Fig. 3.60 Continuation from Fig. 3.59: developing expansion processes in three spaces

3.8 A Process in Three Spaces: Illustration

Fig. 3.61 Continuation from Fig. 3.60: developing expansion processes in three spaces

65

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3 Expanding Forms: the Geometrical Representation

Figures 3.62, 3.63, 3.64 and 3.65 show how this very configuration (of Figs. 3.54, 3.55, 3.56, 3.57, 3.58, 3.59, 3.60 and 3.61) looks at the end of its development. We find here the situation in which all possible encounters between circles and between triangles have become structurally stable, only changing on continuous dimensions and without any new singularity points to be expected in its future.

Fig. 3.62 Final configuration (see Figs. 3.54, 3.55, 3.56, 3.57, 3.58, 3.59, 3.60 and 3.61), structurally stable

3.8 A Process in Three Spaces: Illustration

Fig. 3.63 Figure 3.62 in more detail.

Fig. 3.64 Figure 3.63 in more detail.

67

68

Fig. 3.65 Figure 3.64 in more detail.

3 Expanding Forms: the Geometrical Representation

Chapter 4

The Geometrical Expansions as a Relational Network

The systems of geometrical expansions that we described in Chap. 3 are characterized by a network of relationships between processes, such that there is, in terms of Rosen’s MR systems, a closed loop of efficient causation. It is this closure that, we believe, is of interest to us, as it is the crux of Rosen’s understanding of complexity in organisms. This would mean that our geometrical processes may visualize some of the complexity of the relational networks that characterize life.

4.1 Rosen’s Basic Idea of Closure to Efficient Causation Rosen (1991, 2000) has reused Aristotle’s classification of causal relations in terms of mapping relations between system components. Rosen distinguishes the idea of ‘efficient causation’ from that of ‘material causation’. It means that a producing actor works as an efficient cause by making/producing something out of a material cause. The latter, in a natural system, can be understood as the matter out of which a particular thing is fabricated; the efficient cause as the actor that performs the fabrication. A natural system can be modeled by means of a formal system, where a material cause is encoded as the domain of a mapping relation, and where an efficient cause is encoded as the mapping operation itself. The outcome of the causal relations within the natural system should then correspond to the codomain of the mapping, within the formal system. Our present task does not allow an adequate presentation of Rosen’s work and his basic ideas. These have been described thoroughly by Louie (2009). A historical overview and theoretical context is given by Cornish-Bowden and Cárdenas (2020). Figure 4.1 is drawn after1 Rosen’s (1991) most famous relational diagram. The efficient causation, represented by the open arrows, should be understood as different 1

Gatherer and Galpin (2013) called this the ‘Goudsmit representation’.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Mowitz and A. L. Goudsmit, Movements of Form, Vision, Illusion and Perception 6, https://doi.org/10.1007/978-3-031-44821-8_4

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Fig. 4.1 After Rosen (1991, diagram 10C6). The closure to efficient causation between three components B, F and Φ. Open arrows denote efficient causation, solid arrows denote material causation

from the material causation, represented by the solid arrows. The component A is to be considered as some kind of feed, caused externally. In contrast, the component Φ can be seen not to participate as a domain for a mapping of material causation. The three components B, F and Φ are maintaining a special circularity, in which the following relations hold: F(A)=>B

(4.1)

B (F)=>Φ

(4.2)

Φ(B)=>F

(4.3)

F can be considered the operator that has A (domain) mapped to B (codomain). Likewise, B operates such as to map F to Φ, and Φ such as to map B to F. Rosen calls this a ‘closure to efficient causation’, which means that every component of the closure (a) is an efficient cause for some other component of the system and (b) has some other component of the system as its efficient cause. Rosen takes the closure to efficient causation as the hallmark of the complexity that is supposed to exist in natural systems. Material causation does not contribute to this complexity. The component A, as a material cause of B, does not participate in this closure to efficient causation. The usual way2 to interpret these relations in terms of biological processes is by considering: A as the set of external sources (nutrients); B as the set of outcomes metabolic processes; F as the enzyme catalysts, which maintain the metabolisms; Φ as the cell’s ‘repair component’ that guards the catalytic processes. In the next section we will interpret these terms in terms of geometrical relations. e.g.: Cornish-Bowden et al. (2007), Louie (2009, p. 298) calls Φ a gene; Rosen (1959b, p. 115) calls Φf a ‘portion of the “genetic” material of the abstract biological system under consideration’.

2

4.2 A Geometrical Interpretation of ‘Closure to Efficient Causation’

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4.2 A Geometrical Interpretation of ‘Closure to Efficient Causation’ The closure to efficient causation, pivotal to Rosen’s understanding of complexity, can be recognized in the geometrical expansion processes that we have presented. The terms that we are introducing here will be valid for each of the spaces of circles and of triangles, unless otherwise specified. We will use the symbol Φ for the nucleus, denoting the symmetry axes both in the spaces of triangles and of circles. We will use F to denote the expansion rules. We will use A to denote the feed of points in either triangle spaces or circle space. These points are assumed to preexist in the three spaces that we introduced above (cf. Sect. 3.4). They are the points where a particular expansion process will incept, either initially or after a transmutation from another space. The spaces in which these points A are situated are assumed to be given with the translation of the onesided boundary into expanding forms. However, what we have called in Sect. 3.4 the transient identification between points of the various spaces is of our interest here. The points A are either the initial input for the expansion processes, or the points of termination/inception, i.e. the results of converging expansions that specify the new expansion rules of the incepting forms; as these expansion rules are specified as the new operators, their first operands are the points of transient identification between the spaces. We will use the term ‘substrate’ (cf. Sect. 3.1.1) as a generic term for the domain of validity of an expansion rule, denoting the expanding configurations B both in spaces of triangles and of circles. The substrate B is produced by F out of the points A. B (substrate) is to be understood differently for the spaces of circles and of triangles respectively. For the spaces of triangles, on the one hand, each substrate part is a line segment, each point of which moves in the same direction with a constant velocity (as in Fig. 3.12). The substrate’s linear extension is the length of each line segment. When we say that in the triangle spaces a substrate part is maximally constrained, it means that a line segment has decreased until only one remaining single point has become selected (as shown in Fig. 3.22). For the space of circles, on the other hand, each substrate part is a sector of a circle and each radius is a direction along which the circle’s contour moves outward with a constant velocity. The substrate’s angular extension is the width of the sector’s arc. When we say that in the circle space a substrate part is maximally constrained, it means that an arc has decreased until only one remaining single radius has become selected (as in Fig. 3.9). The three relations of our interest, (4.1)–(4.3) can be interpreted as productions of substrate, nucleus and expansion rules:

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4 The Geometrical Expansions as a Relational Network relation:

production of:

happens at:

F (A) => B

(4.1)

substrate

initial inception points

B (F) => Φ

(4.2)

nucleus

stable trajectories, gain of form

Φ (B) => F

(4.3)

expansion rules

loss of form, transmutation points

Thus, we can map Rosen’s relational diagram to the elementary geometrical relations that we have described. Within the geometry there cannot be causal relations and we do not introduce these. Instead, we maintain that, within this geometry, the three relations (4.1)–(4.3) together make up a closed path of definitions: assuming external feed, expansion rules define the moving substrate; the substrate defines the nucleus; and the nucleus defines the new expansion rules. The definitions relate the various terms as operators, operands and outcomes, and accordingly they correspond to the relations of efficient causation that have Rosen’s focal attention. We find these definition relations distributed over three different circumstances: in the structurally stable trajectories, at the singularity points in general, and at the singularity points that are transition points. We will discuss these now.

