Computational Models in Architecture: Towards Communication in CAAD. Spectral Characterisation and Modelling with Conjugate Symbolic Domains 9783035618624, 9783035618488

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COMPUTATIONAL MODELS IN ARCHITECTURE 

APPLIED VIRTUALITY BOOK SERIES VOL. 12

COMPUTATIONAL MODELS IN ARCHITECTURE  — TOWARDS COMMUNICATION IN CAAD. SPECTRAL CHARACTERISATION AND MODELLING WITH CONJUGATE SYMBOLIC DOMAINS NIKOLA MARINČIĆ

BIRKHÄUSER Basel

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Nikola Marinčić Chair for Computer Aided Architectural Design (CAAD), Institute for Technology in Architecture (ITA), Swiss Federal Institute of Technology (ETH), Zurich, Switzerland SERIES EDITORS Prof. Dr. Ludger Hovestadt Chair for Computer Aided Architectural Design (CAAD), Institute for Technology in Architecture (ITA), Swiss Federal Institute of Technology (ETH), Zurich, Switzerland PROF. DR. VERA BÜHLMANN Chair for Architecture Theory and Philosophy of Technics, Institute for Architectural Sciences, Technical University (TU) Vienna, Austria Acquisitions Editor: David Marold, Birkhäuser Verlag, A-Vienna Content and Production Editor: Angelika Heller, Birkhäuser Verlag, A-Vienna Proof reading / Copy editing: Prof. Dr. Michael Doyle, École d’architecture de l’Université Laval, Canada Layout and Cover Design: onlab, CH-Geneva, www.onlab.ch Typeface: Korpus, binnenland (www.binnenland.ch) Printing: Christian Theiss GmbH, A-St. Stefan/Lavanttal Library of Congress Control Number: 2019933808 Bibliographic information published by the German National Library The German National Library lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in databases. For any kind of use, permission of the copyright owner must be obtained. ISSN 2196-3118 ISBN 978-3-0356-1848-8 e-ISBN (PDF) 978-3-0356-1862-4 © 2019 Birkhäuser Verlag GmbH, Basel P.O. Box 44, 4009 Basel, Switzerland Part of Walter de Gruyter GmbH, Berlin/Boston

987654321 www.birkhauser.com

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TABLE OF CONTENTS Abstract 7 Preface 9 Acknowledgements 11 I An overview: Architecture and computation 13

I background: a quest for coherence 15 — II computers and architecture 33  — III towards a new vision of architectonics 105

II Architectonics of communication: how different natures communicate

125

III An instrument for communication: Self-organizing model

219

IV An experiment: Communication and natures of architectural representation

237

I natural communication model 126 — II glossematics 146

I self-organizing map 220 — II self-organizing model 225

I nature(s) of architectural representation 238 — II computational precedents 243 — III spectral characterisation of an abstract object 247 —  IV modelling with conjugate symbolic domains 271

V EPILOGUE 283 REFERENCES 289 IMAGE AND ILLUSTRATION CREDITS 295

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ABSTRACT This work deals with computational models in architecture, with the ambition of accomplishing three objectives: 1 To position the established computational models in architecture within the broader context of mathematical and computational modelling. 2 To challenge computational models in architecture with contemporary modelling approaches, in which computation is regarded from the perspective of communication between different domains of a problem. 3 To show how within the paradigm of communication, it is possible to computationally address architectural questions that cannot be adequately addressed within the current computational paradigm. The first part of the work begins in the 19th century, delves into the body of thinking from which computation emerged and traces two general attitudes towards mathematical modelling, which will each eventually lead to different interpretations of computation. The first one, described as the logicist tradition, saw the potential of formal, mechanised reasoning in the possibility of constructing the absolute foundation of mathematics, its means of explanation and proof. The second one, the algebraist tradition, regarded formalisation within a larger scope of model-theoretic procedures, characterised by creatively applying abstraction towards a certain goal. The second attitude proved to be a fertile ground for the redefinition of both mathematics and science, thus paving a way for contemporary physics and information technology. On the basis of the two traditions, this dissertation identified a discrepancy between the computational models in architecture, following the first tradition, and those commonly used in information technology, following the second. The Internet revolution, initiated by the development of search engines and social media, is recognised as indicative of the changing role of computers, from “computing machinery” towards the generic infrastructure for communication. In this respect, three contemporary models of communication, proponents of the algebraic tradition, are presented in detail in the second part of the work. As a result, the selforganizing model is introduced as the concrete implementation of the ideas appropriated from communication models. In the last part of the work, the self-organizing model is applied to the problem of similarity between spaces, on the basis of their architectural representation. By applying partition and generalisation procedures of the self-organizing model to a large number of floor plan images,

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a finite collection of elementary geometric expressions was extracted, and a symbol attached to each instance. This collection of symbols is regarded as the alphabet, by means of which any plan created by the same conventions can be described as the writing of that alphabet. Finally, each floor plan is represented as a chain of probabilities, based upon its individual alphabetic expression of a written language, and its values used to compute similarities between plans.

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PREFACE Some might find this doctoral thesis unconventionally written. Instead of circumscribing its scope and concentrating its efforts on accomplishing a single objective within that scope, it engages with an unusually extensive body of knowledge with the aim of providing additional angles to its principal research domain: computational models in architecture. This body of knowledge involves early analytic philosophy, computability and probability theory, formal logic, quantum physics, abstract algebra, computer-aided design, computer graphics, glossematics, machine learning and architecture. However, the reason for such a comprehensive approach and perhaps radical gesture is not to claim any expertise nor mastery over the aforementioned fields of knowledge. To the contrary, it is a matter of methodology, aiming to operate in a more architectural manner, without losing the necessary rigour and consistency required of an academic work. An architect’s effort towards creating a masterful work, whether it is a building or a theory, always involves the integration of a wide variety of aspects laying outside of his/her own area of expertise. I see this apparent difficulty as a potential to enrich my work, and as a source of inspiration towards finding new, unexplored research perspectives. One more reason in favour of such approach can be justified by the very theories cited within this work, especially the concept of communication. To communicate with someone or something involves a responsive spectrum of frequencies on both sides, and tuning oneself to become sensitive to the potential resonances. The wider and richer this spectrum is, the more meaningful communication becomes. In this sense, the aim of this work is to make the spectrum of the research as resonant as possible, hoping to establish a more satisfying communication with the field of computational models in architecture, as well as with the reader. Nikola Marinčić, Spring 2019

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ACKNOWLEDGEMENTS I would like to express my deepest gratitude to Professor Ludger Hovestadt for introducing me to a fantastic, new world of ideas that have been an abundant source of inspiration in the last six years. This work would not have been possible without the continuous support, remarkable patience and absolute freedom he has given me to pursue my ideas. I am especially indebted to Professor Vera Bühlmann who provided me with the means to become a literate person, and so much more. I see this work as a showcase for the intellectual abilities I have acquired or strengthened by taking part in her theory colloquium for five years. I would also like to offer my special thanks to Professor Elias Zafiris for an invaluable introduction to abstract mathematics and quantum physics, and for giving me a deep insight into his natural communication model. His ideas provided a missing link to my thinking and allowed me to complete my thesis with a feeling of great satisfaction. I would like to thank my colleagues from the chair of CAAD for creating a truly unique and stimulating environment. I will always be proud of sharing it with you. My special thanks are extended to Jorge Orozco, Miro Roman, Mihye An, Diana Alvarez-Marin and Vahid Moosavi, who provided me with great feedback, and with whom I had amazing discussions throughout my whole engagement as a researcher. I would also like to thank Dennis Lagemann, Poltak Pandjaitan and David Schildberger for their help with translating the abstract of the thesis into German, and to Mario Guala for his great help with all the administrative aspects that made my stay at ETH much easier. I also wish to express my gratitude to Michael Doyle for doing a great job with copyediting this dissertation for publication. I would also like to thank my lovely wife Vanessa, for her relentless support and love through the challenging time of writing my thesis, and for helping with the proof-reading. Finally, I wish to thank my father Miodrag and my sister Ivana for always having faith in me, and for doing their best to help me in my pursuit of happiness. Nikola Marinčić, Spring 2019

ACKNOWLEDGEMENTS

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AN OVERVIEW ARCHITECTURE AND COMPUTATION

AN OVERVIEW

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I

Architecture and information technology… two species similar in kind, neither of them being in the least disciplinal: both affect everything, both are arts of gathering things. The one, 2,500 years old and dignified, and the other, just fifty years of age and impatient. L. Hovestadt, “Cultivating the Generic” (2014: 9) What today seems to be a passionate love affair between architecture and information technology, is in fact quite a delicate relationship, full of misunderstandings. The first “universal machines,” built in the 1940s, emerged as a side effect of the resolution of the 19th century attempt to ground all formalized, mathematical knowledge in logic. 1 Soon, computation was seen essentially as the mechanised treatment of logic. 2 Nevertheless, computers quickly got the attention of almost every field of human endeavour, including architecture. 3 With a certain amount of scepticism, acquired in the long tradition built upon mastership, architecture did not embrace its new potential “partner” very easily. Early researchers saw a lot of promise in computation, but for the large majority of practitioners, it seemed to be in poor taste 4 to simply embrace “logic” as a means to mechanise their articulations with the promise of greater efficiency and formal clarity. However, with the expansion of personal computers and intuitive computer-aided design software, the resistance became futile. An architecture was born out of generic drafting and modelling solutions, which employed computation to mimic the established modes of design. 5 While information technology started exploring new ideas, architectural research remained on the path of the “logicist” tradition. Today we live in a different world. Computers are omnipresent in our existence, and are no longer about logic. As the old identities slowly dissolve, new ideas are emerging on what computers are all about. These new ideas come from a higher level of abstraction and offer new unexpected vistas. 6 In this chapter, I will give an account of both old and new, with a hope that architecture might just find a very good partner in information technology, and hopefully reinvent itself in the digital. 1

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Turing, “On Computable Numbers, with an Application to the Entscheidungsproblem.” Turing’s seminal paper on computability comes as an answer to Hilbert’s decision problem (Entscheidungsproblem) but it starts with providing a mechanically constructed arithmetic of “computable” numbers. “Kurt Gödel has reduced mathematical logic to computation theory by showing that the fundamental notions of logic … are essentially recursive. Recursive functions are those functions which can be computed on Turing machines, and so mathematical logic may be treated from the point of view of automata.” Burks, editor’s introduction to Theory of Self-Reproducing Automata, 25. Mitchell, foreword to Architecture’s New Media, xi. “Computational methods to support the synthesis of design solutions have fascinated architectural researchers and horrified the practitioners.” Kalay, Architecture’s New Media, 237. Kalay, 181. See: Hovestadt, “Elements of Digital Architecture,” 28–116.

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I

background: a quest for coherence

Necessity or contingency? The story of symbolic computation emerged out of a peculiar state of affairs that started in the 19th century and got its epilogue in the first half of the 20th century. Before that time, mathematics appeared to be intimately linked to our physical reality, a phenomenon that Erich Reck described with the metaphor of an umbilical cord. 7 Geometry was safely grounded in Euclid’s axiomatic method, dating from around 300 BC. This method consisted of three parts: Axioms, or postulates are statements which are accepted without proof, and act as foundations of a theory; theorems are statements that are derived from the axioms and act as a superstructure (of knowledge) built upon the foundations; logic is a formal apparatus used to deduce theorems from axioms. Logic was established in antiquity and can be traced back to Plato and Aristotle. Aristotelian logic introduced three laws of reasoning in the natural language: laws of identity, contradiction and the excluded middle. 8 The interesting thing about logic was that it preserved the truthfulness of the statements it derived from axioms. It was believed that if the axioms were true, everything that was logically deducible from the axioms necessarily needed to be true as well. In fact, geometry and logic were so stable that their link to physical reality was not questioned for thousands of years. 9 It was not recognised for a long time that the truthfulness of axioms of logic and geometry was in fact accepted on the basis of intuition, which could only confirm that such statements are in fact self-evident. 10 An example of such evident truths were Euclid’s four postulates of planar geometry, which seemed to conceptualise our experience of space: 1 Let it have been postulated to draw a straight-line from any point to any point. 2 And to produce a finite straight-line continuously in a straight-line. 3 And to draw a circle with any centre and radius. 4 And that all right-angles are equal to one another. 11 Euclid’s fifth postulate about parallel lines in two-dimensional geometry seems to be of a different kind than the previous four:

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“With this first conception, geometry is firmly attached to physical reality—the umbilical cord between them is still in place.” Reck, “Frege, Natural Numbers, and Arithmetic’s Umbilical Cord,” 431. 8 Encyclopædia Britannica, s.v. “Laws of thought,” accessed September 1, 2017, https:// www.britannica.com/topic/laws-of-thought. 9 “As late as 1787, the German philosopher Immanuel Kant was able to say that since Aristotle formal logic ‘has not been able to advance a single step, and is to all appearances a closed and completed body of doctrine.’” Nagel and Newman, Gödel’s Proof, 30. 10 Burge, “Frege on Knowing the Third Realm,” 1. 11 Euclid, Elements of Geometry, 7.

AN OVERVIEW

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And that if a straight-line falling across two (other) straightlines makes internal angles on the same side (of itself whose sum is) less than two right-angles, then the two (other) straightlines, being produced to infinity, meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side). 12 This postulate is logically equivalent to the assumption that only one parallel can be drawn through a point outside a given line. 13 The fifth postulate introduced a great deal of problems to mathematicians, as it is neither self-evident, nor can be proved within Euclid’s axiomatic system. 14 Nevertheless, it somehow appears to be a correct statement. Such incoherencies were something that science and mathematics of the 19th century were determined to eradicate. Unlike other branches of mathematics, geometry was considered to be the most stable due to its axiomatic method. It seemed natural to ask whether such a secure axiomatic system could also be established elsewhere. Soon, many branches of mathematics were supplied with “what appeared to be adequate sets of axioms.” 15 It was of the utmost importance to establish an adequate axiomatic system of arithmetic, as it would securely ground other branches of mathematics on top of it. 16 In an attempt to use algebra to ground infinitesimal calculus, Cantor, Cauchy, Weierstrass, Dedekind, and others, showed how different notions in analysis could be defined in arithmetical terms. 17 The promise of axiomatisation was great: For each area of inquiry, having such a set of axioms would yield endless amounts of true propositions. In the mid 19th century, the work of Lobachevsky, Bolyai, Gauss and Riemann 18 began to challenge Euclid’s axiomatic system. In 1829, Lobachevsky developed a “geometry” by appropriating the first four axioms of Euclid, asserting that in his geometry the famous fifth 12 Euclid, 7. 13 Nagel and Newman, Gödel’s Proof, 6. 14 “The chief reason for this alleged lack of self-evidence seems to have been the fact that the parallel axiom makes an assertion about infinitely remote regions of space. Euclid defines parallel lines as straight lines in a plane that, “being produced indefinitely in both directions,” do not meet. Accordingly, to say that two lines are parallel is to make the claim that the two lines will not meet even ‘at infinity’.” Nagel and Newman, 6. 15 Nagel and Newman, 3. 16 Nagel and Newman, 3. 17 For example: “instead of accepting the imaginary number ‘−1’ as a somewhat mysterious “entity,” it came to be defined as an ordered pair of integers (0, 1) upon which certain operations of “addition” and “multiplication” are performed. Similarly, the irrational number √2 was defined as a certain class of rational numbers—namely, the class of rationals whose square is less than 2.” Nagel and Newman, Gödel’s Proof, 32. See also: Gauthier, Towards an Arithmetical Logic, 1. 18 “However, the geometric starting point of Riemann was not the non-Euclidean geometry, of which Riemann apparently had not even taken note, but rather the theory of surfaces developed by Carl Friedrich Gauss.” Jost, historical introduction to On the Hypotheses Which Lie at the Bases of Geometry, 26.

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postulate was not a true statement. This will become known as Bolyai– Lobachevskian geometry. Compared to Euclidean geometry, which was considered to mirror physical reality 19, the hyperbolic geometry of Lobachevsky and Bolyai was radically different. This geometry was able to describe a world that could not be observed empirically, while at the same time remaining a perfectly valid mathematical construction. Finally, in 1868, Beltrami demonstrated the independence of the fifth postulate from the other axioms. Implications of these events shook the very idea of mathematical foundations. If the validity of mathematical statements could not be guaranteed by the truthfulness of the axioms, as they need not be self-evident or mirror reality anymore, what remains of the ideas of grounding and validation? Moreover, if mathematics is not about the truths of our world, then what it is about? Gradually, it became clear that the position of necessity within mathematics was to be shifted from the truthfulness of its axioms to the validity of the inferences it employed. 20 Mathematics became abstract and stripped of meaning, as illustrated by the famous quote from Russell: … mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. 21 What replaced the method of validating a system of premises on the basis on its truthfulness, was a new idea of internal coherence, known as consistency. If an axiomatic system was to be consistent, it needed to guarantee that no mutually contradictory theorems can be deduced from the postulates. 22 With that requirement, an important question begged to be asked: Are even the axioms of Euclid’s system consistent? There was no single approach to the idea of creating a consistent system, and the interest in this question by two equally rigorous but ideologically quite distinct schools of thought warrants attention. The approach of the first group of mathematicians, including George Boole, Richard Dedekind and David Hilbert, among others, can be characterised as an algebraic approach to the idea of consistency. On the other side, Gottlob Frege, Bertrand Russell and their school of thought established an approach based on formal logic.

19 “Against Leibniz and Wolff, Kant thus emphasises and elaborates the axiomatic nature of geometry, i.e., that geometry has real axioms and that the propositions of geometry cannot be obtained analytically from definitions.” Jost, 28. 20 “We repeat that the sole question confronting the pure mathematician (as distinct from the scientist who employs mathematics in investigating a special subject matter) is not whether the postulates he assumes or the conclusions he deduces from them are true, but whether the alleged conclusions are in fact the necessary logical consequences of the initial assumptions.” Nagel and Newman, Gödel’s Proof, 8. 21 Russell, Mysticism and Logic, 58. 22 Nagel and Newman, Gödel’s Proof, 10.

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Algebraist tradition The ‘algebraist’ approach heavily relied on abstraction as the operative means to create coherent but contingent frameworks that did not offer a unifying consensual definition of the basis. The objectivity which they sought to establish within algebra was not something they considered as already given, but rather something that needed to be produced. George Boole was the first to revolutionise the study of logic after Aristotle. In his 1847 book The mathematical analysis of Logic, he established the study of logic on a purely algebraic basis. His algebra of logic provided a precise notation “for handling more general and more varied types of deduction than were covered by traditional logical principles.” 23 In 1854, he published his second monograph on algebraic logic, known as An Investigation of the Laws of Thought. The most important invention in his work was the equational treatment of logical statements, which allowed him to assess the validity of logical problems, and to extend their scope. In his book, he demonstrated how to transform any logical problem into an operative algebraic equation. By solving the algebraic equation, the logical problem was able to be resolved. 24 One of the most misunderstood algebraists of the time was the mathematician Richard Dedekind. 25 His approach to the problem of mathematical foundations was to arithmetise mathematics, but without appealing to numbers and the operations on them as naturally given. For Dedekind, natural numbers were a free creation of the human mind and abstraction was a tool to think with. In his essay “On Continuity and Irrational Numbers” (1872), he attempted to rigorously define the notion of a continuous magnitude, which at the time rested upon geometrical intuitions. 26 His method, known today as the Dedekind cut, constructed irrational and real numbers by freeing them from any content. 27 Dedekind considered the application of ordinal numbers as central, which allowed him to identify numbers structurally. He defined the cut as a separation which possesses one property, namely that it separates the domain

23 Nagel and Newman, 31. 24 Boole, An Investigation of the Laws of Thought, 24–38. 25 “… great philosophers, such as Cantor and Dedekind, are treated as philosophical naïfs, however creative, whose work provides, at best, fodder for philosophical chewing. Not only have we inherited from Frege a poor regard for his contemporaries, but, taking the critical parts of his Grundlagen as a model, we in the Anglo-American tradition of analytic philosophy have inherited a poor vision of what philosophy is.” Tait, “Frege Versus Cantor and Dedekind: on the Concept of Number,” 215. 26 “The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner.” Dedekind, Essays on the Theory of Numbers, 2. 27 Tait, “Frege Versus Cantor and Dedekind: on the Concept of Number,” 222.

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of rational numbers into two classes A1 and A2, where every number a1 belonging to A1, is smaller than every number a2 from A2. 28 fig. 1

Dedekind cut (Hyacinth, 2015)

1

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41 29

239 1393 169 985

3363 2378

577 408

99 70

17 12

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√2 His construction of natural numbers goes “beyond logic because it appeals to entities which, although created by the intellect, are nevertheless objectively available to it.” 29 In the chapter “Architectonics of Communication: How Different Natures Communicate,” we will investigate the mathematical framework of category theory in the light of Dedekind’s legacy of free creation of numbers, and his attention to structural properties. David Hilbert’s early work was greatly inspired by the advances in the axiomatic treatment of geometry. In The Foundations of Geometry (1899) 30, Hilbert devised a set of twenty axioms as a foundation of Euclidean geometry. 31 Unlike Euclid’s system, Hilbert’s axioms are not about the physical space, but “rather, they are taken to form the definition or characterisation of a certain abstract structure.” 32 In other words, Hilbert’s axioms are not self-evident truths, but contingent truths, which employ algebra to construct their consistency. 33 Since the algebraic characterisation cannot be “accommodated within any one ideal and elemental order,” 34 Hilbert provided the contingent basis as the source of consistency. 35 Some of the approaches to establish consistency required an infinite number of elements; others simply shifted Dedekind, Essays on the Theory of Numbers, 12–13. Potter, Reason’s Nearest Kin, 282. The original title is Grundlagen der Geometrie. Hilbert, The Foundations of Geometry, 2–16. Reck, “Frege, Natural Numbers, and Arithmetic’s Umbilical Cord,” 431. “The geometric statement that two distinct points uniquely determine a straight line is then transformed into the algebraic truth that two distinct pairs of numbers uniquely determine a linear relation; the geometric theorem that a straight line intersects a circle in at most two points, into the algebraic theorem that a pair of simultaneous equations in two unknowns (one of which is linear and the other quadratic of a certain type) determine at most two pairs of real numbers.” Nagel and Newman, Gödel’s Proof, 15. 34 Bühlmann, “Continuing the Dedekind Legacy Today,” 6. 35 Riemann’s “…idea is that if the metric properties of the space do not necessarily follow from its structure, then the space can carry several possible metrics, and the mathematician then can specify any such hypothetical relations and examine the resulting structures and distinguish them with regard to their characteristics. Hilbert will then raise this as the axiomatic method to a systematic program.” Jost, presentation of the text On the Hypotheses Which Lie at the Bases of Geometry 46. 28 29 30 31 32 33

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the problem of consistency of one system, by placing the responsibility on another system used as its base. Hilbert found these approaches unsatisfactory. In the next twenty years, he became obsessed with the idea of proving the absolute consistency of an axiomatic system, which led to the development known as Hilbert’s program. 36 Logicist tradition The ‘logicist’ approach to consistency emerged into a dominant paradigm whose followers appropriated computation as a child of their own tradition. Its proponents wished to encapsulate an ultimate objectivity within a system of foundations, upon which the whole of mathematics could rest. The implementation of this idea required grounding all of mathematics in logic. The objective was to construct an ideal, foolproof reasoning apparatus on the logical basis, which could, ideally, (in) validate any logical statement. The most prominent member of the logicist party was Gottlob Frege. If Boole’s idea was to ground logic within mathematics by means of algebra, Frege’s idea was quite the opposite. He wished to ground the whole of mathematics in arithmetic by means of the powerful deductive logic. Frege claimed that all the axioms of arithmetic could be “deduced from a small number of basic propositions certifiable as purely logical truths.” 37 In Begriffsschrift (1879), Frege invented quantification theory, which was a first step towards a precise notion of purely logical deduction. 38 The “conceptual notation” he defined allowed him to represent mathematical statements involving, for example, an infinite number of prime numbers. 39 In 1884, in The Foundations of Arithmetic 40, Frege introduced his own number theory, made to emulate formal logic. 41 He wished to show that arithmetic could be reduced to logical fundamentals, without any basis in intuition. Moreover, he regarded arithmetic as a completely objective “realm.” His central claim in The Foundations was that: In arithmetic, we are not concerned with objects which we come to know as something alien from without through the medium of the senses, but with objects given directly to our reason and, as its nearest kin, utterly transparent to it. 42 Today, we can more easily recognise the alarming implications of such a statement. By regarding mathematics as a transparent, objective

Nagel and Newman, Gödel’s Proof, 25. Nagel and Newman, 32. Tait, “Frege Versus Cantor and Dedekind: on the Concept of Number,” 213. Tait, 217. However, he did so by utilising cardinal numbers, which was the misunderstanding of the notion of infinity introduced by Cantor. 40 Originally published as Die Grundlagen der Arithmetik. 41 Gauthier reformulates Frege’s question into: “How far can we go into arithmetic with deductive logic alone?” Gauthier, Towards an Arithmetical Logic, 1. 42 Frege, The Foundations of Arithmetic, §105:115. 36 37 38 39

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reality, 43 which was simply to be accessed by reason, Frege diminished the role of human creativity and invention. His statement completely rejects the possibility of the creative abstraction that Dedekind was fighting for. 44 Despite all this, Frege is considered to be the founder and the “hero” of abstraction until this day. 45 Very soon, a complete new set of problems had emerged out of Frege’s program. In 1901, Bertrand Russell showed that Frege’s logical axioms were inconsistent. 46 He discovered that Frege’s approach could lead to the construction of paradoxical sets, which was named Russell’s paradox. 47 The source of Frege’s inconsistencies lied in its self-referentiality. The logicists sought to remain within the same paradigm, while avoiding self-reference at all costs. In the period from 1910 to 1913, Russell and Whitehead wrote Principia Mathematica, a cornerstone of the logicist paradigm. It was a three-volume work of mathematical foundations that attempted to establish a set of axioms and rules powerful enough to prove all mathematical truths. It was meticulously designed to keep the inconsistencies out “in a most staunch and watertight manner.” 48 Principia Mathematica also appeared to be the final solution for the problem of consistency, as it reduced the problem of consistency of arithmetic to the problem of the consistency of formal logic itself. 49 This was the moment where Russell and Whitehead’s work became closely intertwined with Hilbert’s search for absolute consistency, which consisted in the complete formalisation of a deductive system by “draining” it from any meaning, as described by Nagel and Newman: The postulates and theorems of a completely formalised system are “strings” (or finitely long sequences) of meaningless marks, constructed according to rules for combining the elementary signs of the system into larger wholes. Moreover, when a system has been completely formalised, the derivation of theorems from postulates is nothing more than the transformation (pursuant to rule) of one set of such “strings” into another set of strings. 50

43 Burge, “Frege on Knowing the Third Realm,” 2. He called it “The third realm.” 44 Bühlmann, “Continuing the Dedekind Legacy Today,” 8. 45 “However, more important to me in this paper than the question of Frege’s own importance in philosophy is the tendency in the literature on philosophy to contrast the superior clarity of thought and powers of conceptual analysis that Frege brought to bear on the foundations of arithmetic, especially in the Grundlagen, with the conceptual confusion of his predecessors and contemporaries on this topic.” Tait, “Frege Versus Cantor and Dedekind: on the Concept of Number,” 215. 46 Irvine and Deutsch, “Russell’s Paradox.” 47 Irvine and Deutsch, “Russell’s Paradox.” famous “set of all sets that are not members of themselves.” 48 Hofstadter, preface to Gödel, Escher, Bach, 4. 49 Nagel and Newman, Gödel’s Proof, 33. 50 Nagel and Newman, 20.

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The defining trait of formal systems lies in their simplicity. They include a limited number of signs, a grammar which defines how to create wellformed strings, a set of strings taken as axioms, and a set of transformation rules. 51 This introduces two levels from which a formal system can be considered: The first, “lower” level consists of the “meaningless marks” that are produced mechanically; the second accommodates highlevel reasoning about the processes of the lower level. Hilbert defined the higher level as a meta-language. His goal was to a find method that could prove the absolute consistency of a system. He believed that the solution lied on the “lower” level and was interested in demonstrating the “impossibility of deriving certain contradictory formulas” 52 within it. In other words, Hilbert’s hope was that a purely formal language could be used to prove its own consistency. The main achievement of Principia Mathematica, was that it provided “a remarkably comprehensive system of notation, with the help of which all statements of pure mathematics (and of arithmetic in particular) can be codified in a standard manner.” 53 The book’s notation and deductive system presented themselves to Hilbert as a perfect medium for establishing an absolute proof of consistency. His work seemed to be on the right track until 1931, when Gödel’s theorems proved that neither Principia, nor any other system of that kind, could ever achieve this goal. linguistic turn The philosophy of the early 20th century experienced a crisis similar to the one of mathematics. The accounts that held philosophy as the fundamental discipline responsible for the questions of foundations and knowledge, started to lose appeal in the light of the clarity and the precision demonstrated by modern logic. An idea began to emerge that philosophical facts do not exist per se, but that they are above all language articulations. Accordingly, philosophy should have been dealing with clarification of thoughts on a logical basis by analysing the logical form of propositions. 54 As a consequence, the attention of philosophy turned to language as an operative medium for thought, and to grammar as an apparatus for coherent thinking. Within the relation between philosophy and language, another current emerged that was interested in the relation between grammar and logic. This interest introduced two schools of thought: The first one was established by the Austrian philosopher Ludwig Wittgenstein; the second by Rudolph Carnap and the Vienna Circle. 55 51 Hofstadter, Gödel, Escher, Bach, 35. 52 Nagel and Newman, Gödel’s Proof, 27. 53 Nagel and Newman, 33. 54 Wittgenstein, Tractatus Logico-Philosophicus, 45 (4.12). 55 Potter, Reason’s Nearest Kin, 18.

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Wittgenstein, an associate of Russell and a young admirer of Frege’s work, is considered to be one of the progenitors of the linguistic turn. In 1921, he wrote Tractatus Logico-Philosophicus, which conveyed the idea that philosophical problems arise from an inconsistent nature of the language that is used to construct philosophical statements. 56 In the preface of the Tractatus, he summed up his argument as the following: What can be said at all can be said clearly; and whereof one cannot speak thereof one must be silent. 57 Tractatus was the first philosophical work putting the language at the centre of its inquiry, boldly stating that “the limits of my language mean the limits of my world.” 58 This non-intuitive position was in strong contrast to most of the Western philosophical tradition. To free philosophy from incoherence, Wittgenstein required an ideal language for philosophical analysis, as “ordinary” language was full of ambiguities. Wittgenstein’s conception of a language was not simply an instrument of logic. If this were the case, the argumentation would need to be set up so that it leads to an argument or a proof. It was a philosophical grammar, designed to draw a line separating valid philosophical language from nonsense. 59 By creating a philosophical system as an application of his rigorous grammar consisting of atomic facts, propositions and operators, Wittgenstein believed to have eliminated all philosophical problems. However, in all of his self-proclaimed success, he also realised “how little has been done when these problems have been solved.” 60 Decisive for the linguistic turn in the humanities were the works of yet another tradition, namely the structuralism of Ferdinand de Saussure and the ensuing movement of poststructuralism. 61 Saussure’s general complaint was directed at the lack of systematicity in the study of language. 62 In his university lectures, collected and published only later by his students in Course in General Linguistics (1916), 63 Saussure referred to a number of approaches for studying language, finding them all inadequate. 64 For him, grammar was detached from language and too dependent upon (and limi­ ted by) logic, having the sole purpose of distinguishing between correct Wittgenstein, preface to Tractatus Logico-Philosophicus, 23. Wittgenstein, 23. Wittgenstein, 74. Wittgenstein, 23. Wittgenstein, 24. Wikipedia, s.v. “Linguistic turn,” last modified March 24, 2017, 15:55, https:// en.wikipedia.org/wiki/Linguistic_turn. 62 Saussure, Course in General Linguistics, 3–4. 63 Originally published as Saussure, Ferdinand de. Cours de linguistique générale. Publ. par Charles Bailly et Albert Sechehaye avec la collaboration de Albert Riedlinger. Lausanne: Pavot, 1916. 64 “At the same time scholars realised how erroneous and insufficient were the notions of philology and comparative philology. Still, in spite of the services that they rendered, the neogrammarians did not illuminate the whole question, and the fundamental problems of general linguistics still await solution.” Saussure, 5.