4.2.1 Structurally Stable Trajectories The substrate B produces nucleus Φ, according to (4.2), when adjacent configuration parts coalesce in the course of their expansions. In both the spaces of triangles and of circles the nucleus is defined as a bisector that is a symmetry axis between the expansion rules. The type of bisector is different for either type of space, as is the type of expansion rule. The nucleus Φ develops in time; at each point it is the outcome of the coalescence of two adjacent substrate parts. In the spaces of triangles the nucleus is defined as the angular bisector of the expansion rules (F), where the expanding substrate parts (B) are triangle (or polygon) sides (cf. Sect. 3.3). The direction of the expansion rules is paramount in determining whether a nucleus is specified as an internal angular bisector or as an external angular bisector (cf. Sects. 3.3.5 and 3.3.6). In the space of circles the expansion rules are the midpoints of the circles. Here too the nucleus is defined as the symmetry axis (bisector) between the expansion rules (F). The nucleus takes shape at the points of encounter between the radii (B) of adjacent circles (e.g. as in Fig. 3.5). As this nucleus develops within the space of circles, it takes the shape of a (generalized) Voronoi diagram (cf. Sect. 3.2).

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In brief, both in the space of circles and in the triangle spaces B (F) => Φ (4.2) holds at the stable trajectories, i.e. the nucleus Φ is defined by the substrate B as a function of the expansion rules F, i.e. as their symmetry axis/bisector.

4.2.2 Regular Singularity Points Like in structurally stable trajectories, at singularity points the nucleus Φ delimits the substrate B. We distinguished two types of regular singularity points that we have called ‘gain of form’ and ‘loss of form’. Gain of form For the singularity points of the type 1 ⇒ 2 (for triangles/polygons, as in Fig. 3.12), there is a situation in which the expanding substrate B develops new adjacency relations, and hence defines new nucleus Φ, in accordance to relation (4.2). This also happens at the coalescence of two circles, when a single nucleus originates, as in Fig. 3.4. We called this a type 0 ⇒ 1 singularity. Loss of form At the singularity points of the type ‘loss of form’ the nucleus Φ can be seen to specify the shrinking domain of a particular expansion rule f s ∈ F (using the subscript ‘s’ for ‘shrinking’). As we said, Φ is defined by B, and as such it works as a demarcation of B. It is this delimiting role which evolves now into a role of specification, when a singularity point of the type ‘loss of form’ is approached. At the moment of loss of form, i.e. at the singularity point, the shrinking constituent part bs ∈ B has proceeded unto a zero extension. This means that an expansion rule f s is no longer at work for this bs . Thus, the nucleus specifies here the shrinking domain of f s and accordingly its termination at the singularity point. Here the relation Φ (B) => F (4.3) holds. This is shown for circles in Fig. 4.2. We recognize the shrinking sector (bs ) as the shrinking arcs (two stages of which shown in cyan) that end at Z. We recognize its expansion rule (f s ) as the circle midpoint B. At Z this rule terminates. Loss of form is shown for triangles in Fig. 4.3. Here the shrinking structure (bs ) is the middle contour line segment (shown in pink) that terminates at P, specified by the two bisectors ϕ1 and ϕ2 , which is where its expansion rule f s terminates. The zero extension means that the expansion rule f s ∈ F that is associated to bs , has terminated at the singularity point. Thus, the courses of the two converging nucleus parts ϕ1 and ϕ2 specify the position of the singularity point, and accordingly they determine the extinction of f s , in accordance with (4.3). At singularity points of the type ‘gain of form’ the relation B (F) => Φ (4.2) is applicable. Furthermore, at singularity points of the type ‘loss of form’ the nucleus Φ can be seen to act as selector which expansion rule will terminate. It delimits the shrinking substrate part and thus determines the termination of the expansion rule related to the shrinking substrate part. Hence: Φ (B) => F (4.3).

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Fig. 4.2 An extension of Fig. 6: loss of form (of the type 2 ⇒ 1). The circle with midpoint B has one shrinking sector (bs , two stages shown in cyan) of which the angular extension goes to zero at the singularity point Z; and one sector (six stages shown in orange) of which the angular extension will tend to a stable nonzero value. For the shrinking sector bs the point B is the expansion rule f s that terminates in Z, as specified by the two hyperbolic bisectors ϕ1 and ϕ2

Fig. 4.3 Adaptation from Fig. 3.17. Loss of form (of the type 2 ⇒ 1) in a triangle configuration. The shrinking substrate bs is shown in pink. Its expansion rule f s terminates in P, as specified by the two bisectors ϕ1 and ϕ2

4.2.3 Special Singularity Points: Transmutations Transition points are different from regular singularity points, in that they are both the points of termination for a shrinking configuration, and subsequently the points of inception for a new expanding configuration (i.e., an incepting circle or pair of triangles). Both for circles and for triangles there is a loss of form and consecutively a gain of form in the spaces of triangles or circles respectively. The loss of form is of the type 3 ⇒ 0. Here, as in loss of form of the type 2 ⇒ 1, it is the nucleus Φ that delimits the

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three terminating constituent substrate parts bt ∈ B, which can be called bt1 , bt2 and bt3 (using the subscript ‘t’ for ‘terminating’). Hence the nucleus Φ specifies until when the associated rules f t1 , f t2 and f t3 will remain operational and when they will terminate, as in the general case of ‘loss of form’. As in Sect. 4.2.2 with respect to ‘loss of form’: Φ (B) => F (4.3). It is at the termination points that the terminating parts of the nucleus ϕt1 , ϕt2 and ϕt3 ∈ Φ define the incepting expansion rule f i ∈ F (using the subscript ‘I’ for ‘incepting’) afresh from the terminating structure parts bt1 , bt2 and bt3 . This means that the above relation (4.3) is here of a constructive kind. The new value of the incepting f i is derived from B in its final and maximally constrained extension, as specified by the terminating nuclear parts ϕt1 , ϕt2 and ϕt3 , where the terminating bt1 , bt2 and bt3 have obtained zero extension. Figure 4.4 borrows from Fig. 3.22 and shows three coalescing triangle sides that progressively include a shrinking triangle up to a single point M, from which then a new expanding circle incepts. Thus, the bisectors JM, KM and LM are the terminating parts of the nucleus ϕt1 , ϕt2 and ϕt3 . They delimit the domains bt1 , bt2 and bt3 of the three terminating expansion directions f t1 , f t2 and f t3 . Thus the termination point M is specified, being the terminating structure of zero linear extension and becoming the new circle expansion rule f i for the incepting circle, as in Fig. 3.37. Likewise, three coalescing circle sectors progressively include a shrinking deltoid up to a single point, from which then a new inverse pair of expanding triangles incepts. This is the case of transmutations from the circle space to the triangle spaces, as in Fig. 3.33. Here the same relation Φ (B) => F (4.3) holds. Unlike (cf. Sect. 4.2.2) Fig. 4.4 Adapted from Fig. 3.22. The three substrate parts (terminating triangle sides in pink) bt1 , bt2 and bt3 are delimited by the bisectors ϕt1 , ϕt2 and ϕt3 up to the termination point M, which is the new incepting expansion rule f i , the midpoint of an incepting circle, as in Fig. 3.37. At L the enclosure occurs and the polygon’s expanding contour is split into an outer contour and an inner (shrinking) contour that terminates at M

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the role of selector performed by Φ, here the nucleus Φ not only specifies when a triple of expansion rules terminates. It is also constructive as an efficient cause in the specification of new expansion rules for the new forms that incept after the termination of shrinking enclosed forms. It is this specification that we may also consider in terms of writing. This will have our attention in Sect. 4.3.

4.2.4 On the Construction of Φ Rosen (1991), in building his arguments for closure to efficient causation as the pivotal quality of the living, relies on an auxiliary idea of an external component β that acts as an efficient cause that produces Φ, as in β(F) => Φ. The problem with assuming such additional external component is that it should be explained in its turn: why β? In order to avoid an infinite regress (about how β is produced by some other external component β’, which is produced by β”, etc.) he maintains that β can be subsumed under B, hence the relation B(F) => Φ (4.2) completes the closure to efficient causation. Many authors3 have investigated how this subsumption of β under B should be understood, such that an infinite regress can be avoided. The general opinion is that such β is ‘logically possible’ (Letelier et al. 2006, p. 954) and that β is a property of B, not the same as B (Cárdenas et al. 2010, p. 80). The latter agree with Mossio et al. (2009)4 in that it is possible for the closure to efficient causation to be simulated in terms of procedures B, F and Φ that recursively activate each other. On the other hand, in our geometrical representation the answer to the problem how Φ is produced (caused efficiently) by B almost seems to come for free: a nucleus (Φ) is defined as the symmetry axis of the expansion rules, the ‘substrate’ B (polygon line segments or circle sectors) acts as the operator that has a pair (or triple) of expansion rules F as its operands. Thus B defines Φ as a function of F (4.2). Furthermore, we can now interpret β in geometrical terms as a property of B, namely as the limits of the extension (angular or linear) of the structure parts. The points of coalescence between substrate parts are the points where expanding forms (coalescing circles or triangles/polygons) delimit each other. We have described this in terms of ‘mutual constraints’ (cf. Sect. 3.5.1). The nucleus, the structure parts and the expansion rules are related such as to define each other in a circular way, the ‘closure to efficient causation’. In simple triangles this circularity is an almost trivial and obvious quality of the triangle. However, it is not so much this quality of obvious coherence that should surprise us, as the fact that this coherence is preserved during the expansions and coalescences of all types.

3

Letelier et al. (2006, 2011); Gutiérrez et al. (2011), Louie (2011). cf.: “In a word, Rosen’s definitional infinite regress is perfectly handled by recursion, in particular as formalized in λ-calculus. At the same time, those programmes simulating the closure may potentially activate an operationally infinite behaviour.” (Mossio et al., 2009, p. 494).

4

4.3 Expansions, as-if Reading and Writing

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4.3 Expansions, as-if Reading and Writing As points of termination of enclosed forms, the transition points are the outcomes of operations of selecting the terminal substrate parts, maximally constrained (cf. Sect. 4.2). They are the points at which these substrate parts become defined as the sources for future expansion rules after the transmutation. Accordingly, the expansions that produce the shrinking configurations can be formulated as processes of defining, hence as if a code is being written. As points of inception of newly developing triangles or circles, the points are the sources from which the expansion rules are taking effect. Accordingly, the newly incepting expansions that take place can be formulated as processes of executing the expansion rules, hence as-if they are processes of reading (executing/interpreting) a code. The qualification ‘as-if’ means that the expansion processes do have an appearance of reading and writing, however without coded instructions that exist at the transition points for longer than a single transient moment. They are a kind of proto-writing and proto-reading, expansion processes that bear a potentiality of writing and reading, however without a real written code as their outputs or inputs. The processes of writing the future expansion rules start at the moments of enclosure, when an expanding configuration touches upon itself, as described in Sects. 3.2.3 and 3.3.4. Here no sudden switches are occurring. The ongoing expansions can be understood as the processes of reading (interpreting, executing) the expansion rules, and in the course of these expansions enclosed configurations occur that terminate at one or more points. It is at these termination points that in the triangle spaces the shrinking triangles attain a zero linear extension (cf. Fig. 3.37), and, likewise in the circle space the shrinking deltoids attain zero angular extension (cf. Fig. 3.33). These decreasing extensions (cf. Sect. 3.4.2) can be understood in terms of the specification of the new expansion rules. The most explicit moments of writing are those of final selection of substrate (final position in shrinking triangles, final directions in shrinking deltoids). It is also at these points of termination that the expansion comes to an end, locally, no longer forestalling the paradoxality it was meant to postpone. This kind of crisis is resolved by the transition into another space. Here the ultimate substrate, being constrained maximally, turns into new expansion rules within a new space, one that is organized differently from where it was specified. Thus, the points of transmutation between the various spaces are most explicit as switches from (a) processes of writing the future expansion rules to (b) processes of reading the expansion rules that have become effective (actual). On the other hand, in contrast to the points of transmutation, at the moments of enclosure (for instance at point L in Fig. 4.4) the distinction between the reading and the writing is least explicit and the course of the shrinking configurations is least recognizable as writing. It is at these moments of enclosure that one contour splits into an outer contour and an inner contour, as we have seen in Sect. 3.5.2.3. This split tags the onset of a development of writing, one that only gradually becomes recognizable, in the course of the shrinking inner contour, as the gradual specification of a new expansion rule.

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Notice that the onset of the development of writing takes place in the course of reading, which means that here writing is nested in reading and that the two are not so much inverse processes in the way Löfgren5 defined ‘description process’ and ‘interpretation process’), as related in various ways, ranging from nesting to full opposition. The specification of new expansion rules at the moments of transmutation is where a semantics (substrate) is converted into a new syntax (expansion rule).6 The transition points themselves are transient, existing only in the moments of the transmutation. If one would wish to consider them as some kind of code or written instruction, they should exist beyond the dynamics of the system (which they do not) and have a particular nonzero duration (which they do not have). A veritable (not ‘as-if’) code must have an existence beyond the dynamics of the system in which it partakes and it must have a particular nonzero duration. These conditions are not met in the geometrical representation and it is beyond the scope of this book to explore them here.

5 Löfgren (1990) uses ‘description process’ for the transition from a dynamic process towards a written code, and ‘interpretation process’ for the other transition from a written code to a dynamic process. ‘Description process’ and ‘interpretation process’ are technical terms for the acts of writing and reading, respectively. Equivalent terms for description within this context are: program, definition; and for interpretation: application, execution. 6 cf. Matsuno (1993, p. 275) on the transition from semantics to syntax.

Chapter 5

The Geometrical Representation and Its Computability

5.1 Introduction The primordial idea is full self-reference (cf. Chap. 2). Whether or not this does contain its own implementation into the real material world remains a question beyond our capacity to understand. We faced paradox in terms of the one-sided boundary (cf. Sect. 2.1) and we found a way to open it and turn it into a geometrical idea of outward movements of forms, which brought it within our reach of thought. The geometrical idea does not contain its own implementation into the real world. It is incomplete, if only for that reason. It does not contain full self-reference, but instead the expansions of configurations of circles and of triangles have been introduced as unfoldings of self-referential indications. The performer of the acts of indication is left beyond the description of the expansions. The idea was: to unfold the self-referential indication so as to avoid and forestall paradoxality. We introduced outward movement and we had two options for the translation of these unfoldings into expansions (cf. Sect. 2.2): as ruled by position (circles) or by direction (triangles). The contour lines’ expansions (direction rules position) can be taken as enabled by the acts of indication; the acts of indication can also be taken as enabled by these expansions (position rules direction). We investigated the various ways in which coalescences occur between circles (cf. Sect. 3.2) as well as between triangles and polygons (cf. Sect. 3.3). Then we looked for connections between the spaces of circles and triangles and we described the transmutations (cf. Sect. 3.4). Comparable to Stuart Kauffman’s notion of the ‘adjacent possible’, we are facing the ‘adjacent thinkable’, which means: we can shape the geometrical configurations and their spaces according to how far our imagination may bring us. The expansions of the configurations of triangles and of circles have been introduced as postulates in a discourse initiated by the present authors. As our discourse unfolds, there is a confluence of perspectives: we describe not only the various expansions and coalescing configurations, their terminations, inceptions and transmutations etcetera, but also