56 57 58 59 60 61

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and incorrect forms. He found that philology was not about language at all, but about the interpretation of texts “as a means to literary and historical insight.” 65 He recognised some potential in comparative philology, and in its task of finding similarities and differences between languages. 66 Saussure’s vision of linguistics was that it should be able: • to describe and trace the history of all observable languages, which amounts to tracing the history of families of languages and reconstructing as far as possible the mother language of each family; • to determine the forces that are permanently and universally at work in all languages and to deduce the general laws to which all specific historical phenomena can be reduced; and • to delimit and define itself. 67 For Saussure, language was a “system of signs that express ideas” 68 and was part of a larger whole, of a “science that studies the life of signs within society,” 69 which he termed semiology. 70 His concept of a sign challenged the traditional view, which considered words as mere labels attached to concepts. He defined sign as an entity that united a concept of a thing, the signified, and its sound image, the signifier. Since there cannot be a concept without it being named, the signified and the signifier necessarily exist as a pair. For Saussure, language was about symbolic manipulation, thus the “real things” did not play any role in the constitution of a sign. Another crucial view that he held was that signs possess differential, not natural, identity. In other words, a sign is being a sign only by the virtue of not being any other sign: … the concepts are purely differential and defined! Not by their positive content but negatively by their relations with the other terms of the system. Their most precise characteristic is in being what the others are not. 71 mechanisation of articulation The advent of digital computers was rapid, overwhelming, and its development is still underway. There is no room here to mention every important contributor. For the purposes of this dissertation, the focus will be on the figures who have established the main computational paradigms and on those whose work has had the biggest influence on computer-aided architectural design.

65 66 67 68 69 70 71

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Saussure, 1. Saussure, 4–5. Saussure, 6. Saussure, 16. Saussure, 16. Saussure, 16. Saussure, 117.

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Recursion and computability In the 1920’s, Hilbert’s program crystallised the main expectations of formal systems for the purpose of axiomatization, most notably ideas of: • Consistency: No mutually contradictory theorems should be deducible from the axioms. • Completeness: Axioms of a deductive system are “complete” if every true statement that can be expressed in the system is formally deducible from the axiom. 72 At an international conference in 1928, Hilbert introduced a famous challenge that illustrated his hope of the potential of formal systems, what he referred to as the Entscheidungsproblem 73 (decision problem). He asked whether an algorithm could be made that takes two inputs: (i) a description of a formal language (for example Principia Mathematica (PM)) and (ii) a statement expressed in that language (for example, a theorem of PM), and outputs either true or false, depending whether the statement ii is provable within the formal language i. All that remained to settle the question of foundations once and for all was to solve the problem. Unfortunately for Hilbert, the complete opposite happened. In his 1931 paper “On formally undecidable propositions of Principia Mathematica and related systems,” Austrian logician Kurt Gödel proved that Hilbert’s requirements of consistency and completeness could not both be achieved in a formal system. Moreover, he exposed the fundamental limitations of all axiomatic systems, including those of arithmetic and logic. 74 The pinnacle of Gödel’s paper are the two theorems known as Gödel’s incompleteness theorems, which he proved in an ingenious way. The first theorem states that any system of a certain complexity—in which, for example, arithmetic can be developed—is essentially incomplete. In other words, true statements that cannot be derived from the axioms could be expressed in such a system. The second theorem shattered Hilbert’s hope of achieving absolute consistency by showing that a formal system alone cannot be used to prove its own consistency. In his proof, Gödel applied the idea of recursive enumerability, and demonstrated how arithmetic, defined by recursive functions, could be made to emulate logic.

72 Nagel and Newman, Gödel’s Proof, 73. 73 German for “decision problem.” 74 “What is more, he (Gödel) proved that it is impossible to establish the internal logical consistency of a very large class of deductive systems— elementary arithmetic, for example—unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves.” Nagel and Newman, Gödel’s Proof, 3.

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Fig. 2

Wir definieren nun nach dem Rekursionsschema (2) eine Funktion | (x,h) folgendermaßen:

A recursive definition from Gödel’s famous paper. (Gödel, 1931)

| (0,h) = 0 | (n +1, h) = (n +1) $ a +| (n, h) $ a (a) wobei

a = a [a (t (0, h))] $ a [| (n, h)]

In 1936 and 1937, only five years after Gödel’s paper, Alan Turing and Alonzo Church delivered another blow to Hilbert’s hope that the Entscheidungsproblem could be solved. Both mathematicians proved that Hilbert’s question cannot be positively answered. 75 In order to make Hilbert’s notion of the algorithm operative, and its operations explicit, Turing conducted an experiment involving a hypothetical machine which operated with “empty” symbols mechanically. By mechanical means, the machine could automate the operations of finitistic formal systems. Like Gödel, Turing attempted to solve the Entscheidungsproblem within arithmetic, and in doing so introduced a novel constitution of arithmetic, utterly different from one based on deductive logic. Turing introduced his paper with the notion of a computable number: According to my definition, a number is computable if its decimal can be written down by a machine. 76 The ‘computable’ numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. 77 The computing machine consisted of an infinite tape divided into a number of discrete elements called “squares.” Each square could be empty but was also capable of bearing a symbol, for example 0 or 1. The machine could carry out only four actions: read the symbol from the square, write the symbol to the square, erase the symbol from the square, or move the tape one step left or right. Like Hilbert’s formal system, such a machine was completely described by a finite number of conditions that he defined as “m-configuration.” 78 Depending upon the symbol that was read from the square, the configuration assigned actions to be taken. For example, one such m-configuration c1 would instruct the machine to write a symbol to the current square, move one step to the left and change its current configuration to c2. Such simple procedures were to be repeated indefinitely. At any moment the machine was “directly aware” only of one symbol: the “scanned symbol” from the “scanned square.” But the tape is what allowed the machine to

75 In other words, it is impossible to decide algorithmically whether statements in a finitistic formal system are true or false according to the description of the formal system. 76 Turing, 230. 77 Turing, “On Computable Numbers, with an Application to the Entscheidungsproblem,” 230. 78 Turing, 231.

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“effectively remember some of the symbols which it had seen (scanned) previously,” thus serving as its memory. 79 It was shown later that such a simple mechanical machine could in fact emulate any possible formal system, but was prone to the same limitations discovered by Gödel. 80 By explicating the machine’s capabilities, Turing had effectively reduced the class of real numbers to the class of computable numbers, a class whose sole property was that it could accommodate for a finite mechanical calculation of all of its members. He showed that large classes of numbers were in fact computable, but not all numbers. 81 Those that were computable, were also necessarily enumerable 82, which was proved by Gödel. The same holds true for computable sequences. By instrumentalising them, however, Turing turned what appeared to be an inherent limitation into a new perspective: It is possible to invent a single machine which can be used to compute any computable sentence. 83 With this statement, Turing turned the attention from the necessity implied in the limits of mechanically computing numbers, towards the contingency implied in the infinity of possible sequences. He implicitly pointed out that limitations are inherent to any formal system, but not to the creativity of an individual having such a system at his or her disposal. His statement transcended the computable machine into the universal or any-machine. Digital computers and algorithms According to Turing, the first mechanical computers, namely Babbage’s difference and analytical engine were invented as early as the beginning of 19th century but failed to surpass the prototypical stage. 84 Inspired by Turing’s early work, the first digital computers 85 came to existence in the 1940s. Hungarian-American polymath John von Neumann was one of the pioneers who streamlined the

79 Turing, 231. Or what we would call today—stored program. 80 “In short, it has become quite evident, both to the nominalists like Hilbert and to the intuitionists like Weyl, that the development of a mathematico-logical theory is subject to the same sort of restrictions as those that limit the performance of a computing machine.” Wiener, Cybernetics, 13. 81 “Computable numbers include all numbers which could naturally be regarded as computable.” - large classes of numbers are computable including PI, e, etc. “The computable numbers do not, however, include all definable numbers, and an example is given of a definable number which is not computable.” Turing, “On Computable Numbers, with an Application to the Entscheidungsproblem,” 230. 82 “…able to be counted by one-to-one correspondence with the set of all positive integers.” Turing, 230. 83 Turing, 241. 84 Turing, “Computing Machinery and Intelligence,” 439. 85 Burks, editor’s introduction to Theory of Self-Reproducing Automata, 6–10. For example, ENIAC (1943–45) and EDVAC (1945).

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design of digital computers 86 and wrote the first successful algorithms. 87 For him, computation was part of a larger umbrella of automation–theory that seeks “general principles of organisation, structure, language, information and control.” 88 Such theory was meant to explain the processes inherent to natural systems by means of both analogue (natural automata) and digital computers 89 (artificial automata). However, Turing’s construction in which arithmetic (thus logic) could be reduced to computation inspired von Neumann to think of them as one and the same thing. He introduced logic at the heart of the theory of automata, often referring to it as a “logical theory of automata.” 90 Arthur Burks illustrated this point in his introduction to von Neumann’s book Theory of Self-Reproducing Automata: To conclude, von Neumann thought that the mathematics of automata theory should start with mathematical logic and move toward analysis, probability theory, and thermodynamics. When it is developed, the theory of automata will enable us to understand automata of great complexity, in particular, the human nervous system. 91 The early work of Claude Shannon has firmly established the technical foundation of digital computers in logic. In his famous 1937 master’s thesis named “A Symbolic Analysis of Relay and Switching Circuits,” he investigated the correspondence between Boolean algebra and electrical relays, which were the building blocks of electrical components of the time. He advanced the design of electrical switches by proposing that they be implemented as binary switches. 92 The logical basis of electrical switches became the cornerstone for the design of electronic digital computers 93, but its further development to transistors and computer chips was made possible only with development of quantum physics. In 1948, Shannon published his paper “A Mathematical Theory of Communication,” which is considered to be the founding work of information theory. In this paper, Shannon defined entropy as the quantitative measure of information within the set-theoretical paradigm (which will be discussed and challenged in the part II).

86 …by separating data from instructions, analogous to the Turing’s tape, so that “by changing the program, the same device can perform different tasks.” Kalay, Architecture’s New Media, 28. 87 “He devised algorithms and wrote programs for computations ranging from the calculation of elementary functions to the integration of non-linear partial differential equations and the solutions of games.” Burks, editor’s introduction to Theory of SelfReproducing Automata, 5. 88 Burks, 21. 89 Burks, 21. 90 Burks, 25. 91 Burks, 28. 92 Shannon, “A Symbolic Analysis of Relay and Switching Circuits,” 3–4. 93 “In other words, the structure of the machine is that of a bank of relays, capable each of two conditions, say “on” and “off”; while at each stage the relays assume each a position dictated by the positions of some or all the relays of the bank at a previous stage of operation.” Wiener, Cybernetics, 119.

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German mathematician Norbert Wiener pushed the idea of automation further within an emerging thermodynamic understanding of the world, which, as he wrote, operated on the principles of conservation (and degradation) of energy 94, information and control. 95 Within such a paradigm, the distinct processes of both natural (animal) and artificial (machine) entities could be addressed from the perspective of energy and information exchange. Since the input and the output of each component of the system were necessarily interconnected, each event affected the state of the whole environment. Wiener used the example of patients suffering from ataxia, whose muscles were completely healthy, but their brain was not able to establish control over their actions. 96 His hypothesis was that the brain was not simply an organ that gives orders to other organs, but also a monitoring device, that continuously, and in real time, adjusts its “outputs” according to the “inputs” it receives from the senses. Such a continuously adapting control process he called the chain of feedback 97, and named the entire field of “control and communication theory, whether in the machine or in the animal,” cybernetics. 98 Accordingly, every system conceived upon the principle of feedback chains, is a cybernetic system. For Wiener, the basis of a feedback chain lies in the anatomy of the brain. He considers neurons to be the elements of the human computation system “which are ideally suited to act as relays.” 99 If the brain was using computation to control its own feedback chain, then digital computers had potential for controlling any system: It has long been clear to me that the modern ultra-rapid computing machine was in principle an ideal central nervous system to an apparatus for automatic control; and that its input and output need not be in the form of numbers or diagrams but might very well be, respectively, the readings of artificial sense organs, such as photoelectric cells or thermometers, and the performance of motors or solenoids. 100 Wiener did not stop at defining the program for cybernetics, but rather developed it into a mathematical model that could be computationally implemented. He saw an enormous potential for the optimal governance of systems, going as far as proposing its use by psychopathologists for the control of physiological diseases. 101 94 “The living organism is above all a heat engine, burning glucose or glycogen or starch, fats, and proteins into carbon dioxide, water, and urea. It is the metabolic balance which is the center of attention.” Wiener, 41. 95 “the present time is the age of communication and control.” Wiener, 39. “…the present age is as truly the age of servomechanisms as the nineteenth century was the age of the steam engine or the eighteenth century the age of the clock.” Wiener, 43. 96 Wiener, 8. 97 Wiener, 96. 98 Wiener, 11. 99 Wiener, 120. 100 Wiener, 26. 101 In the chapter: “Cybernetics and Psychopathology” Wiener, 144–154.

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Wiener’s work has greatly influenced computer science, with his neuron model as a precursor to the neural network perspective to machine learning. 102 However, it is important to remember that cybernetics is primarily a control paradigm that models the world as a closed thermodynamic system. Its constitution, once it is set up, is, in principle, fixed. The social parallel to such a paradigm appears today as a tyrannical form of governance and care should be taken in proposing its application to systems more complex than a thermostat. Computation and language Around forty years after the Principia Mathematica and the peak of Hilbert’s programme, researchers began to consider natural language from the perspective of formal systems. The most prominent figure in this respect was Noam Chomsky, whose most important work on the topic is Syntactic Structures (1957). Chomsky’s interest in language was purely formal and pragmatic, focusing on two elementary notions: • Syntax: “the study of the principles and processes by which sentences are constructed in particular languages.” 103 • Grammar: “device of some sort” for producing sentences of such a language. 104 For Chomsky, grammar plays the role of a mechanical “judge,” making efficient binary decisions whether a given sentence is correct or not, independent of any semantics. He gave a famous example: Sentences (1) and (2) are equally nonsensical, but any speaker of English will recognize that only the former is grammatical. (1) Colorless green ideas sleep furiously. (2) Furiously sleep ideas green colorless. 105 Before presenting his generative model, Chomsky described two models that he considered incapable of dealing with the complexity of grammar: the Markov model, and the phase structure model. The fact that Chomsky dismissed the Markov model, which later became the de facto standard for machine translation (as well as search engine technology) well illustrates his misunderstanding of and disregard for mathematics. 106 Chomsky presented his own generative transformational model as the most adequate and “natural” model for addressing linguistic structures. The model consists of three stages where simple transformations of strings are applied in succession. In the first stage, Chomsky applied 102 Wikipedia, s.v. “Cybernetics,” last modified August 30, 2017, 17:55, https://en.wikipedia. org/wiki/Cybernetics. 103 Chomsky, Syntactic Structures, 11. 104 Chomsky, 11. 105 Chomsky, 15. 106 “I think that we are forced to conclude that grammar is autonomous and independent of meaning, and that probabilistic models give no particular insight into some of the basic problems of syntactic structure.” Chomsky, 33.

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the phase structure model, which he had characterised as inadequate if applied in isolation. In the second stage, he introduced “simple and generalised transformations of the structure.” In the third stage, he applied “morphophonemic rules.” 107 Without being acquainted with 19th century mathematics, one might fail to notice that these three stages of transformations are in fact an appropriation of Hilbert’s formal systems, decontextualised and put to pragmatic use. Chomsky reduced the complexities of a language to pure syntax and defined language as a set of sentences constructed out of a finite number of strings and rules. 108 Although his attempt to define natural language as a recursively defined structure of string replacements failed to capture the complexity of human languages, it has greatly influenced artificial languages used to program computers. These languages were seen as “a conceptual leap above BASIC, FORTRAN and all the engineering-based languages, which were ill-structured, inconsistent and clumsy.” 109 Soon, transformation grammars and state machines entered other fields of science. Computational approaches in biology had a special relevance for computer-aided architectural design, as they utilised generative grammars for the development of form. In 1968, the Hungarian biologist Aristid Lindenmayer developed a special type of formal system, for the purpose of modelling the cellular interactions in the development of plant filaments. 110 Today, we call such grammars L-systems. 111 In the beginning, Lindenmayer’s language was mathematical and showed traces of inspiration from cybernetics and the theory of computation, where the main elements—inputs, outputs and states— described and encapsulated the model. We assume a “black box,” or in more recent terms a “sequential machine,” which has buttons or levers through which inputs can be given to the machine, dials or slots from which outputs can be read out, while the machine is thought to pass from state to state in consecutive discrete moments of time. Under a sequence of inputs the machine, therefore, undergoes a sequence of changes in state, and produces a sequence of outputs, which is why it is called a “sequential” machine. 112

107 Chomsky, 32–33. 108 Chomsky, 13. 109 Coates, programming.architecture, 27. 110 “…filaments are composed of cells which undergo changes of state under inputs they receive from their neighbours, and the cells produce outputs as determined by their state and the input they receive. Cell division is accounted for by inserting two new cells in the filament to replace a cell of a specified state and input.” Lindenmayer, “Mathematical Models for Cellular Interactions in Development I.,” 280. 111 Pruskinkiewicz and Lindenmayer, The Algorithmic Beauty of Plants, 1. 112 Lindenmayer, 281.

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The design of his sequential machine promoted the idea of cell division, which allowed it to easily model growth. Such growth, he observed, bears similarity to the growth of plants 113, which he illustrated with a diagram. fig. 3

Row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Lindenmayer’s diagram of cellular growth. (Lindenmayer, 1968)

0 0 1 0 1 1 01 1 0 0 1 0 11 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1

1

1 0 1 1 1 0 0 0 1 1 1 1 1 0 1 0 0 0 0 1 1 0 0 1 10 1 01 1 01

1 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 0 11

1 0 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 1 10 1 1 1 1 01 11

1 1 0 1 1 0 1 11 00

1 0 1 1 1 1 0 0 1 0 0 0 1 11 1 11 00 0 00 10

1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1

In his later work, as his interest shifted from numerical patterns to forms, Lindenmayer’s method became less mathematical, and more grammatical in the sense of Chomsky. Ultimately, it became a simple formal system aimed to explain the complexity of living organisms: The main question we wish to address is whether a ‘program’ or ‘grammar’ can be defined and used profitably in the study of multicellular development “from egg to adult organism. 114 BEyoND THE liMiTS of forMal logic Turing’s discovery answered the question posed by Hilbert’s programme, and, in a certain way, does speak the language of the 19th century. Nonetheless, I argue that the viewpoints inspired by the interpretations of Gödel’s theorems, insisting that computation is not more than formalised logic are misleading. Computers, whose inherent capabilities logically equate the capability of a Turing machine 115 will never be able to prove their own

113 “…this is in spite of the fact that new cells are continually inserted, and old ones are being pushed to the right or disappear by division. Thus, a stable pattern is generated, moving from the left to the right, while the cells participating in this pattern are continually replaced or displaced. This is rather reminiscent of the growing apical regions of plant organs, like those of shoots and roots.” Lindenmayer, 287. 114 Lindenmayer and Jürgensen, “Grammars of Development,” 4. 115 “Kurt Gödel had reduced mathematical logic to computation theory by showing that the fundamental notions of logic (such as well-formed formula, axiom, rule of inference, proof) are essentially recursive (effective).” Burks, editor’s introduction to Theory of Self-Reproducing Automata, 25.

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consistency by means of syntax or to compute certain numbers. But we have to ask, is this really what computation is all about? If we embrace computation from this perspective, our curiosity will be confronted by inevitable formal limits, and our creativity bounded by necessities. If, on the contrary, we embrace computation from the ‘algebraist’ perspective, in which computers are any-machines, our interest will be in model-theoretic procedures that computation allows to articulate. 116 These procedures will be neither objective nor absolute, neither necessary nor contingent. The essence of computation lies beyond necessities of formal logic and calls upon our capacity to think. By following the legacy established by Dedekind, we will use computers to “engender cases out of abstraction.” 117 This chapter concludes with a hypothesis: The kind of mathematics (thus abstractions) we decide to set in motion will set the limits of what can be done with computers. II

computers and architecture

Leon Battista Alberti, the Renaissance man credited with establishing the architectural profession in the 15th century, codified architectural design into parallel projection scale drawings. 118 As for the physical models of buildings, Smith refers to the texts of Vitruvius to acknowledge the importance of small-scale models for builders already in Roman times. 119 The two-dimensional—graphical—and three-dimensional— volumetric—representation of architectural models remain until today the standard encoding of architectural design, produced on a daily basis by the large majority of practicing architects around the world. In their first encounters, architecture and computers began to medialise the technique of drawing. A prominent example is the work of Ivan Sutherland, often acknowledged as the grandfather of interactive computer graphics, graphical user interfaces and (indirectly) computer-aidedarchitectural design. 120 In 1963, he created Sketchpad, the first interactive computer-aided drawing program operating in three-dimensions. It used a light pen as the input device and a modified oscilloscope as the output device. 121 Such experiments gradually introduced a generic model of space 116 Bühlmann, “Continuing the Dedekind Legacy Today,” 17. 117 “The key assumption he has to make, thereby, is to consider abstraction as an act of engendering.” Bühlmann, 11. 118 Carpo, The Alphabet and the Algorithm, x, 31. 119 “First, the Romans seemed to be well aware of the persuasive application of the smallscale model. Second, the small-scale model built by Callius permitted a population untrained in architecture to easily view the possibilities of a full-scale mechanism. Third, the Roman small-scale model presented a mechanism granting the architect and the population an opportunity to perceive a possible future.” Smith, Architectural Model as Machine, 15. 120 Salomon, The Computer Graphics Manual, 10. 121 Salomon, 10.

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to computer screens and seemed to settle an unspoken agreement—that a computer-generated space should mimic our “natural” intuition of threedimensional space. Introducing the mathematical model of the camera to the space clearly suggested placing a human observer within the computer. The major technical challenge at the time was how to model and display realworld or imaginary objects. One way of achieving this was to approximate real-world objects with the numerical geometric models 122 created from idealised geometrical components such as lines, curves, surfaces, solids, and to project them onto the two-dimensional surface of the screen. This approach evolved into a discipline of mathematics called geometric modelling 123, and to its pragmatic counterpart, computer-aided-geometric design. 124 Two more prominent terms of the time merit clarification: The first is computer-aided design (CAD) 125, which indicates a practice of furnishing the infrastructure of geometric modelling with user-friendly graphical interfaces, used to model digital geometric objects; 126 the second is computer graphics 127, which appropriated CAD modelling to create virtual objects, but with a primary focus of creating a realistic simulation of the real world. 128 The first two parts of this chapter investigate how the ideas behind computer graphics and computer-aided design influenced the way we think about modelling architecture with computers and give an overview of their mathematical and technical infrastructure. The third part outlines the prominent computational models in architecture, situating them within the previously covered mathematical traditions and the more general models of computation. The last part introduces the current state of the art and identifies its inherent limits. mathematical bases of geometric modelling Many scholars consider the analytic (Cartesian) geometry to be the mathematical foundation of computer graphics and computer-aided

122 Golovanov, Geometric Modeling, 11. 123 Golovanov, 11. 124 “CAGD is based on the creation of curves and surfaces and is accurately described as curve and surface modelling.” Rockwood and Chambers, Interactive Curves and Surfaces, 2. 125 “Computer-aided design was one of, maybe the first application to make use of computergenerated curves and surfaces.” Peddie, The History of Visual Magic in Computers, 47. 126 “Geoscience used CAGD methods to represent seismic horizons; computer graphics designers modelled their objects with surfaces, as did molecule designers for pharmaceuticals. Architects discovered CAGD, word processing and drafting programs based their interface protocols on free-form curves (PostScript), and even moviemakers discovered the power of animating with such surfaces, beginning with TRON, continuing through Jurassic Park, and beyond.” Rockwood and Chambers, Interactive Curves and Surfaces, 7. 127 “The first use of the phrase Computer Graphics is generally attributed to William Fetter … who used the phrase in the early 1960s in the development of first computer model of a human body.” Peddie, The History of Visual Magic in Computers, 101. 128 Hoffmann, Geometric and Solid Modeling, 3–4.

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design. 129 It is productive to scrutinise this statement within the context architect and computer scientist Ludger Hovestadt introduces in “Elements of Digital Architecture” (2015), where he describes geometry as the “rationalisation of thought patterns amid known elements.” 130 He distinguishes between three principal geometries, each of them “unlocking a new world”: Euclidean geometry, operating in space; analytic geometry, operating in time; and digital code, operating in values. 131 In his 1637 work La géométrie, French mathematician and philosopher René Descartes introduced analytic geometry. Using algebra 132, he parametrised the Euclidean paradigm and captured time 133, which in turn fixed a new reference system for all physical processes. 134 Unlike Euclidean geometry, described by text, analytic geometry operates with numbers, making its character formulaic and principally non-visual. To represent it outside of equations, they must be drawn and thereby need to be parametrised and evaluated. Only in this sense can one say that Cartesian geometry is the foundation of CAD and computer graphics. However, this leaves us with a dilemma: What kind of modelling would correspond to Hovestadt’s third geometry of digital code? An attempt at an answer will be presented later in this work. Polynomial functions Euclid’s geometry rests upon constructions with straight lines and circles. 135 Descartes showed that for the analytic geometry to capture circularity (thus curvature), its algebraic elements must be raised to a power, which can be accomplishing using polynomials. Mathematically, the simplest form of a polynomial is described as a function of a single variable t on its domain: a n t n + a n —1 t n —1 + … + a 1 t + a 0

Where an, an-1, a0 are constants called coefficients. 136 A product of a variable and a coefficient constitutes a monomial. 137 Thus, every polynomial can be expressed as a finite sum of monomials, whose variables

129 For example: “Computer-aided geometric design has mathematical roots that stretch back to Euclid and Descartes.” Rockwood and Chambers, Interactive Curves and Surfaces, 6, or: “The Cartesian system and geometry is the basis of all computer graphics.” Peddie, The History of Visual Magic in Computers, 33. 130 Hovestadt, “Elements of Digital Architecture,” 35. 131 Hovestadt, 35. 132 “Often it is not necessary thus to draw the lines on paper, but it is sufficient to designate each by a single letter.” Descartes, The Geometry of Rene Descartes, 5. 133 Hovestadt, “Elements of Digital Architecture,” 35. 134 Jost, historical introduction to On the Hypotheses Which Lie at the Bases of Geometry, 14. 135 Euclid, Elements of Geometry, 7. It is reflected in his elementary postulates. 136 Barbeau, Polynomials, 1. 137 Rockwood and Chambers, Interactive Curves and Surfaces, 28.

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are raised to a nonnegative integer power. 138 The monomial whose variable is raised to the highest power within the polynomial determines its degree. According to their degree, polynomials of the first degree are considered linear polynomials, of the second degree as quadratic polynomials, of the third degree as cubic polynomials, of the fourth degree as quartic polynomials, and so on. 139 From the perspective of the elements that could be substituted by the variable, a polynomial is often mapped to a polynomial function 140, such as in our case of a single variable t: p (t ) = a 0 + a 1 t + a 2 t 2 + … a n t n

Polynomial functions are important because they are simple, easy to compute, highly applicable and have a finite evaluation schema. 141 The evaluation of a polynomial can be accomplished by simply replacing its variable with a number and carrying out the computation. 142 To obtain inequalities or to graph a wide variety of functions, it is important to know the value of t for which the polynomial takes the value 0—a value referred to as the zero of a polynomial. 143 On a more abstract level, operating with polynomials and vectors is quite similar: Both can be added, multiplied with constants or with each other, etc. 144 Furthermore, by means of the elementary algebraic structure known as a field, it is possible to establish an isomorphic mapping from the space of polynomials to the space of vectors. 145 In this case, to a polynomial of degree n corresponds a vector of a dimension n+1. n+1 pn ~ – V

For example, we can use { to symbolise a mapping of the quadratic polynomial 146 to its corresponding space of vectors: {: p 2 → V 3

In this case, the coefficients of the quadratic can be represented as the three-dimensional vector:

138 Barbeau, Polynomials, 1. 139 Barbeau, 2. 140 Broida and Williamson, A Comprehensive Introduction to Linear Algebra, 255–56. 141 Farouki, “The Bernstein Polynomial Basis: A Centennial Retrospective,” 382. 142 Barbeau, Polynomials, 2. 143 Barbeau, 2. 144 Barbeau, 2–3. 145 See chapter 4.2. “The Algebra of Polynomials” in Hoffman and Kunze, Linear Algebra, 119–23. 146 Polynomial of the second degree.

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a0 { ( a 0 + a 1 t + a 2 t 2) = a 1 a2

In the light of this mapping, the collection of variables: B = {1,t,t 2 }

gains an interesting geometrical interpretation: It can be seen as a set of “atomic,” linearly independent vectors called basis vectors, whose linear combination spans the space of the same dimensionality. 147 Thus, in our example we can represent every quadratic polynomial as the linear combination of the basis set B. From elementary linear algebra, we know that there can be (infinitely) many bases spanning a vector space, and by means of linear transformations we could rewrite them in another form. 148 The choice of the appropriate basis depends on the task at hand. Through the aforementioned mapping, any newly obtained basis will also yield its polynomial form. One important application of polynomials was interpolation. In mathematics, to interpolate means to “estimate values between given known values,” 149 and in polynomial terms this corresponds to finding a simplest 150 polynomial passing through a set of points. For example, to interpolate between two points, it is sufficient to use a first-degree polynomial (maps to a line in linear algebra); three points would require a quadratic polynomial (maps to a parabola); four points a cubic polynomial. 151 At the end of 18th century 152, French mathematician Lagrange introduced a special form of polynomial that could approximate all the given points. 153 Polynomials in what later became known as the Lagrange form 154 are those that are mapped to standard basis vectors 155 in their corresponding vector space. For example, to approximate four points on a 147 If monomials are considered as a collection P containing all the variables raised to the degrees ranging from 1 to k, “and if by means of this base we can represent any polynomial of degree k as a sum, then the question may be asked whether any polynomial of degree k can be written as a summation of terms, each a product of a coefficient and a basis function from P. If yes, then P is said to span the set of polynomials of degree k. Further, if P is the smallest set to span, then it is a basis for the polynomials of degree k. The set P is a basis called a power basis. If any monomial is eliminated from P, then not all polynomials of degree k can be written in terms of P.” Rockwood and Chambers, Interactive Curves and Surfaces, 29. 148 Rockwood and Chambers, 29. 149 Rockwood and Chambers, 60. 150 Barbeau, Polynomials, 206. Simplest meaning—of the lowest possible degree. 151 Finding a polynomial interpolating a set of points is a linear algebra problem. It is an equation in which polynomial coefficients are the variables to be solved for. 152 “Lagrange formula appears in the fifth lecture of his “Leçons élémentaires sur les mathématiques” delivered in 1795 at the École Normale.” Gautschi, “Interpolation Before and After Lagrange,” 347. 153 Rockwood and Chambers, Interactive Curves and Surfaces, 62. 154 Rockwood and Chambers, 60. 155 For each basis vector, one of its coordinates will be equal to one, while all the others to zero.