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Mowitz and A. L. Goudsmit, Movements of Form, Vision, Illusion and Perception 6, https://doi.org/10.1007/978-3-031-44821-8_5

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our own expanding explorations of these processes. The expansions are not only objects of our geometrical imagination, but they are equally also our own movements as internal observers, traveling with finite velocity along the processes that we imagine to take place1 In the latter quality the expansions are our own diversions and digressions into the self-referential indication, which we have opened by traveling towards its withdrawing boundaries. Then it is us, observers, who are actively exporting the paradoxality, and accordingly it is us who engage in a search for how the expansions can continue. So that, in the end, the searches for continuation may appear to us as the expressions of our own intentionality. Now that we have elaborated in the previous chapters all relevant ingredients of the geometrical representation, our next task will be to pay attention to the processes in action. In order to visualize them, several arbitrary delimiting choices were made (see Sect. 5.1) and some problems of predicting the context sensitive expansions had to be faced (see Sect. 5.2), in order to keep the task workable of writing a computer program that simulates the expansion processes (see Sect. 5.3). The emphasis of the present chapter is on the impossibility to correctly compute and program these expansion processes (see Sect. 5.4), unlike the software program that simulates them, and unlike cellular automata (see Sect. 5.5).

5.2 Arbitrary Properties of the Geometrical Representation Our way of opening the self-referential indication contains several arbitrary decisions. We can open the self-reference and make a geometrical representation with any amount of initial inception points. Which initial inception points will be chosen is not determined by the very opening of the selfreferential indication. We are not specifying criteria for deciding about these points. Instead, we are facing many options: not only options for the number of points, but also for properties of their spatial distribution, such as their patterns and their degree of divergence from a regular pattern; also options for the number of times these initial inception points are fed into the geometrical spaces, as well as temporal patterns of their time intervals. Those options that have been used in our presentation of the geometrical representation are not the only possible or the ultimate ones. Furthermore, there have been arbitrary choices of speed, dimensionality, the number of spaces and their relations. The speed is equal in all directions and constant. This is an arbitrary decision, made for the ease of understanding, but not based on theoretical arguments. Unequal expansion speeds would give rise to other forms such as ellipses or hyperbolas. Likewise, as to dimensionality, there are no reasons why the processes described here should exist only in 2D. The one-sided boundary (cf. Sect. 2.1) was defined as related to the validity boundary of its own definition and this does not involve a special number of spatial dimensions. However, practical considerations convinced 1

cf. footnote 15.

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us that two dimensions would be an enterprise of sufficiently crispy and spicy bite, let alone the task of programming a visualization of it in higher dimensions. Although not demanded or implied by something intrinsic to the processes described thus far, we introduced one space of circles and two spaces of triangles. The ‘green’ and the ‘blue’ triangle spaces were introduced because we did not find a compelling argument for deciding between the ‘green’ and the ‘blue’ triangles, as if we had to decide between the two roots of a quadratic equation. The presented constellation, therefore, contains an element of arbitrariness, based on the aesthetics of the project. This means that other choices might also have been possible, for instance those in which more than one circle space would play, and/or more than two triangle spaces. This is to say: the constellation of three spaces that we have presented here may not be the whole story, it may be just the first element from an entire class of possible constellations.

5.3 Limits to the Predictability of Expansions Given an expanding configuration that is structurally stable at a particular moment, and given a particular set of expansion rules according to which the expansions are taking place, then it is possible to foresee only the most immediate developments, and only if no sudden inceptions occur due to the termination of enclosed configurations in one of the other spaces. Due to the system’s radical context sensitivity it cannot be predicted how and where an incepting form will affect the domains of validity of expansion rules. The only way to discover that is to observe how the expansion processes take place. There is no shortcut, no way to establish which expansion rule is to be applied where, and for how long, except by following the expansions and see what will happen. The configuration itself is its own criterion of development. This will return in Sect. 5.4.4. The system’s context sensitivity is also manifest in that the slightest change of position or change of angle in a configuration may radically change its course of expansions. Furthermore, there is no way to predict, whether, when or how such radical changes will follow. This is of importance for one of our arguments, to be presented in Sect. 5.3, that the approximation errors made by any simulation algorithm may lead to radical errors in comparison to the processes (as conceived), described in this book. Given the situation of Fig. 5.1, it can be seen that the three polygon vertices, all of which have very acute angles, will meet and coalesce. If their expansions develop according to Fig. 5.2, it is evident that no self-enclosure will occur and no concomitant shrinking triangle will be specified. On the other hand, if the relative expansion speeds of the three vertices turn out to be slightly different from the situation depicted in Fig. 5.3 (due to slightly different angles), then the expansions may develop according to Fig. 5.3, where such enclosure does occur, and this time it will lead to a new termination/inception point. This, indeed, is a radically different result due only to a tiny angular difference.

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Fig. 5.1 Three vertices are close to coalescence

Fig. 5.2 The development of Fig. 5.1, first possibility: no enclosure occurs

Thus, we find that though the expansions are rule governed, we cannot be certain by which rules exactly, until these are shown to us in the course taken by the expansion processes! There is no warranted way to determine these rules beyond following the expansion processes themselves in their course, although, in retrospect they turn out to have developed with full determination. We will return to this in Sect. 5.4.

5.4 On Simulation: the Program Zigzag

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Fig. 5.3 The development of Fig. 5.1, second possibility: an enclosure arises with a shrinking triangle

5.4 On Simulation: the Program Zigzag Our initial ideas were on expansion processes, first of triangles, later also expansions of circles, circles that incept simultaneously or sequentially, giving rise to symmetry axes that were straight line segments or branches of hyperbolas. It seemed an exciting perspective for us to have a visualization of these expansion processes. Initially we did not make a veritable distinction between the geometrical representation, qua purely mathematical idea of expanding configurations, and its visualization by means of a computer program. The latter, as became clear to us in the course of our work, could be understood as a simulation of the mathematical idea, by means of a program that makes computations that are useful for practical reasons, i.e., not for the sake of producing an analogy to the circular relations (4.1) to (4.3) underneath the geometrical representation. The mechanical model underneath the simulation program consists of concatenated parts; the geometrical representation, on the other hand, cannot be decomposed along the lines of these parts, in view of the circular relations between these parts: these are loops of processes in which definitions are constantly being constructed, mutually and circularly. These processes cannot be represented as summands that correspond to structure parts. A simulation, such as a computer visualization, is meant to produce a particular impression or object and it does not include the underlying production processes, as does an adequate model of a living system. This is one of the major points of Rosen (1991) with respect to models of life. Rosen developed his major ideas on the complexity of living beings in a kind of opposition (cf. Rosen, 1959a) against von Neumann’s formal model (19662 of 2

Pattee has pointed out on many occasions (e.g. 2007, p. 2290) that Rosen’s opposition against von Neumann was partially misplaced and unnecessary, as von Neumann would have agreed with several of Rosen’s major tenets. Although some of the formulations in von Neumann’s extensive oeuvre do expire some postwar enthusiasm on simulations by means of logical machine operations,