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37

plane requires computing a sum of four polynomials, each constructed on the unique set of their x coordinates. The first polynomial p1 should evaluate to 1 in the point 1, and evaluate to 0 in all other points (2, 3, 4). The second polynomial p2 should evaluate to 1 in the point 2, and evaluate to 0 in all other points (1, 3, 4), and the pattern continues for other points. 156 l1(x)

Fig. 4

Lagrange interpolation polynomial as a sum of scaled basis polynomials. (adopted from Glosser.ca, 2016)

l2(x)

l3(x)

l4(x)

L(x)

8

6

4

2

0

–2

–4 –5

0

5

Since we know the zeroes of the four polynomials—and we know them because we have specified them by the definition—it is easy to derive the actual polynomials. These polynomials form a Lagrange basis. The Lagrange polynomial would be a linear combination of this basis and the set of their y coordinates. This polynomial simply solves the Lagrange interpolation problem. Using interpolation to approximate a set of points can lead to unpredictable results because the polynomial must pass through the all points. Even a slight change of a single point can dramatically alter the interpolating polynomial. 157 If the set contains many scattered points, as is often the case with any series of measurements, approximation becomes highly inaccurate. An alternative approach to this problem, known as regression, would be “to give up the requirement of exact agreement at some points in favour of gaining some flexibility for making the approximation close everywhere.” 158 In this case, approximation cannot be done globally, but needs to be restricted to an interval 159, where it can be approximated using methods such as least squares. 160

156 Rockwood and Chambers, 61. 157 Barbeau, Polynomials, 213. 158 Barbeau, 213. 159 Barbeau, 213. 160 Barbeau, 214.

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Fig. 5

An example of a non-linear regression.

0.50

0.45

0.40

0.35

0.30

0.0

0.2

0.4

0.6

0.8

Another important application of polynomials is the approximation of continuous functions. The Weierstrass approximation theorem (1885) states that polynomials can uniformly approximate any function that is merely continuous over a closed interval. 161 In 1912, Russian mathematician Sergei Bernstein proved the Weierstrass theorem using only basic algebraic operations. He introduced a parameter t that replaced variable x, within the interval [a, b] with the equality t=(x-a)/(b-a). He was thereby able to map any continuous function from the interval [a, b] to the parametric domain t ![0, 1] by means of a Bernstein polynomial of degree n without any loss of generality. 162 Unlike the basis of Lagrange polynomials, which needs to be computed for the specific points it interpolates, Bernstein basis B is a collection of functions that are computed for a desired accuracy of approximation n and is independent of coordinates: 163 n ( 1 – t ) n—k t k, k

B nk (t ): =

k = 0, 1, … , n

where t is a parameter. 164 Therefore, a Bernstein polynomial p(t) is the approximation of a given continuous function f constructed as a linear combination of parametric Bernstein basis functions: 165 p n (t ): =

n

∑ f ( k / n) B

n k

(t )

k= 0

This approximation schema is very simple and elegant, but it converges very slowly, requiring a high polynomial degree n to approximate the

161 Farouki, “The Bernstein Polynomial Basis,” 383. 162 Farouki, 383. 163 Rockwood and Chambers, Interactive Curves and Surfaces, 36 and Farouki, “The Bernstein Polynomial Basis,” 383. 164 Bellucci, “On the explicit representation of orthonormal Bernstein Polynomials,” 2. 165 Farouki, “The Bernstein Polynomial Basis,” 383, 385.

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39

given function precisely. 166 For that reason, Bernstein polynomial approximants have remained theory rather than practice. 167 Fig. 6

Bernstein polynomial approximation. (Farouki, 2012)

However, approximation was not the end for Bernstein polynomials. The generality of its basis functions allowed for a free creation of polynomials with arbitrary coefficients in Bernstein basis. Such constructions are called polynomials in Bernstein form and, in the 1960s, became a cornerstone of computer-aided geometrical design. 168 Algebraic functions and their graphs Given an arbitrary polynomial and its geometric interpretation, it is sometimes possible to extract valuable information about it, such as its zeros or degree, by simply graphing it. 169 Depending on the particular choice of equation used to represent it, different properties can be investigated, for example the generalised parabola xp-yq=0 or the folium of Descartes x3+y3+axy=0. 170 When learning algebra at school, students are often first presented with the notion of a function that comes in a form y=f (x). As x varies, the value of y is computed by the function f and, when graphed, a pair of coordinates (x, y) gradually forms a curve. 171 Variable y is presented as dependent on x and the form of their dependence is captured in an explicit form. In analytic geometry, however, an equation of the unit circle is presented as x2+y2=1. Here, the focus is on the interdependence of x and y, which is equated with the constant representation of a radius of a circle, capturing the relationship between x and y in an implicit form.

166 Sometimes the required degrees exceeded hundreds (or even thousand), which was impossible to compute at the beginning of the 20th century without computers. 167 Farouki, “The Bernstein Polynomial Basis,” 385. 168 Farouki, 380. 169 Barbeau, Polynomials, 71. 170 Brieskorn and Knörrer, Plane Algebraic Curves, 88. 171 Rockwood and Chambers, Interactive Curves and Surfaces, 10–11.

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Let us look at this from the perspective of graphs. Both implicit and explicit formula can be plotted, and their graph can represent either a curve or a surface. However, the explicit form will be quite deficient in this respect, due to the nature of the relation it embodies. Since f (x) is a functional mapping, only a single value of y can be assigned to each value of x by definition. Therefore, a vertical line cannot be represented, as a function cannot have infinite slope; 172 the derivative f ' (x) is not defined parallel to the y axis; a circle cannot be defined due to the intersection of two points for each x, etc. 173 Another form of equations is not subjected to such limitations. Parametric forms of equations offer advantages to the implicit form 174, including a method of parameterisation, which defines motion on a curve 175 and allows for easier differentiation. 176 Parametric forms remove the direct dependency of the variables y and x, as in the function y=f (x) by introducing a third variable t called a parameter, upon which both x and y are made dependent. x = f (t ) y = f (t )

To represent a vertical line in this form is trivial, as well as the unit circle x = cos(t ) y = sin (t )

where the value of the parameter t traces the circle while varying in the domain 0#t#2r. To represent a polynomial—as we have seen with the example of the Bernstein form—the formula does not change and can be written as: p (t ) = a 0 + a 1 t + a 2 t 2 + … a n t n

Here p(t) takes a form of a vector-valued function of a scalar parameter t whose vectors are unrestricted to a specific dimensionality. 177 If the vectors are three-dimensional, the function p(t) contains three coordinate functions x(t), y(t), and z(t). 178 The function p(t) is called a curve 179, the slope of which is given by the tangent line at any point and computed by finding the derivative vector [x’(t), y’(t)] at any value of t. 180As the

172 Rockwood and Chambers, 11–12. 173 Rockwood and Chambers, 11–12. 174 Rockwood and Chambers, 12. 175 “Motion on the curve refers to the way that the point (x, y) traces out the curve.” Rockwood and Chambers, 12. In this respect parametric representation allows us to control position and speed of an object moving along a curve, or tells us when the object will be at a particular location. The parameter gives us direction and speed. 176 “Vertical tangent vectors can be defined by differentiation, for instance, which is not possible in explicit Cartesian form.” Rockwood and Chambers, 6. 177 Rockwood and Chambers, 17. 178 Rockwood and Chambers, 17. 179 Golovanov, Geometric Modeling, 13. 180 Rockwood and Chambers, Interactive Curves and Surfaces, 15.

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parameter t changes, the derivative vector computed at each point corresponds to the speed at which the point traces out the curve. 181 Lissajous curves were investigated in the 19th century in connection with vibration problems and with the motion of a double pendulum. 182 In the early days of computer graphics, parametric representation allowed displaying these curves on an oscilloscope for the first time. 183 The parametric equation of one such a curve can be written as: x = cos(3t ) y = sin (2t )

Variation of the value of the parameter t in the domain 0#t#2r draws the Lissajous curve. Sampling the values of the parameter t at certain steps reveals the direction in which the curve unfolds, corresponding to the motion of the point that traces it. Fig. 7

A parametric Lissajous curve evaluated in 20 steps.

t = 0.05

t = 0.10

t = 0.15

t = 0.20

t = 0.25

t = 0.30

t = 0.35

t = 0.40

t = 0.45

t = 0.50

t = 0.55

t = 0.60

t = 0.65

t = 0.70

t = 0.75

t = 0.80

t = 0.85

t = 0.90

t = 0.95

t = 1.00

181 Rockwood and Chambers, 15. 182 Brieskorn and Knörrer, Plane Algebraic Curves, 65. 183 “Lissajous figure on an oscilloscope, displaying a 3:1 relationship between the frequencies of the vertical and horizontal sinusoidal inputs, respectively.” Peddie, The History of Visual Magic in Computers, 295.

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Another advantage of parametrically defined curves is that they generalise more easily to higher dimensions. For example, by adding another equation for the coordinate z we obtain a Lissajous curve in three dimensions: x = cos(3t ) y = sin (2t ) z= t

Fig. 8

A parametric Lissajous curve in three dimensions.

6

4

2

0 -1.0

-0.5

0.0

-1.0

0.5 -0.5 0.0 0.5

1.0 1.0

Again we can see how the “analytic base allows interesting curves to be defined through mere generalisation of the form of equations.” 184 Splines: Pragmatics of mathematical curves The discussion of curves thus far has included only analytic curves 185, which have a limited practical value. The period after the Second World War saw the pragmatic revival of mathematical curves following the development of 184 Brieskorn and Knörrer, Plane Algebraic Curves, 87. 185 “We will designate curves as analytic curves if their coordinates in some local coordinate system can be described by analytic functions without using points, vectors, or other curves.” Golovanov, Geometric Modeling, 17.

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43

computers. This pragmatism was embodied in the idea that curves could be used as a geometric modelling tool to not only represent but also design arbitrary shapes. 186 The breakthrough happened in the 1950s with the work of French engineers Pierre Bézier at Renault and Paul de Casteljau at Citroën—who independently developed a smooth type of a curve, created on a Bernstein polynomial basis 187 that could be constructed on a freely specified set of points. 188 The curve became known as the Bézier curve 189 and the family of parametric curves as splines. 190 De Casteljau and Bezier turned the approximation inefficiency of Bernstein polynomials to their advantage, converting it into a mathematical tool that would allow them “to create complex shapes for the automobile industry, to replace clay models.” 191 Geometric modelling is characterised by a progression of geometric elements from simple primitives, such as points and lines towards more complex shapes. 192 To illustrate this, we will demonstrate how to derive Bézier curves by using only elementary algebra. The demonstration starts with the parametric equations of a line in two dimensions, passing through two points (x0,y0) and (x1,y1): x = x 0 + ( x 1 – x 0) t y = y 0 + ( y 1 – y 0) t

This equation can be rewritten as: t

x = (1 – t ) x 0 + t · x 1 y = (1 – t ) y 0 + t · y 1

These equations are in fact the same. Rewriting them exposes their polynomial basis {1-t, t} that multiplies coordinate points 193 and shows that the coefficients are precisely those discrete values that will produce the desired

186 “The initial use of CAGD was to represent the data as a smooth surface for numerical control. It soon became apparent that the surfaces could be used for the design.” Rockwood and Chambers, Interactive Curves and Surfaces, 7. 187 “Ultimately, the work of de Casteljau and Bézier lead to adoption of the Bernstein form, typified by what is now called a Bézier curve.” Farouki, “The Bernstein Polynomial Basis,” 386. 188 Farin, Curves and Surfaces for CAGD, xvi. 189 “By taking a linear combination of Bernstein polynomials we can define a generalized parametric curve over the interval [a, b], which is known as the Bézier curve.” Bellucci, “On the explicit representation of orthonormal Bernstein Polynomials,” 3. 190 Computationally speaking, splines are parametric curves, whose computation “may be broken down into seemingly trivial steps—sequences of linear interpolations.” Farin, Curves and Surfaces for CAGD, 25. 191 Farouki, “The Bernstein Polynomial Basis,” 386. 192 Rockwood and Chambers, Interactive Curves and Surfaces, 3. 193 Rockwood and Chambers, 17.

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interpolation in a form of a continuous line equation. 194 In that respect, the coefficients (x0,y0) and (x1,y1) are called control points of the line. 195 In a more general form, the parametric equation of the line can be defined as: l (t ) = (1 – t ) p 0 + t p 1  196

Here the parameter t takes value in the range (0#t#1) defining the line segment; p0 and p1 can be vectors of arbitrary dimensionality: pi =

xi yi

but in our specific case they are: p0 =

x0 y0

and

p1 =

x1 y1

The expressions 1-t and t that make up the polynomial basis are also called blending functions, as each of them influences its respective coordinate, acting as a weight. blend [2p]

x

y

1– t

x0

t

x1

y0 y1

Constructing a line between points is the simplest example of linear interpolation, which is perhaps the most fundamental concept in geometric modelling. 197 However, even for this simplest case, other forms of linear interpolation are possible, also resulting in the straight-line interpolation of two points. Choosing a different basis/blending function can produce the same result with a different parametrisation—that is, the motion of a particle at t tracing a line will be different. 198 Let us give a practical example of a curve interpolating the two points A={1, 2}; B={8, 5}. Replacing the variables with numbers in the previously given equations and tables, gives: x = (1 – t ) 1 + 8t y = (1 – t ) 2 + 5t 194 “Although a parametric curve or surface (a vector-valued function of one or two variables) is an infinitude of points, its computer representation must employ just a finite data set. The mapping from the finite set of input values to a continuous locus is achieved by interpreting those values as coefficients for certain basis functions in the parametric variables.” Farouki, “The Bernstein Polynomial Basis,” 386. 195 Rockwood and Chambers, Interactive Curves and Surfaces, 34. 196 Golovanov, Geometric Modeling, 27. 197 “All subsequent curves and surfaces are defined by repeated linear interpolation in some form.” Rockwood and Chambers, Interactive Curves and Surfaces, 28. 198 Rockwood and Chambers, 28.

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45

blend [2p]

x

y

1– t

1

2

t

8

5

Rewriting the previous equations gives the equation of the line interpolating between points A and B defined parametrically in the range (0#t#1): t

x = 1 + 7t y = 2 + 3t

By replacing the variable t with numbers ranging from 0 to 1, we can derive the corresponding x and y coordinates sweeping the line. t

Fig. 9

t

x

y

0

1

2

1

8

5

6

Graph of a parametrically defined line segment.

{8, 5}

5

t=1 4

3 {1, 2} 2 t=0 1

0

2

4

6

8

The successive composition of multiple first-degree curves defines a polyline. It is the simplest compound curve constructed on a set of points, where line segments sequentially connect the given points. 199 The next level would be to interpolate between three points on a plane: (x0,y0), (x1,y1), and (x2,y2). The interpolation consists of three steps: First we interpolate between points (x0,y0) and (x1,y1): B 01 = (1 – t ) p 0 + t · p 1

Then we interpolate between points (x1,y1) and (x2,y2): B 12 = (1 – t ) p 1 + t · p 2

199 Golovanov, Geometric Modeling, 19.

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Lastly, we interpolate between the two newly obtained interpolating functions, just as we would between points: B 012 = (1 – t ) · B 01 + t · B 12 = (1 – t ) ((1 – t ) p 0 + t · p 1) + t ((1 – t ) p 1 + t p 2)

When we distribute the terms in the equation and rewrite it to expose the polynomial basis we get the interpolation formula for three points. B 012 = (1 – t ) 2 · p 0 + 2t (1 – t ) · p 1 + t 2 · p 2

Points pi define the control polyline. 200 This is expressed in the following table: blend [3p]

x

y

(1 – t ) 2t (1 – t )

x0 x1

y0 y1

t2

x2

y2

2

It is interesting to notice that the three basis (blending) polynomials that we have just derived are in fact the same expressions that would be the result of evaluating the previously given Bernstein basis function for n=2: B nk (t ): =

n ( 1 – t ) n—k t k, k

k = 0, 1, 2

In fact, Bézier curves are simply a linear combination of the Bernstein basis with arbitrary vectors, where n reflect the number of both basis polynomials and control points. 201 Let us give a practical example of a Bézier curve interpolating the three points: A={-2, 2}; B={0, -1}; C={2, 2}. When we replace the numbers in the previously given equations and tables, we obtain: x 012 = (1 – t ) 2 · x 0 + 2t (1 – t ) · x 1 + t 2 · x 2 y 012 = (1 – t ) 2 · y 0 + 2t (1 – t ) · y 1 + t 2 · y 2 blend [3p] (1 – t)

2

2t (1 – t) t2

x

y

–2

2

0 2

–1 2

When we solve the equation for t, we obtain a parametric equation of the line interpolating points A, B and C defined in the range (0#t#1):

200 Golovanov, 28. 201 Bellucci, “On the explicit representation of orthonormal Bernstein Polynomials,” 3.

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x = –2 + 4t y = 2 – 6t + 6t 2

By replacing the variable t with numbers ranging from 0 to 1, we derive the corresponding x and y coordinates sweeping the line. t

fig. 10

x

y

0

–2

2

0.2

–1.2

1.04

0.4

–0.4

0.56

0.6

0.4

0.56

0.8 1

1.2 2

1.04 2

{–2, 2}

Graph of a parametrically defined parabola sampled in 6 intervals.

{2, 2} 2.0

t=0

t=1

1.5

1.0

0.5

–2.0

–1.0

1.0

2.0

–0.5

–1.0

{0, -1}

If we wish to interpolate a curve in a higher resolution, we sample the parameter t in smaller increments yielding additional (x, y) points: fig. 11

{2, 2}

{-2, 2}

Graph of a parametrically defined parabola sampled in 21 intervals.

2.0

t=0

t=1

1.5

1.0

0.5

-2.0

-1.0

1.0

2.0

-0.5

-1.0

48

{0, -1}

TOWARDS COMMUNICATION IN CAAD

I will conclude with the interpolation between four points on a plane: (x0, y0), (x1, y1), (x2, y2), and (x3, y3), which is the standard form of the Bézier curve. By repeating the procedure from the example of three points, or by simply solving the Bernstein function, we obtain the polynomial basis represented in a table: blend [4p] (1 – t ) 3

x x0

y y0

3t (1 – t ) 2

y1

3t (1 – t )

x1 x2

t

x3

y3

2

3

y2

Finally, let us show a practical example of a Bézier curve interpolating four points: A={1, 2}; B={3, 7}; C={8, -5}; and D={10, 8}. Depending on the sampling increment of the parameter t, the Bézier curve can be visualised in a required resolution. Fig. 12

{10, 8}

Graph of a parametrically defined cubic sampled in 11 intervals.

{3, 7}

5

{1, 2}

0

2

4

6

8

10

-5 {8, –5}

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49

{10, 8}

Fig. 13

Graph of a parametrically defined cubic sampled in 51 intervals.

{3, 7}

5

{1, 2}

0

2

4

6

8

10

-5 {8, –5}

The first and the last basis function in the table have the highest polynomial degree, thus have the most influence on their respective points. Because of that, the first and the last are the only points through which the curve will pass and through which the curve will be cotangent to the control polygon. 202 The curve can approach the other two points with lesser influence, but it can never pass through them. 203 To illustrate the mathematical concepts behind it, we have evaluated the Bézier curve by direct substitution, that is by replacing the variable t with numbers ranging from 0 to 1 and deriving the coordinates. Technically, this is probably the least effective method of evaluating a point on the curve. 204 Raising small floating-point numbers to high powers by using computers is a very error-prone operation. 205 One fast and numerically stable method to evaluate the Bézier curve is de Casteljau algorithm. This algorithm allows us “to calculate any point of the Bezier curve using the control points without knowing anything about the Bernstein functions.” 206 Furthermore, it is applicable for computing derivatives 207 and subdividing the curve. 208 Most classes of parametric curves in geometric modelling are defined linear combinations of certain base functions. 209 An extension of Bézier curves, rational Bézier curves, have additional weight attributes attached to their control points, which allows them to accurately represent conic

202 Rockwood and Chambers, Interactive Curves and Surfaces, 34. 203 Rockwood and Chambers, 4. 204 Rockwood and Chambers, 42. 205 Rockwood and Chambers, 42. 206 Golovanov, Geometric Modeling, 30. 207 Derivative can be defined as: “Vector directed along the tangent to the curve in the direction in which the parameter increases.” Golovanov, 13. 208 Rockwood and Chambers, Interactive Curves and Surfaces, 42–43. 209 Hoffmann, Geometric and Solid Modeling, 168–69.

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sections in ways standard Bézier curves cannot. 210 Another type of curve that generalised 211 and further advanced rational Bézier curves is the B-spline212, a “highly controllable spline curve, a favourite of industrial designers.” 213 A B-Spline is a piecewise polynomial, meaning that it can be described as a smoothly connected succession of low-degree Bézier curves. 214 This makes it more versatile than a Bezier curve 215, while in other aspects remaining similar: a B-spline, like a Bézier curve, does not pass through its control points, except at the endpoints. B-splines further developed into rational B-spline curves, also known as NURBS curves, which became the “standard curve and surface description in the field of CAD and graphics.” 216 Just like rational Bézier curves, their shape depends not only on the location of its control points, but also on the weight values attached to them. 217 All the aforementioned curves with different degrees (flat and curved) can be joined into a composite constructions known as contours. 218 0

1 2

4 3

A

fig. 14

D

C 5 6

9

A piece-wise NURBS curve. (Rutten, 2007)

10 11

7 8

B

PragMaTicS of MaTHEMaTical SurfacES Surfaces are the main descriptive elements of a modelled object’s shape. 219 Research in geometric modelling and computer-aided geometric design “has discovered many useful classes of parametric surfaces and has developed a large repertoire of algorithms for their design, analysis, and manipulation.” 220 The explicit form of a surface in three dimensions, takes the form z=f (x, y) and it defines an elevation surface or terrain. 221 Explicitly defined surfaces expose the same problems inherent to explicitly defined curves. 222 Yet again, the parametric form of the surface corrects them. The simplest way to define a parametric surface is to start with a regular two-dimensional parametric curve where x and y

210 211 212 213 214 215 216 217 218 219 220 221 222

Golovanov, Geometric Modeling, 36, 39. “They are generalisations of Bernstein functions.” Golovanov, 40. Golovanov, 40. A “B” standing for Basis spline. Rockwood and Chambers, Interactive Curves and Surfaces, 32. Rockwood and Chambers, 32. Farin, Curves and Surfaces for CAGD, 120. Farin, 120. Golovanov, Geometric Modeling, 65. Golovanov, 83. Golovanov, 85. Hoffmann, Geometric and Solid Modeling, 3. Rockwood and Chambers, Interactive Curves and Surfaces, 17. “They must be single-valued for any point on the plane; They cannot have vertical tangent planes; Transformations may cause the above two difficulties.” Rockwood and Chambers, 17.

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coordinates depend on the parameter t, and introduce another parameter s223 to sweep it in the third dimension by using translational or rotational motion. 224 This family of surfaces includes evolution, extrusion, revolution and swept surfaces. 225 Another way to construct a surface is to use multiple curves. In 1958, Stephen Coons created a mathematical model of the surface, a Coons patch, by interpolating a set of four boundary curves in three dimensions. 226 The early work of Bézier and de Casteljau on curves and Coons on surfaces was crucial in establishing the fields of geometric modelling and computer-aided design. In general, a surface has a complex boundary, described by twodimensional curves in the parametric space of the surface. 227 The surface r can be written in the general form as: x ( u,v ) r ( u,v ) = y ( u,v ) z ( u,v )

where u and v are parameters, which vary freely within the two-dimensional connected domain (0#u, v#1) defining a surface patch. 228 Partial derivatives in respect to each parameter define two vectors and a plane parallel to these vectors at a given point defines a tangent plane to the surface. 229 V

fig. 15

0.1

R parameter space. 2

(Rutten, 2007)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.9 0.8 0.7 0.6

(0.2, 0.4)

0.5 0.4 0.3

O

0.2 0.1

U

223 224 225 226

Rockwood and Chambers, 17. Golovanov, Geometric Modeling, 11. Golovanov, 100. Coons, “Surfaces for Computer-Aided Design of Space Forms,” 1. For the generalisation of Coons patches see: Golovanov, 111–17. 227 Golovanov, 85. 228 Golovanov, 85. 229 Golovanov, 86.

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For the purpose of design, the most intuitive way to define a surface and operate with it is to construct it on a set of points. 230 The simplest type of surface that allows this can be created by extending a Bézier curve to three dimensions; it is known as Bézier surface or Bézier patch. 231 fig. 16

Bézier surface. (Wojciech mula, 2010)

A generalised surface in three dimensional space can be described as the tensor product of Bernstein polynomials. 232 For explanatory purposes, we need to recall the operation of interpolating a curve between three points, when we replaced control point vectors with interpolating functions of vectors. To construct a Bézier polynomial by the tensor product is to replace the same control points in a Bezier formula with the Bézier curves. 233 In this way, we can think of a surface as a nesting of one set of curves inside another. 234 Rockwood and Chambers described this extension into three dimensions in an illustrative way: Imagine moving the set of control points of the Bezier curve in three dimensions. As they move in space, new curves are generated. If they are moved smoothly, then the curves formed create a surface, which may be thought of as a bundle of curves. If each of the control points is moved along a Bezier curve of its own, then a Bezier surface patch is created. 235

230 231 232 233 234 235

Golovanov, 117–20. Rockwood and Chambers, Interactive Curves and Surfaces, 5. Bellucci, “On the explicit representation of orthonormal Bernstein Polynomials,” 3. Rockwood and Chambers, Interactive Curves and Surfaces, 134. Rockwood and Chambers, 135. Rockwood and Chambers, 134.

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Bézier surfaces inherit many properties of Bézier curves, such as endpoint interpolation, tangent conditions, affine invariance, etc. 236 For curves, the set of control points defines a control polyline; for surfaces, it defines a control polyhedron. 237 In the same way that Bézier curves are extended to Bézier surfaces, B-splines are extended into B-spline surfaces, commonly known as NURBS surfaces. 238 By the same analogy, NURBS surfaces inherit many properties of B-spline curves. They employ the same algorithms to calculate shape, normals, tangents, curvatures, etc. The main difference is that “curves have tangent vectors and normal planes, whereas surfaces have normal vectors and tangent planes,” 239 meaning that “curves lack orientation while surfaces lack direction.” 240 The possibility of rendering NURBS surfaces on a screen required a significant amount of computational power, which was only possible in 1989 with the development of Silicon Graphics stations. 241 Operations on curves and surfaces Another aspect of geometric modelling deals with operability and interaction of geometric objects, aiming to extend their modelling possibilities. Such operations include finding a minimum distance point to curve or surface, approximating gaps between curves, trimming a curve at a point, pinning curve endpoints, projecting curves to surfaces, finding parameters associated with three-dimensional points, etc. Some of these operations can be calculated analytically, but only to a certain extent. 242 In order to handle non-analytic cases, geometric modelling utilises differential calculus, most notably Newton’s method. 243 There are many geometric algorithms named after Newton. What they all assume at some point is that the geometry is relatively flat, therefore the first order approximation of the geometry can be used to find a solution. I will illustrate this with an example of projecting a point on a three-dimensional free-form curve. The process starts by roughly approximating the solution, which is usually done by sampling the curve/ surface and/or consulting the object’s bounding boxes. In our specific case, the approximation is made by projecting the point on a tangent line of the curve at its first control point. Finding the tangent line requires 236 Rockwood and Chambers, 136–37. 237 Golovanov, Geometric Modeling, 121. 238 Golovanov, 122. 239 Rutten, RhinoScript 101, 101. 240 Rutten, 101. 241 Peddie, The History of Visual Magic in Computers, 45. 242 In a number of cases we can find algebraic solution (projecting a point to line, conic, sphere, plane, etc.) yielding the solution in the Cartesian space. However, as our objects are defined parametrically, we still need to find the parameter t for curves, and u, and v parameters for surfaces, which often requires non-analytical approach. 243 Golovanov, Geometric Modeling, 160.

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computing the first derivative of the curve’s parametric equation at that point. The actual change of the parameter t along the tangent line gives us the estimate for the required change of the parameter t along the curve, which brings the point corresponding to t closer to its correct location. This process is repeated, iteratively improving the solution, until the desired tolerance is reached. 244 Many other operations in geometric modelling can be solved by applying the same principle. modelling schemata in computer-aided design Computer-aided design (CAD) is a spatial-centric paradigm. It models objects by describing the parts of the space they occupy. 245 CAD utilises a number of modelling schemata to represent space and objects. Different schemata may utilise different notions of space to encode objects, but what they all have in common is the Euclidean notion of space in which the user creates and manipulates the object and in which that object is rendered. The encoding of the space in a modelling schema imposes the encoding of objects. If the implementation of space is discrete, as with voxel modelling, the objects will be discrete as well. However, if the space is encoded as a continuous field, objects can be encoded either as continuous, for instance with solid objects and objects made out NURBS surfaces, or discrete, like with polygonal modelling. In this work, computer-aided modelling schemata have been classified according to the physical fidelity of the objects they encode. In this respect we can discriminate between: modelling with curves: encoding only the outline of the object; modelling with faces: encoding only the skin of the object; solid modelling: encoding the entire volume of the object. Modelling with curves 2D: Computer Drawing Computer drawing makes use of mathematical two dimensional curves. It makes use of their polynomial coefficients by treating them as the control points of the curve 246, which allows the user to intuitively create curves representing her design, while “satisfying prescribed aesthetic or functional requirements.” 247 Such a modelling process is very much in line with the cybernetic idea of

244 Barbeau, Polynomials, 166. See: “Newton’s method according to Newton.” 245 Golovanov, Geometric Modeling, 11. 246 …the curve passes through the endpoints of the control polygon, and the curve is cotangent to the control polygon at these endpoints. These observations hold for any coefficients. Rockwood and Chambers, Interactive Curves and Surfaces, 34. 247 Farouki, “The Bernstein Polynomial Basis,” 386.

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feedback chains that Norbert Wiener introduced in the 1940s. As the points of the control polyline are dynamically changed, the polynomial coefficients are instantly recalculated and “the curve changes in an easy to understand way.” 248 The movement of the hand naturalises polynomial coefficients into “shape handles” 249 and establishes the direct connection between the hand and the eye in a manner similar to drawing with a pencil. Sutherland’s Sketchpad was one of the first examples of modelling with curves that set the foundation for future developments. The subsequent applications of digital-drafting tools adopted the infrastructure of geometric modelling without much question: They simply “replaced the traditional paper and pencil with electronic implements, but did not fundamentally change the task of drawing or modelling.” 250 First-generation CAD Systems were developed in the 1970s as a collaboration between the large aerospace and automotive manufacturers and universities. These collaborations resulted in products like CADAM, which later evolved in the famous parametric modeller known as CATIA. The end of the decade saw a plethora of different 2D drafting and 3D volumetric modellers developed for the large engineering companies. With the advent of personal computers—most notably the IBM PC in 1981—CAD software entered architectural offices. Companies including Summagraphics, MicroStation and Autodesk began developing specialised software for personal computers to support architectural drafting. Autodesk released AutoCAD Release 1 in 1982, which until now remains one of the most popular drafting tools. 3D: Wireframe modelling Following these initial developments, modelling with curves was extended from twodimensional to the three-dimensional representations of space. What became known as wireframe modelling represents only the vertices and the edges of the modelled object. 251 Since wireframe models appeared visually highly abstracted, the user was required to intuitively infer the volume of the object from its edges. 252 Although wireframe modelling is obsolete today, the wireframe representation is not. Contemporary modelling software still employs the wireframe model to construct a vector image of a modelled object. 253

248 Rockwood and Chambers, Interactive Curves and Surfaces, 34. 249 Farouki, “The Bernstein Polynomial Basis,” 386. 250 Kalay, Architecture’s New Media, 75. 251 Hoffmann, Geometric and Solid Modeling, 2. 252 Kalay, Architecture’s New Media, 142. 253 Golovanov, Geometric Modeling, 276.