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self-reproducing automata. The latter model departs from a written code that is to be processed in various ways (copied qua text, executed qua program). Rosen’s criticism focused on the paradox of a self-reproducing automaton and he pointed at the impossibility to define the automaton before its output and to define the output before the automaton, if the automaton’s output is meant to exist as a material realization, a physical entity capable of reproducing itself. A major difference between von Neumann’s work on self-reproduction and Rosen’s relational model is the presence versus absence of a written code that is part of the model. Rosen wants to stipulate the difference between models of the living and simulations of it; the latter being ‘simple’ and produced by processes that are external to their output, the former being ‘complex’ and containing their own production processes.3 Rosen’s major stake was to distinguish strictly between a system that contains its own production processes on the one hand, and on the other hand a simulation of such system. A simulation relies on a production process that is beyond its own scope; it does not deal with how it is produced. Although von Neumann explicitly did not claim to have a solution for connecting his model to material reality, Rosen’s criticism was that von Neumann’s model would contain a paradox as soon as a material realization of it were built; hence the latter would be a simulation, not containing its own construction processes. The relational diagrams4 that Rosen put forward were meant to be models, not simulations. There is no written code involved, and these diagrams explicitly do not deal with matters of material realization. Likewise, our geometrical representation contains a preservation of several geometrical relations, as expressed in the relations (4.1) to (4.3), hence a preservation comparable to Rosen’s ‘closure to efficient causation’,5 but it does not contain coded instructions, self-reproduction or any clues for material realization. Furthermore, the geometrical representation is explicitly meant as a way to forestall paradox by spatial expansions. If ever a material realization might be made of it, then the forestallment of paradox will have to be one of its features. he does emphasize, when read scrutinously, the importance of understanding the organization of a system, such as the central nervous system (e.g. von Neumann, 1963, p. 321). This emphasis is actually a thing Rosen could have appreciated as a recognition of his pivotal distinction between simulations and models. Pattee was right in pointing out the agreement that could have existed between Rosen and von Neumann on the limitations of algorithmic formal models of life, as well as in pointing out that in this respect Rosen has misunderstood and hence missed von Neumann’s model. 3 Louie (2017), continuing on Rosen’s lines, has described the processes of reading and writing in terms of Rosen’s relational diagrams. Louie (ibid., pp. 221ff) also introduces mappings in order to situate the genetic code within the relational networks, notably in a paragraph named ‘The word became flesh’. Thus, the contrast between von Neumann’s position and Rosen’s may have become less insuperable, in the end. 4 Such as in Fig. 4.1. 5 cf: “Rosen’s ‘replication’ is the efficient cause of an (M,R)-system’s inherent ‘autonomy’, a kind of ‘relational invariance’ in terms of its entailment pattern. This is a concept of relational biology that has no obvious counterpart in molecular biology.“ (Louie 2009, p. 269).

5.4 On Simulation: the Program Zigzag

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Our simulation of the geometrical representation, on the other hand, is based on a software program. The program Zigzag (2018, see References) was written by Joachim Mowitz between 1988 and 2018. The code is in dense C++, to be compiled as an executable, and originally meant to fit into 640 K ram MSDOS systems as were customary in the late 80s. Thus, as of 1995, gradually a visualization of moving forms became available in which all varieties of encounter between expanding forms could be observed and reconsidered, and thanks to which many of our initial misapprehensions came to light.6 Between 2011 and 2013 our long term friend Markus The, a brilliant Amsterdam software architect, provided it with a new C# interface so that it could also run on Windows 64-bit systems. The program Zigzag produces the visualizations of the expansion processes. It is organized in a way that does not resemble the circularity between definitions in the geometrical representation (corresponding to the efficient causes of Rosen’s diagrams). The geometrical representation was our way to open the self-referential indication and avoid it as a locus of paradoxality (cf. Sect. 2.1). The program, on the other hand, does not deal with these definitions and their circularity corresponding to Rosen’s closure to efficient causation; it just computes the approximated values of encounter points between expanding forms and their (angular or linear) distances to the relevant expansion rules. Figure 5.4 shows the possible encounter points between circles, as well as how easily they may become outdated by new developments. Likewise, Fig. 5.5 shows two stages of expanding circles and the predictions about points of encounter between circles and about the endpoints of circle segments (the predicted singularity points of the type 2 ⇒ 1, such as the point Z in Fig. 3.6). Clearly some of the latter predictions have become outdated and are corrected after value updates by the program. The programmed visualizations have been an essential instrument for us in studying the geometrical representation. However, it became clear that there are some vital differences between the geometrical representation and its simulation, that reflect the different ways in which they are organized. The expansions: continuous versus discrete The geometrical representation has the expansions as continuous processes, but the program takes discrete steps and computes the values of the expanding configurations from previous values, in discrete steps. The expansion speed: constant versus hardware dependent The idea of the geometrical representation is a growth of all geometrical forms that is a simultaneous expansion of constant speed. The program aims at simulating this constant and simultaneous expansion speed for all parts, but, obviously, it works sequentially.7 For simple expansions this will not be notable, but above a 6

Professor Otto Rössler (personal communication, Jena 1999) proposed the term ‘transmutation’ for the transitions between the spaces of circles and triangles, appreciating our software of those days (“good programming!”). 7 cf. Gunji (1992, p. 287). Gunji also refers to Conrad (1984). See for instance there, p. 353.

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Fig. 5.4 Detail of a screenshot from the program Zigzag. The midpoints of the dark red line segments are the predicted encounter points between circles of similar generation. At subsequent moments these possible encounter points may be dropped and others may be found, for instance due to newly incepting circles. It can be seen that new and untouched circles (shown in white) recently did bring already such new possible points of encounter, thus outdating and invalidating previously foreseen encounter points

certain number of constituent parts it unavoidably will lead to delays and pauses. The hampering visualizations remind us at those moments that the computer simulation is taking place in real time and accordingly has only finite computational powers at its disposal. Of course, the simulation itself can be simulated in its turn, for instance by producing a movie of subsequent images of short intervals, so that smooth and continuous expansions are portrayed and so that nothing will unveil the fluctuating degrees of computational power required for the images in these various intervals. The values: irrational versus rational All computed values are numerical approximations. The geometrical representation is mainly dealing with irrational values. For instance, the coordinates of termination points of the shrinking deltoids in the circle space correspond to the centerpoints of the circle of Apollonius’ tenth problem (cf. Sect. 3.6). The program approximates these points with rational values. An analytic solution would yield irrational values.8

8

cf. Gavrilova and Rokne (2000).

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Fig. 5.5 Zigzag screenshots: two stages of a circle configuration. The midpoints of dark red line segments show the predicted points of encounter between circles; the white line segments show the predicted end points of circle sectors. The screenshot on the right shows that some of the predictions (left) had to be corrected

The unavoidable differences between irrational values and their numerical approximations make the simulation susceptible to radical divergences from the ideal course of developments.