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Fig. 17

Wireframe representation of Corbusier’s Ronchamp chapel. (Mitchell, 2016)

Polygonal modelling Polygonal modelling is a three-dimensional modelling schema that approximates the surface of an object by a collection of flat faces, each a polygon. 254 The most common choice for a polygon is a triangle, as its three points always lie on the same plane. 255 Surfaces created by polygons are often called polygon meshes or, in the case of triangles, triangle meshes. 256 Triangle meshes are the simplest way to approximate a surface. They are defined locally and have both topological and geometrical definitions. The geometrical part of the definition consists of vertices that encapsulate points in three-dimensions. The topological part consists of faces that encapsulate vertices and are defined by minimum three vertices. 257 By means of this simple structure, a polygonal mesh can have an arbitrary topology, as its faces may or may not need be connected at all. 258 Because of their local 254 Salomon, The Computer Graphics Manual, 487. 255 Salomon, 487. 256 “Surface based on triangulation … connected polyhedral surface formed by a set of triangular plates joined to one another along their common sides.” Golovanov, Geometric Modeling, 158–60. 257 Rutten, RhinoScript 101, 88. 258 Rutten, 88. For example, you can delete any number of polygons from the mesh and still have a valid object

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definition, meshes can store more information inside the mesh format, such as normals, texture-coordinates and colours. 259 All of these features make them arguably the most flexible and most common of all three-dimensional representations. 260 Curves and surfaces explored in the previous chapter were defined algebraically. As their parameters take value from the field of real numbers, they can be thought of as ideally smooth. Displaying them requires approximation 261, which is accomplished by sampling the parameter space in a certain resolution. In this regard, the very features that account for the flexibility and simplicity of polygonal modelling can become its major weaknesses. Polygonal modelling involves only one level of representation, where creation of an object is in fact the creation of its geometrical rendering. To overcome this limitation, polygonal modelling employs methods that join additional levels to its structure. Such methods are classified under subdivision surface modelling 262, as they transform its object—a coarse polygonal mesh, into a smooth surface. In 1978, computer scientists Edwin Catmull and Jim Clark devised the most prominent subdivision algorithm that was named after them. 263 Their approach to modelling involves creating and editing a low-polygonal model, while taking into account its conjugate subdivided high-polygonal shape. Such modelling requires both skill and experience and has evolved into two practical modelling techniques used in computer-generated imagery: hard surface modelling and organic modelling.

259 Rutten, 88. 260 Rossignac, “Solid and Physical Modeling,” 7. 261 “Graphics workstations usually accept triangular or quadrilateral facets, which are shaded and displayed by hardware. The resolution of the facets can be controlled to optimize the speed or quality of display.” Rockwood and Chambers, Interactive Curves and Surfaces, 148. 262 Rossignac, “Solid and Physical Modeling,” 9. 263 Salomon, The Computer Graphics Manual, 39.

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a

Fig. 18

b

Subdivided polygonal mesh. (Widen Media Blog, 2011)

Polygonal meshes play a role in both computer-aided design and computer graphics. In the former, they are used as auxiliary representations that approximate curved solids in most solid modellers for rendering and exporting the models to various applications. 264 When used for approximation of parametric surfaces 265, a triangular mesh can be defined as a “mapping of the two-dimensional triangulation in the parametric domain onto the three-dimensional space.” 266 In computer graphics, meshes are used as primary object representations “in most virtual reality, animation, entertainment, architecture, and other applications.” 267 The most prominent use of polygonal meshes in architecture is in architectural visualisation and animation. Solid modelling Solid modelling is a mathematically precise modelling schema that distinguishes between the interior, the boundary, and the exterior of the modelled object. This distinction is considered important to manufacturing or construction applications as it enables the computation of different properties of the model, such as the volume, area, centre of gravity, as well as physical simulations. A solid model is a complete and unambiguous representation of an object, whose topological properties allow classifying each part of the space as an object’s exterior, interior or 264 Rossignac, “Solid and Physical Modeling,” 7. 265 Golovanov, Geometric Modeling, 159. 266 Golovanov, 160. 267 Rossignac, “Solid and Physical Modeling,” 7.

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its boundary. For example, “integral properties of a solid may be approximated by classifying randomly sampled points and by adding the properties of points that fall in the interior.” 268 The main difference between solid modelling and the other modelling schemata lies in the topological restrictions that the model must satisfy. However, this restriction can be satisfied in a variety of seemingly different approaches. 269 In this respect, the distinction between terms solid modelling and surface modelling can be confusing, as these two terms do not belong to the same level of abstraction. Albeit having a different restrictions imposed on the geometry of the bounding faces, a surface modelling schema such as boundary representation should be classified under solid modelling as it still satisfies the same topological restrictions required from a solid object. 270 The more precise way of classifying solid modelling would be to distinguish between discrete modelling schemata, constructive modelling schemata, and surface modelling schemata. Preliminary mathematics Solid modellers, independent of their implementation, must satisfy a certain number of topological restrictions. Solid modelling employs the correspondence between Boolean set theoretic operators (complement, union, intersection, and difference) and their logical counterparts (not, or, and, and and-not) to “define sets in terms of logical expressions of predicates (membership classification) of the points they contain.” 271 A + B

union

set of points that belong to A or to B

A – B

difference

set of points that belong to A but not to B

AB

intersection

set of points that belong to A and B

!S

negation

set of all points that are not in S

By means of this correspondence, operations on solids inherit the properties of their logical counterparts, and a geometrical interpretation. As such, all the laws of formal logic are preserved and extended to solid geometry, such as: !S = S ! ( A + B ) = !A + !B

268 Rossignac, 2. 269 Rossignac, 5. 270 Golovanov, Geometric Modeling, 225. 271 Rossignac, “Solid and Physical Modeling,” 3.

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The example equations are the set theoretic equivalents of the two important logical statements: The first one is known as the double negation, and the second one as De Morgan’s law expressing the negation of a conjunction. Moreover, many geometrical operations that are considered useful in classical model making can be represented logically. Classifying the points within a solid requires a concept of adjacency. 272 A point p is adjacent to a set S “if all balls of centre p and strictly positive radius contain at least one point of S.” 273 The set S decomposes space into three parts. The first part is the boundary of the point set S, named bS, which is the closed, lower dimensional set of points that are adjacent to S and to its complement !S. 274 For example, the boundary of a circle is a closed set in topology, thus defined with the algebraic equality x2+y2=r2. The second part is the interior of the point set S, named iS which is an open set of points in S that are not in the boundary bS. For example, the interior of a circle is open set defined with the algebraic inequality x2+y2#r2. The third part is the exterior of the point set S, named eS, which is an open set comprising the remaining points. In topology, the boundary bS is usually further decomposed into: skin sS—points in S adjacent to both iS and eS; wound wS—points in !S adjacent to iS and eS; hair hS—points in S not adjacent to iS; and cut cS—points in !S not adjacent to eS. 275 Exterior

Fig. 19

Skin

The decomposition of a two-dimensional point-set into its interior iS, skin sS, hair hS, exterior eS, wound wS and cut cS.

Wound

Hair

Interior

Cut

This representation is considered unnecessarily complex for practical applications, which is why it is replaced with the representation of regularised solids. A solid rS is said to be regularised, when its wound becomes part of its boundary, its hair becomes part of its exterior, and its cut becomes part of its interior. 276

272 adjacent: lying near, close, just before, very close. 273 Rossignac, “Solid and Physical Modeling,” 3. 274 In other words, the sphere with the centre at p that contains both interior and exterior points. 275 Rossignac, “Solid and Physical Modeling,” 3. 276 Rossignac, 4.

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A regularised solid is considered bounded if it fits inside a ball of finite radius. 277 This proves beneficial in that the dangling elements are no longer represented and when the boundary is bounded, a solid “S may be represented without ambiguity by its boundary bS.” 278 Now it is possible to define the term solid more precisely as a subclass of all three-dimensional sets containing all possible Boolean combinations of volumes that are regularised and bounded. In respect to a solid S, a point p is described as being: IN the solid, when p is a member of iS (interior); OUT of the solid, when p is a member of eS; and ON the solid, when p is a member of bS. Determining the actual membership starts with classifying a point in respect to the boundary. If a point p lies on the boundary bS, then it is by definition ON S. If not, then its membership must be established with respect to its interior iS, as illustrated in Figure 20. 4: OUT

Fig. 20

The parity of the number of boundary crossings by a ray from a point establishes the point’s membership.

5: IN 4

5 4 3

3

2 1 2

1 1 2

3

3: IN

A point p will be IN (the interior iS) a solid, if any arbitrary path that starts from p stretching to infinity will cross the boundary of the solid bS an odd number of times. Similarly, the point p will be OUT (the exterior eS) of the solid, if any arbitrary path that starts from p stretching to infinity will cross the boundary of the solid bS an even number of times. For the sake of simplicity, if a path touches the boundary without crossing it then it will be excluded.

277 By bounding a solid, one avoids the “need to compute and represent intersection points or curves that are infinitely far from the origin.” Rossignac, 4. 278 Rossignac, 4.

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Once the point-set membership of solids is determined, it is easy to perform different operations on them by means of logical operators. For example, we can consider a membership of a point p with respect to two solids A and B. If the point is not ON at least on one of the solids, we can logically deduce the membership classification of p with respect to AB, A+B and A-B without any further information. For example, “if p is ON A and IN B, it is ON AB, IN A+B and OUT of A-B.” 279 Once the topological requirements of a solid object are satisfied, solid modelling can be further classified on the account of the restrictions that their bounding faces impose on its geometry. In the constructive solid schemata, such as constructive solid geometry (CSG), the boundary of an object is always regularised, thus every solid is considered a manifold. This can be an advantage, but it is not a necessary condition to satisfy the topological restrictions of a solid. In surface modelling schemata such as boundary representation, the boundary of a solid is a non-regularized set, thus the modelled object can also be a non-manifold. Thus, “the modelling systems that manipulate such non-regular features must construct, store, and exploit explicit information about the full-dimensional (open) cells and their bounding lower dimensional shells.” 280 The next chapter will give an account of the different solid modelling schemata according to the restrictions imposed on the geometry of the bounding faces. Discrete modelling schemata The simplest schema that can be used to model solid objects is based on the idea of spatial decomposition. 281 In its simplest implementation, three-dimensional space is subdivided into discrete elements of a specified size in the form of a three-dimensional lattice. 282 An object is represented as a binary mask that indicates which cells of this lattice are IN the solid. 283 This is known as spatial occupancy enumeration model, or voxel model. 284

279 Rossignac, 5. 280 Rossignac, 4. 281 Hoffmann, Geometric and Solid Modeling, 61. 282 Hoffmann, 61–62. 283 Rossignac, “Solid and Physical Modeling,” 5. 284 Kalay, Architecture’s New Media, 143–144.

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Fig. 21

A parallel algorithm for octree generation from a polyhedral shape representation. (Kitamura and Kishino, 1996)

The simplicity of the data structure that is usually used to represent voxels renders the problem of determining the connectivity between elements trivial, but at the same time, introduces a lot of redundancy, because each voxel is computationally given the equal importance. There are several technical solutions for the problem of redundancy. A prominent one is to aggregate voxels with identical mask values into a hierarchical structure known as an octree. 285 In spite of many disadvantages, the voxel model can be quite useful in cases when the focus of modelling is to precisely characterise each element of space with measurements. Such is the case with medical imaging, which is the primary use of the voxel model. 286 A recent example of using voxel models in architecture can be found in the work of Dillenburger and Hansmeyer at the ETH Zurich, where the voxel model was used as a data structure for large scale 3D printing. 287 Constructive modelling schemata If computer drawing models emulate the traditional technique of drawing, then constructive schemata of solid modelling resemble traditional additive/subtractive model making. 288 The analogy is

285 Rossignac, “Solid and Physical Modeling,” 5. 286 Kalay, Architecture’s New Media, 143. 287 Digital Grotesque, “Printing,” accessed March 1, 2017. https://digital-grotesque.com. 288 “Digital technologies for design and fabrication may in such cases still be seen as instrumental mediators, but functionally they are more akin to material utensils, like hammers and chisels, than to traditional notational vectors such as blueprints or construction drawings.” Carpo, The Alphabet and the Algorithm, 35.

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achieved by means of operations combining primitive objects or the results of previous constructions into a solid object. 289 The operations themselves can be easily edited and parametrised. 290 The most prominent constructive schemata are constructive solid geometry (CSG), binary space partitioning (BSP) and procedural solid models, with the latter generalising the first two. 291 Constructive solid geometry is the most prominent among constructive solid modelling schemata, which involve methods of representation encapsulated by tree structures; design methodologies, defined by point-set operations; and standard sets of primitive objects, such as cylinders, pyramids, cubes, etc. 292 Solids are represented as “set-theoretic Boolean expressions of primitive solid objects,” which implicitly define both surfaces and interiors.” 293 The solid must be regularised, which can be done after each Boolean operation, or performed on the result produced by combining several solids. 294 As Hoffman notes: “CSG representations are concise, always valid, (i.e., always define a solid or the empty set) and easily parameterized and edited.” 295 From the mathematical perspective, all primitives used in CSG are algebraic halfspaces; that is, “point sets defined as {(x, y, z}|f(x, y, z)#0} where f is an irreducible polynomial.” 296 They are created by sweeping a contour—a closed curve, defined analytically or by joining different curves together—along another space curve. Both the contour and the space curve are parametrically defined, meaning that a solid cylinder with a radius r and height l can be formed by sweeping a disk of radius r along a line segment of length l, where r and l are polynomial coefficients. 297 Primitives includes cylinders, spheres, cones, and tori. 298 “With each primitive object there is associated a local coordinate frame,” and “these local coordinate frames must be related to one another, by placing them with respect to a common world coordinate frame.” 299 The basic solid operations that CSG must implement include “classifying points, curves, and surfaces with respect to a solid; detecting redundancies in the representation; and 289 Rossignac, “Solid and Physical Modeling,” 13. 290 Rossignac, 13. 291 Rossignac, 6. 292 Hoffmann, Geometric and Solid Modeling, 21. 293 Hoffmann, 13. 294 Rossignac, “Solid and Physical Modeling,” 4. 295 Rossignac, 13. 296 Hoffmann, Geometric and Solid Modeling, 15. 297 Hoffmann, 14. 298 Hoffmann, 14. 299 Hoffmann, 21.

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approximating CSG objects systematically.” 300 As a design methodology, CSG relies on transformations and Boolean operations. Primitives and their groups can be transformed by means of rigid body motions: transformations combining rotation, translation and scaling. 301 “The transformed instances may be combined through regularized Boolean operations such as unions, intersections and differences to form intermediate solids or the final solid.” 302 The process of regularisation, follows every operation and applies “the topological interior operation followed by the topological closure,” thus transforming the result into a solid. 303 Hoffman gives a conceptual outline how to compute the intersection AkB of two solids: 1) We compute AkB in the set-theoretic sense. The result is a collection of volumes, and additional faces, edges, and vertices. These additional faces, edges, and vertices are lower-dimensional structures that we will eliminate. 2) We now take the interior of AkB. The interior consists of all those points p!AkB, such that an open ball of radius e, centred at p, consists only of points of AkB, for a sufficiently small radius e. 3) We form the closure of this interior, by adding all boundary points adjacent to some interior neighbourhood. A point q that is not an interior point of AkB is adjacent to the interior if we can find a curve segment (q, r) of sufficiently small length e, between q and another point r of AkB, such that all points of this segment are interior points of AkB, except q. Note that the lower-dimensional structures do not enclose volume and are therefore not adjacent to the interior of AkB. 304 The resulting solid is the regularised intersection. 305 Figure 22 shows a solid S, constructed with five simple primitives, where A, B, and E are rectangular cuboids, and C and D are cylinders.