5.5 Unprogrammability As the programming efforts proceeded, it became increasingly clear that, for any area, the predictions made about future developments of expanding configurations would outdate quickly and radically at the occurrence of any new singularity point; but for which areas was this the case, and for which areas was it not? Could these areas be demarcated a priori? It seemed they could not. The predictions had to be adapted in the course of the expansion processes. It was only during these developments that clarity could be obtained whether the predicted events of encounter and coalescence would become realized or whether they would be prevented by other upcoming developments that would take priority. For instance, the precise sequence of events, or the expansion speeds of acute vertices, as in Figs. 5.1, 5.2 and 5.3, could radically determine the course of developments and still become apparent only upon closer inspection, that is, not very long before they take place. This also meant: it became

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clear that the program could give clarity only during its run, and not in advance. Any prediction of upcoming developments would have to be reassessed unremittingly.9 Given an expanding configuration that is structurally stable at a particular moment, and given a particular set of expansion rules according to which the expansions take place, then a structural stability will only remain intact until, somewhere in the entire configuration, a singularity point is encountered. We already (cf. Sect. 5.2) mentioned the context sensitivity of the expansion processes. Due to this context sensitivity it cannot be predicted how and where a new singularity point will affect the validity domains of other expansion rules. The transitions between spaces of circles and triangles enhance this context sensitivity even further: at any moment in any of the spaces of circles or triangles a new form may incept, suddenly and anywhere, due to a termination in an enclosed and shrinking configuration elsewhere in a different space. This means: whenever the prediction of a certain development of coalescing forms seems obvious within one space, a sudden inception, due to a transmutation from another space, may radically invalidate the predicted course of events, for example as in the series of Figs. 3.48–3.51. In our explorations of the expansion processes we found the peculiar situation that though each of the expansions is rule governed at any moment, we could not be certain by which rules precisely, until these are shown to us in the course taken by the expansions. These rules are selected by the expanding configurations within a particular space and they are newly defined at the transition points between spaces. Could there be a shortcut, a way to decide in advance which expansion rule is to be applied when on which substrate; which substrate is specifying which nucleus; which nucleus is selecting or defining which expansion rule? The major question is whether the only way to discover such developments is to have all the expansion processes take place in due time and oversee them, or whether there may be a shortcut, one that would enable us to compute the expansions in advance without having to run them first. We offer an intuitive argument in Sect. 5.4.1 why such analytical solution cannot exist. We do not offer a formal proof, nor an argument about the possibilities of such a proof. Then we will argue in Sect. 5.4.2 that the expansion processes have their outward movements as an a priori assumption that is not implied or produced by the geometrical relations they underlie. In Sect. 5.4.3 we will argue that the speed of these expansions must be finite because there is only a limited control of boundary conditions possible. In Sect. 5.4.4 we will propose that the expansions are more like constructions rather than computations.

9

Within the context of ‘internal measurement’ Matsuno (2002) mentions the necessity of a ‘constant update’ of internal descriptions. See also footnote 3 of Chap. 2. Such internal measurement, as an ongoing activity that Matsuno prefers to compare with the present progressive tense, is an alternative to what he calls the ‘pathological context dependence’ within causal descriptions in physics (Matsuno, s.a.).

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5.5.1 An Informal Argument about the A Priori Incomputability The circularity between the relations (4.1)–(4.3) unfolds as a recursion over the components (substrate, nucleus, expansion rules) that change and switch roles of defining operators and their operands. This unfolding recursion takes place within a particular time interval. It cannot be replaced by a computation that immediately predicts the expansion developments, without having to follow the recursions. Furthermore, during the recursions the process itself builds the rules that specify which operators apply to which operands, and when; therefore, these developments cannot be known beforehand. It means that the set of operators and the set of operands that are at stake at a particular moment in a particular configuration will be changing at every singularity point encountered. This is the argument. We may wish to take the operands as presenting the system’s states and the operators as the state transition rules. Then, however, if it is at all possible here to dissect the idea of ‘state’ from that of ‘state transition’, such dissection can be performed at best only for small intervals. No a priori criteria can be given for the duration of those intervals, nor for the determination of which expansion rules will extinguish, which ones will recombine, and which symmetry axes between which substrate parts will together determine which singularity point of any type (such as a point of termination inside an enclosure) and which newly defined expansion rules. This, in fact, is a reason for understanding the state transition rules as included in the set of state parameters, instead of being distinct and external to these.10 Only locally can we distinguish the transition rules from the system states, and even so only for short intervals of unknown duration. More globally, such distinction is not very meaningful, as the transition rules change so quickly and so often. These rules and their domains of validity cannot be specified beforehand.11 Furthermore, if an analytical solution could exist so that the developing expansions could be predicted in advance, then it would have to deal with all expansions simultaneously and it would have to oversee the network of their interactions. Such solution would contain an internal a priori organization of commands and task specifications, with the purpose of preventing our dependence on a posteriori observations of expansions that will be computed only in the course of their unfolding recursions. However, 10

Governments tended to regulate prices of oil and natural gas, after free market mechanisms had allowed unforeseen price surges (autumn 2022): suddenly the ‘rule’ of the market itself became just a state parameter among others, and no longer an untouchable regulative principle. Such context dependence of rules also makes up, in a hilarious way, the famous dictum by Groucho Marx: “Those are my principles, and if you don’t like them …. well, I have others.”. 11 Departing from a different position, Rosen (1985) stipulates something related, namely that the incompleteness of predictive models of a living system is a necessity: the ‘span of time over which a modelling relation remains stable’ (p. 336) is necessarily limited, due to the occurrence of new interactions between the system and its environment that are not included or foreseen in the model. Furthermore, ‘… the domain of stability of a given modelling relation is finite.‘ (p. 337). As a result, after some finite time span the model’s predictions will have to diverge, radically and unavoidably, from the actual behaviors of the modelled system (p. 305).

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such solution cannot be organized in a way simpler than the events to be computed. If a theory should be simpler than the data it explains, as Chaitin12 maintains after Leibniz, then that would not be the case here.

5.5.2 Expansions as A Priori Pattee, carrying further on von Neumann’s ideas, uses Newton’s strict distinction between laws of nature and boundary conditions, such as the initial state of a system. Pattee is interested in the material realization of models of living systems, in particular those of the living cell’s coded instructions and how these interact with the cell’s physical processes. Pattee13 stipulates that the genetic code does not contain instructions for protein folding, nor needs to contain these. That is, once the protein has been built as a linear sequence of amino acids, the foldings occur spontaneously, i.e., as the effects of natural laws, not as the effects of coded instructions, so that the folded proteins become functioning enzyme catalysts. Thus, the cell’s genetic instructions do not have to cover all processes related to cell maintenance and reproduction. This means that the genetic code does not need to be complete in its description of all cell processes, i.e., the coded instructions do not need to contain a complete self-referential description.14 The important point for us is that Pattee recognizes the value and functionality of this incompleteness. We may imagine a huge configuration of domino stones that serves as an intricate initial condition, waiting for a first impulse, so that gravity will do the rest. The point is: gravity itself is not part of the initial condition. It is taken for granted as a law of nature. Its presence a priori is comparable to how Pattee values the laws of nature in protein folding. A parallel between Pattee’s emphasis on the absence of genetic instructions for protein folding and our geometrical representation is that the latter does not account for the expansions. The expansions were introduced (cf. Sect. 2.1) artificially, as a way for us to temper and forestall the paradoxality intrinsic to the idea of a self-referential indication. Thus, as the expansions were given, they underlie the geometrical relations beforehand, without being specified by them. This is how far the parallel between the expansions and the laws of nature can be stretched. For the expansions are assumed a priori. Furthermore, let us notice that the very purpose of tempering and forestalling the paradoxality implies the assumption of a finite expansion speed. This assumption does have a pragmatic appearance, but we will see (cf. Sect. 5.4.3) that it is a necessity for theoretical reasons as well.

12

e.g. Chaitin (2008). e.g. (1976, p. 178), (2001, p. 353), (2007, p. 2290). 14 e.g. (1977, p. 263). 13

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5.5.3 Limited Control of Boundary Conditions The a priori incomputability of the expanding configurations is compatible with what Matsuno (1989) describes in terms of a limited theoretical controllability of boundary conditions. Matsuno presents a kind of criticism of classical Newtonian science, namely with respect to the absolute distinction between boundary conditions (system states) and laws of nature (rules for state transitions). Whereas Pattee (e.g. 1977, 2007) maintains just this distinction throughout his entire oeuvre and focuses on the complex relationship (named semantic or semiotic closure) between the laws of nature and the boundary conditions within living beings, Matsuno is questioning this very distinction, not conceptually but in its realization in situations of (living) systems in which the speed of the interaction processes between components cannot be ignored as a factor relevant for the system’s organization.15 He argues that this speed is not taken into account in classical physics as a relevant factor, as if the classical laws, such as those of motion, always are obeyed in nature without the slightest time delays, i.e. with infinite speed. Instead, he points out, the finite speed of interactions cannot be simply ignored, as it contributes substantially to the system’s organization and its workings. As a result, theoretical control of the system’s boundary conditions is limited, as the laws of nature are not being obeyed in ways that are fully context independent. Instead, their workings are not fully distinct from the boundary conditions upon which they are supposed to act. Accordingly, there are interactions between processes which are overlooked if a classical model assumes an infinite value of the speed at which laws of nature are being constrained by boundary conditions.16 Likewise, as we have argued (cf. Sect. 5.4.1) that an a priori analytical solution to compute and predict the developments of the expanding configurations is impossible and that we are bound to wait and see the expansion processes develop with finite speed, we argue that this lack of a priori computability implies a certain indistinguishability between system states and their state transition rules. We are facing thus, in the expansions of the geometrical representation, a situation of limited theoretical control of the boundary conditions, one that resembles Matsuno’s arguments that such limited theoretical control is due to the nontriviality of finite interaction speeds. We already introduced the finite expansion speed as an assumption a priori (cf. Sect. 2.1). It was not assumed there for theoretical reasons, but its purpose was to afford the translation of the self-referential indication into a domain of spatial relations. On the other hand, an infinite expansion speed would be tantamount to the availability and possibility of a clear distinction between system states and state transition rules. Therefore, the finite expansion speed that we introduced turns out to be a necessity and not just a pragmatic auxiliary assumption. 15

Matsuno (1989, e.g. pp. 36ff.) considers the propagation velocity of interaction changes as a pivotal constituent of internal measurement. Likewise, Gunji (e.g. 1995, p. 35) emphasizes the velocity of observation propagation. It would not come as a surprise if this notion of internal measurement will eventually turn out to be consistent with Pattee’s semiotic closure. 16 cf. footnote 11 on Rosen’s (1985) ideas on the incompleteness of predictive models.

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5.5.4 Constructions and Sensitive Movements Now that we gave an argument for the impossibility of analytical solutions that can predict the expansions, we may face the question whether the expanding configurations should be understood in terms of computations at all. If, to speak with Chaitin, there is not a theory available that is simpler than the data it has to explain, then the reverse may be true: each configuration at a particular moment may be more or less as complicated as the configurations that derive from it in the course of the ongoing expansions. The circular relations between the various definitions that shape the expansions do not so much specify the subsequent values of the positions and directions involved, as subsequent ways to construct these positions and directions, that is, they specify how to find the positions and directions and how to use them for further developments, though without measuring them as such. The expansion processes, in fact, are not so much computations as a continuous flow of geometrical constructions that enable subsequent (adjacent) constructions, without having to compute the irrational values of the various symmetry axes and singularity points. Each configuration at an arbitrary moment in time is a template for constructing the configurations that will follow, and so in a way that adapts to the latest developments, both continuously at structurally stable intervals and discretely at singularity points. This flow of continuous constructions amounts to a quality that we may call sensitive movement. This, of course, is not a property of the expansions themselves, but a qualification by someone who observes the running process.

5.6 Differences from Cellular Automata It has been suggested that the expanding configurations of the geometrical representation are basically ‘just’ cellular automata (CA). Indeed they may seem to be akin to the behaviors of CA and indeed they may have some properties in common with CA. Although CA of the so called classes III and IV do also display chaotic and unpredictable17 behaviors, it is obvious that for CA of any class the execution of computations is well possible. CA are designed just for being computable step by step. Then what can be of our interest with respect to differences between the geometrical representation and CA? First, in CA the clear distinction between states and state transitions is beyond any debate. The system states are computable at every step, given the transition rules; the latter are well defined a priori and well distinguished from the former. On the other hand, in the geometrical representation this clear distinction is absent (cf. Sect. 5.4.1), or at best it is present for short intervals only, due to the fluid circularity of the defining relations between substrate, symmetry axes and expansion rules (cf. Sect. 4.2). 17

Toole and Page (2010).

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In the second place, CA are defined on a grid with predefined dimensions. In the constructions of the geometrical representation, on the other hand, no pregiven grid is available, nor is there a fixed timestep. At best, not the geometrical representation itself but its simulation, such as the visualizations produced by Zigzag, could be said to take place on a grid, as do CA. The simulation’s grid could perhaps be set to that of screen pixels, or else be defined by the level of the floating point resolution of the computing medium. However, in any case such grid, irrespective of its resolution, would not be of much help in programming a simulation of the expansion processes and their changing positions and directions. In the third place, CA are defined with a predefined timestep. A timestep cannot be imposed at all onto the geometrical representation, as it would have to be one that varies according to the duration of intervals of structural stability of the expansions. Next to that, the discrete timestep in CA is of a fixed value 1 or some other integer value, not an unpredictable value and not an irrational value such as those of the structurally stable intervals. Furthermore, the geometrical representation, not its simulation, contains continuous expansions. These are best understood as geometrical constructions, not analytical computations, nor numerical approximations. On the other hand, a simulation such as Zigzag can be given a timestep, but any predictions based on it will have to be checked and corrected incessantly, as mentioned before.18 Therefore, in the expanding configurations of the geometrical representation we are facing a type of unpredictability that not only defies a priori computations, but a type that also effectively frustrates the valid a posteriori computation of spatial relations and time intervals.

18

cf. footnote 9.

Chapter 6

Conclusion

Self-reference is an entanglement of logical types that occurs when these are folded back upon themselves and fuse, like when an indication comes to point at itself. Selfreferential situations produce paradoxical results easily, as in Epimenides’ dictum of the lying Cretans or as in Russell’s barber paradox. An indication that refers to itself has been introduced in our definition of a one-sided boundary and this led us to the idea that paradox can be forestalled if boundaries are allowed to move outwardly. We argued that the expansion speed of these boundaries is necessarily finite and that the many encounters and interactions between expanding forms cannot be computed beforehand. Their course of development is predictable only for relatively short intervals. The duration of these intervals cannot be predicted either. The expanding forms are the outcomes of geometrical constructions rather than of computations, which gives the expansion processes an appearance of sensitive movements. Our subject matter was a network of relations between expansion processes in a geometrical domain of triangles and circles. We introduced expanding forms and their components (structure, expansion rules, nucleus, singularity points) that were seen to define each other in the course of the expansions, so that they illustrate Rosen’s (1990) concept of closure to efficient causation, a concept considered by him as the hallmark of the complex organization of life. While the forms themselves changed both in continuous and discrete ways, their network of mutually defining relations remained intact. The entire project was strictly formal; there was no obvious connection to material realizations such as biological phenomena. Nevertheless, we related our work to ideas from the domain of theoretical biology. Could the geometrical representation contribute to our understanding of life and its origination? This question pertains to various issues. At this place we mention some of these, hoping that our work be useful in further explorations. First, the a priori assumed expansion movements of the geometrical representation underlie the recursive definitions between structural components of the geometrical configurations. Is this a logical exercise for us in order to avoid paradoxes in

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Mowitz and A. L. Goudsmit, Movements of Form, Vision, Illusion and Perception 6, https://doi.org/10.1007/978-3-031-44821-8_6

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constructing our theoretical models of life, or are the recursions between the definitions of the geometrical components adequate as a metaphor of the recursions in the mutual specification of biological parameters? If the latter, then are there also events in living beings that are comparable to what we have called the transmutation between the geometrical spaces, a kind of crisis in which some of the expanding forms have created shrinking enclosures that terminate in a single point, from which new expansions incept in one or more other spaces? Second, can the forestallment of paradox be a property of living beings? This question relates to the semiotics of coding and decoding, writing and reading the genetic information within living beings. To the extent such genetic information may tend to become self-referential, the unfoldings will be helpful to prevent logical accidents, such as paradoxes. Hence, it may be of interest to see if there are processes in living beings that actively prevent paradox. If so, then is such prevention always effective? What would happen when it is not? Finally, how can the coded information that is omnipresent in living beings be understood in terms of the expansion processes that we presented? Is it possible to complement the as-if writing and as-if reading, so that a veritable writing and reading of a real code can be related to the geometrical expansions? Would random material imperfections or disturbances play a role that is far more constructive for the origination of coded instructions in living beings than would be possible within a pure geometry of expanding forms?

Appendix A

The Shrimp

One of our first forms that we used extensively for studying polygon bisectors is shown in Fig. A.1. For reasons unclear to us now, this form was dubbed by us ‘the shrimp’ (and we still have warm feelings for it). By the late 80s we had not yet dropped the incorrect assumption (cf. footnote 4 of Chap. 3) that polygons like this one had a nucleus that was both internal and external (Figs. A.2 and A.3).1

Fig. A.1 The ‘shrimp’, one of our first forms that we used for studying polygon bisectors. The polygon here is in blue, the bisectors are in greenish gray. The image is derived from a plotted overhead transparency sheet; notice the irregularities of the ink at the far line ends

1

Mowitz and Goudsmit (1989). This ‘shrimp’ polygon was even used by the publisher of the edited volume as a decorative illustration to their advertisement notifications.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Mowitz and A. L. Goudsmit, Movements of Form, Vision, Illusion and Perception 6, https://doi.org/10.1007/978-3-031-44821-8

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Fig. A.2 Early studies of the shrimp: its expansions (both inwards and outwards); the ‘strata of radial lines’ (the conceptual predecessors of the expansion directions, cf. Mowitz and Goudsmit 1988); the nucleus (both internal and external)

Appendix A: The Shrimp

Fig. A.3 As in the previous figure: studies of the shrimp

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Appendix B

The Workgroup

The story begins in 1985 as the authors of this book met in a workgroup that started in 1984. The original aim was to prepare a course in System Dynamics at the University of Amsterdam. Soon there was a growing interest into paradoxes and what in those days was called second-order cybernetics. The geometry was sneaking in almost unnoticedly as innocent portrayals of vague notions, as we tried to arrive at some understanding of complex ideas such as autopoiesis, a hot topic by that time. Questions of the type ‘what does this or that particular line segment denote?’ gave rise to bitter debates and eventually departures, mostly without slamming doors. Some of those who left in fact considered such questions about the denotation of individual lines most appropriate. Those who stayed and those who joined in later found themselves puzzled by the unclear idea that if it were not so much the lines themselves that mattered, then what it could be that were so important in their relatedness (Fig. B.1). This, in fact, was the kickoff of our explorations into questions why the relations between geometrical properties were so much more interesting to us than the individual nodes. These questions were congruent with our interest in biological theories on organizational closure and self-referential networks, in which paradoxality was hiding around almost every corner. Especially the almost hermetic closure of the relations between the bisectors and the contour lines of triangles and polygons gave us an intuitive sense of the internal coherence of living beings. We explored biological theories such as those of Maturana and Varela, later also those of Rosen, Pattee and Matsuno. There were various theoretical ways in which the coherence of living beings could be approached and we felt that ‘our geometry’ might shed some light on it, but it was less obvious what light and how. Thus, given our geometrical intuitions, a subjective appreciation of the aesthetics was always present in the background, and sometimes in the foreground as well.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Mowitz and A. L. Goudsmit, Movements of Form, Vision, Illusion and Perception 6, https://doi.org/10.1007/978-3-031-44821-8

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Fig. B.1 The workgroup that for several decades served as an incubator for our ideas, such as those of this book. Clockwise from upper left: Jan Hendriks, Arno Goudsmit, Jan-Gerrit Schuurman, Joachim Mowitz. The image is a screenshot from a video call session of 22 march 2020, in a time when live meetings were not allowed due to Covid-19 quarantine restrictions. The question at hand was: ‘should the recording of this session be deleted afterwards, and if so, will it?’

The geometry developed into ideas about symmetry axes of triangles and polygons, later also symmetry axes between circles. Initially we watched and studied static images of polygons and static images of circles. It took us some time to realize that the various symmetry axes could be understood as the points of encounter between the contour lines of expanding forms. This was true for a single triangle as well as for a large configuration of forms. The symmetry axes thus came to presuppose the movement of the geometrical forms. First we had them shrink inwardly and expand outwardly, as in Fig. A.1 of Appendix A. Then we preferred the idea of expansion only. Circles came next; they could expand too, developing symmetry axes of a different type. Circles of equal size, that is, of equal moment of inception, had straight lines or line segments as their symmetry axes; pairs of circles of unequal size had hyperbolic symmetry axes. As of 1995 the first versions of the program Zigzag came to assist our imagination. We spent many hours of gazing at it. By having the circles expand, and likewise the triangles, configurations arose which made enclosed areas that shrank as the contributing forms continued to expand. We wondered what would happen at the termination points of these shrinking forms. Could the terminal position of a shrinking triangle be of value as a new inception point for an incepting circle? Thus, we came to think of two spaces, one of triangles and one of circles. The reverse was not so obvious: how could a shrinking deltoid define a new triangle? Here we came to consider the distinction between rules and substrates and to understand the spaces of triangles and of circles as inverse with respect to the roles of positions and directions, namely as substrates or as rules.

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Enclosures in the space of circles, which lead to one or more shrinking deltoids, left a legacy of three directions, viz. those specified by the terminal (zero extension) sectors of the contributing circles. Here the authors had diverging preferences for how the three ultimate circle directions should be received in the space of triangles. Should they be reflected so as to point back in the direction of the circle midpoints, or should they continue in the opposite direction, away from those midpoints? There was no obvious criterion, whereas aesthetical criteria could be found on either side of the debate. We tried the possibility that both inverse triangles were incepting simultaneously, whereas only the expanding triangle that was first to meet and coalesce with other triangles or polygons would be allowed to continue. We were satisfied with this solution, although it felt as a kind of compromise, until we found the solution presented in this book. The solution we eventually found in 2015 is the introduction of two disjoint triangle spaces that are informed by the same terminating circle enclosures and that each contain one interpretation: a reflected one (green) and a continuous one (blue). Here the aesthetics of the solution sealed our appreciation of it, but it is obvious that other solutions could have been found and may be found in the future, for instance related to choices different from those mentioned in Sect. 5.1.

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