300 Hoffmann, 14. 301 Rossignac, “Solid and Physical Modeling,” 13. 302 Rossignac, 13. 303 Rossignac, 13. 304 Hoffmann, Geometric and Solid Modeling, 22. 305 Hoffmann, 22.

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Fig. 22

C

A

B

A solid model created by applying Boolean point-set operators on five primitives.

D

E

The procedure defining the construction of object is defined in set-theoretical notation and represented as: 1. C , D 2. B – [C , D ] 3. A – E 4. [A – E ] , [B – [C , D ]]

The application of Boolean logical point-set operators used to create a solid can be conveniently drawn as a tree. “This tree is considered to be the representation of the object and is customarily called a CSG tree.” 306 Fig. 23

,

CSG tree representing primitives and Boolean point-set operators.

A

, E

B

C

D

306 Hoffmann, 23.

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SurfacE MoDElliNg ScHEMaTa Surface modelling refers to parametric solid modelling schemata, in which the spatial shells 307—constructions made of mathematical curves and surfaces—define the boundary of a solid object. Its main assumption is that the most variable part of the object is its surface, with the interior occupying the same volume: It is full inside and empty outside. Surface modelling differentiates between parametric surface modelling, (parametric) trimmed surface modelling and solid trimming models. 308 The term parametric comes from curves and surfaces and indicates that the boundary of the solid is a “patchwork of biparametric patches, each defined by a mapping M(s, t) from the unit square [0, 1]2 to a connected portion of a surface,” where M(s,t) is a low-degree (or rational) polynomial with two parameters s and t. 309 Trimmed surface models incorporate a trimming model that turns curve segments into edges to bound the parametrically defined host surface. 310 Finally, solid trimming is a hybrid approach between constructive and surface models, usually used to convert from one schema to another. 311 The major representation schema in surface modelling is boundary representation (B-rep). 312 It is a parametric trimmed surface modelling schema that uses shells to construct bounding surfaces separating the modelled object from the rest of the space. 313 The boundary representation data structure stores only the boundary of an object by uniting mathematically defined curves and surfaces and all their associated connections. fig. 24

Utah teapot made of Bézier curves. (O’Shea, 2017)

307 308 309 310 311 312 313

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Golovanov, Geometric Modeling, 225. Rossignac, “Solid and Physical Modeling,” 6. Rossignac, 6. Rossignac, 6. Rossignac, 6. Hoffmann, Geometric and Solid Modeling, 13. Golovanov, Geometric Modeling, 204.

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Boundary representations represent a solid unambiguously, “by describing its surface and topologically orienting it such that we can tell, at each surface point, on which side the solid interior lies.” 314 Its description contains two parts: topological—defining the connectivity and orientation—and geometric—mapping the topological part to space. 315 There are three geometric parts of the B-rep definition: point, line and surface. The corresponding topological entities are: vertex, edge and face. 316 All three topological entities of boundary representation are mutually interdependent. A vertex is the simplest topological entity that connects multiple edges together and maps to a point in three-dimensional space. Edges are finite curves defined by a parameter t and a direction 317 and create a skeletal frame held together by the vertices. When the parametrically defined curve is trimmed to serve as an edge of a surface, its segment defines a starting point and an end point. Coordinates at these points correspond to vertices of the edge. Defining a parametric surface requires two parameters, usually defined as u and v. Two-dimensional parameter spaces form a rectangular region bounded by the parameter values. fig. 25

v

parameter space

The curved surface of a cylinder represented as a biparametric trimmed region.

v

valid inputs

u

u

Faces are finite patches of a surface associated with the direction of a normal 318, with each face being a trimmed region defined by a loop of all of its boundary edges. As such, trimming a surface does not require the topological definition of the surface to change: A new face is created by simply replacing the old set of edge loops with a new one. 314 315 316 317 318

Hoffmann, Geometric and Solid Modeling, 36. Hoffmann, 36. Hoffmann, 36. Golovanov, Geometric Modeling, 205. Golovanov, 205.

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On the next structural level, boundary representation uses shells to represent internal voids within an object. 319 Shells are “separate closed manifold surfaces” 320 comprised of connected faces, which are defined by edges, vertices, and loops. 321 A set of all the shells defines a solid object 322 or a shape. In that respect, a boundary representation serves as a “topological glue” that joins different shells together into a shape. A shape is considered a valid solid only if its surface is completely closed and has no self-intersections. If at any point on the shape’s boundary we can find a sphere adjacent with both its interior and exterior, such a shape is a manifold (and could be physically constructed); otherwise, the shape is a nonmanifold. Boundary representation has some important topological characteristics. The most important is the Euler characteristic, a topological invariant that is computed by taking into account the number of faces, edges, vertices and loops that make up the object’s surface. 323 It is defined as: H = F – E + V + (F – L)

where H stands for the Euler characteristic, F for the number of faces, E for the number of edges, V for the number of vertices, and L for the number of loops. 324 Two shells have the same topology, if by deforming one we can obtain another. Another way of saying that is that two shells have the same topology if they have the same Euler characteristic. 325 If the Euler characteristic of a shell is known, the Euler-Poincare formula 326 “allows to determine the topological type of the shell.” 327 To change the topological characteristic of the shell, boundary representation uses Euler operators. “In particular, they can create closed surfaces, and modify these surfaces by adding or deleting faces, edges, and vertices.” 328 By adding or deleting handles, they allow to “modify the genus of the surface.” 329

319 Hoffmann, Geometric and Solid Modeling, 41. 320 Hoffmann, 41. 321 Golovanov, Geometric Modeling, 204. 322 “Solid is a connected set of points located on the inner side of one outer and several inner shells which are located inside the outer shell together with the points of these shells.” Golovanov, 211. 323 Golovanov, 205. 324 Golovanov, 206. 325 Golovanov, 210. 326 Hoffmann, Geometric and Solid Modeling, 39. 327 Golovanov, Geometric Modeling, 210. 328 Hoffmann, Geometric and Solid Modeling, 42. 329 Hoffmann, 42.

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Constructive solid geometry and boundary representation, have different strengths and weaknesses. Hoffmann notes that “a CSG object is always valid in the sense that its surface is closed and orientable and encloses a volume, provided the primitives are valid in this sense.” An advantage of a B-rep object is that it is easily rendered on a graphic display system. 330 In order to take advantage of the strong points of both representations, some modellers implement multiple representations, i.e. both CSG and B-rep. Such modellers are called dual-representation modellers. 331 Anatomy of a geometric modeller Before the turn of the 21st century, technical implementations of different modelling schemata were seldom distributed in a form of code. More often, the geometric model of a particular schema was consolidated into a commercial software package. The packages—described and advocated as design tools 332— were furnished with a graphical user interface, included technical documentation and best user practices. The architecture of the majority of geometric models usually consists of four parts. 333 The first is a geometric description, which can be expressed as a boundary representation, a constructive-solid model, or as any other geometric modelling schema. The second part lists the geometric constraints imposed on the model. For the purpose of coordinating the elements of a model, these constraints extend the principle of parametrisation to the relation between the geometric elements. The third part provides a build history, which is the “list of manipulations reflecting the algorithmic sequence of the model construction.” 334 The fourth part contains the model attributes, which define and store properties of geometric elements. 335 For many technical disciplines, the ability of these tools to create a precise geometrical model of a desired object was invaluable. Such a geometrical model permitted the setting up of “numerical experiment[s] to determine the stress-strain state, frequencies and modes of natural oscillations, stability of structural elements, thermal, optical, magnetic and other properties of modelled object[s].” 336 The computer-aided design modelling

330 Hoffmann, 13. 331 Hoffmann, 13. 332 Kalay, Architecture‘s New Media, 75. 333 Golovanov, Geometric Modeling, 274. 334 Golovanov, 274. 335 Golovanov, 274–75. For example, area, volume, lengths, angles, etc. 336 Golovanov, 275.

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paradigm was praised as widely applicable. 337 Before discussing the ways in which these developments have influenced architectural modelling, we need to first address the same developments from another perspective—that of computer graphics. computer graphics: towards a perfect simulation Experts of different technical disciplines tend to agree about the aim of computer graphics. For example, Jon Peddie opens his book on computer graphics with the words: It was difficult to write this book because it is impossible to trace a single line of development to generating beautiful realistic 3D images with a computer. 338 Three pages later, he makes a resolute statement: The whole idea of 3D on a computer is to create a realistic simulation of the real world. 339 A similar assessment comes from Christoph Hoffmann, an expert in the field of “solid-modelling systems”: 340 Indeed, the primary focus of computer graphics is to render realistic images of objects, and this can be done from data structures that do not represent complete solids. 341 It is important to note that the development of modelling in computeraided design had not occurred independently from the development of computer graphics. Both fields have been influencing and supporting each other for years. The research in CAD had been consistently developing means for achieving the goal set by computer graphics to objectively and realistically simulate the world. 342 Roughly at the same time as Sutherland’s Sketchpad, researchers began exploring cathode ray tube (CRT) technology, which led to the development of the first television sets. 343 The second breakthrough in screen technology came in 1973 with the invention of raster graphics. 344 Discretisation of the display introduced a new technical means of storing images in a buffer

337 “Geoscience used CAGD methods to represent seismic horizons; computer graphics designers modelled their objects with surfaces, as did molecule designers for pharmaceuticals. Architects discovered CAGD, word processing and drafting programs based their interface protocols on free-form curves (PostScript), and even moviemakers discovered the power of animating with such surfaces, beginning with TRON, continuing through Jurassic Park, and beyond.” Rockwood and Chambers, Interactive Curves and Surfaces, 7. 338 Peddie, The History of Visual Magic in Computers, 1. 339 Peddie, 4. 340 Hopcroft, Foreword to Geometric and Solid Modeling, iii. 341 Hoffmann, Geometric and Solid Modeling, 3–4. 342 “Other elements of math were developed; all have contributed to creating realistic 3D worlds.” Peddie, The History of Visual Magic in Computers, 40. 343 Peddie, 39. 344 This is attributed to Richard Shoup at “Xerox PARC” company.

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called a bitmap. 345 Prior to being displayed on a screen, each image was to be subdivided into a raster grid 346 comprising cells (pixels) able to store one or more values. A monochrome screen stores only the brightness of each pixel, whereas a colour screen stores the brightness values of each of the three basic colour components: red, green and blue (RGB). 347 This allowed displaying rich imagery on the computer screens and influenced the development of graphical user interfaces. 348 The ability to display (coloured) areas of a computer model also simplified the rendering of lines and made computer drawings more legible. By rendering the image in “layers” the invisible lines were hidden, according to their distance to the observer. This development drastically changed the appearance of computer models and allowed them to be shaded or even textured. 349 After the initial development of flat shading, Gouraud and Phong improved existing shading techniques, which allowed smooth representation even for low polygon meshes. 350 In 1974, Edwin Catmull pioneered digital texture mapping, which could project and correctly orient a texture on top of a geometric face. 351 Fig. 26

Bernard Tschumi: Architectural rendering using shaded polygons. (1994)

345 Salomon, The Computer Graphics Manual, 33. 346 Wikipedia, s.v. “Display Resolution,” last modified August 29, 2017, 03:02, https:// en.wikipedia.org/wiki/Display_resolution. This grid is defined by the resolution of the screen. For example, grid of today’s HD screens is subdivided into two million cells or pixels (1920 × 1080 pixels) and 4K screen into 8.3 million pixels (3840 × 2160). 347 Salomon, The Computer Graphics Manual, 34. 348 “GUI, graphical user interfaces, appeared in 1984 with the release of the first Macintosh computer.”, “The Microsoft Windows 3.0 operating system was first shipped in 1990 and, of course, gave a tremendous boost to the concept of GUI.” Salomon, 12. 349 Kalay, Architecture’s New Media, 162. 350 Peddie, The History of Visual Magic in Computers, 39. 351 Kalay, Architecture’s New Media, 167.

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The need to simulate effects of light and transparency of geometric models led to a number of illumination models. In the 1980s, Turner Whitted of Bell Labs developed a highly sophisticated rendering method, called ray tracing 352. In comparison to other production stages, this method was significantly more demanding in computationally and therefore not practical at the time. This changed in 1982 with the founding of Silicon Graphics Incorporated (SGI) and the introduction of high performance graphics computers, which made ray tracing and other methods more feasible (and incidentally were used to create some of the first fully computer-generated short films at by Graphics Group, the predecessor of Pixar Animation Studios). 353 These early efforts have greatly influenced architectural visualisation and are considered common practice today. Fig. 27

Loris G. Macci: An early example of raytracing in architectural visualisation. (1994)

The invention of the frame buffer had another important implication. If the buffer could be calculated fast enough, computers would be able to display objects moving in real time, in a manner similar to the technique of stop-motion. 354 Buffering had made possible real-time movement in two dimensions, which was later employed in the development of computer games. However, animation in three-dimensional space was much more complicated. With the rendering of a single image (single frame of the animation) already computationally costly—each second of animation required a minimum of 24 images for the smooth effect of transition—the animation seemed to be practically impossible. Large companies like Pixar and Industrial Light and Magic, specialising in

352 Salomon, The Computer Graphics Manual, 852. 353 Salomon, 12. 354 “In a computer, the local storage is the frame buffer, because it holds one (or more) frame(s) of the image. The term frame comes from the movies where each frame of the move is stored on film and run rapidly in front of the projector.” Peddie, The History of Visual Magic in Computers, 298.

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special effects for movies, finally made it possible. Even today, animation is something that is rarely attempted on personal computers, but rather sent to render farms linking thousands of powerful computers. Determining the required spatial position of the animated object according to given parameters 355, or computing those parameters in order to obtain the required position 356 of an object, required more functionality within the simulation systems of computer graphics. A mathematical solution to these problems was found in kinematics: a branch of classical mechanics, often referred to as “geometry of motion.” Depending on the direction of the calculation, we either speak of forward or inverse kinematics. 357 The implementation of kinematics that did not account for object masses or forces that caused the motion, seemed to be adequate at first. However, as the complexity requirements of simulations increased, animation software vendors realised that their products would be more competitive if they accommodated a larger scope of real-world simulations. This introduced simulations of dynamical systems, including effect of gravity, object collision and other physical forces. 358 Finally, the development of particle systems 359 allowed for simulating the highly chaotic systems or phenomena including “hair, fur, grass, dust, fire, sparks, smoke, fog, and water spray, as well as glowing trails and snow storms.” 360 This overview provides enough information to let us investigate the extent to which computer graphics and its modelling schemata have influenced architectural modelling. In this regard, the crucial notion to consider is the notion of space. It was demonstrated that the internal representation of the modelling schemata used in computer-aided design relies not only on Cartesian 361 but also topological notions of spatiality. 362 However, the way this internal representation is employed in the modelling process is surprisingly limited. An even greater surprise comes when one considers that this limitation reflects foundational issues of the 19th century. To follow the aim of computer graphics (of providing a realistic simulation of the world), modelling in computer-aided

355 “For example, if the object to be animated is an arm with the shoulder remaining at a fixed location, the location of the tip of the thumb would be calculated from the angles of the shoulder, elbow, wrist, thumb and knuckle joints.” Wikipedia, s.v. “Forward kinematics,” last modified May 18, 2017, 17:09, https://en.wikipedia.org/wiki/Forward_kinematics. 356 “For example, … artist can move the hand of a 3D human model to a desired position and orientation and have an algorithm select the proper angles of the wrist, elbow, and shoulder joints.” Wikipedia, s.v. “Inverse kinematics,” last modified August 29, 2017, 10:19, https://en.wikipedia.org/wiki/Inverse_kinematics. 357 Wikipedia, s.v. “Forward kinematics,” last modified May 18, 2017, 17:09, https:// en.wikipedia.org/wiki/Forward_kinematics. 358 Salomon, The Computer Graphics Manual, 881–882. 359 Salomon, 881–882. 360 Salomon, 881. 361 Golovanov, Geometric Modeling, 313. 362 Golovanov, 12. Geometric modeling heavily relies on differential geometry and numerical methods.

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design seeks to be truthful to the world it models and does so by means of intuition. By emulating the Euclidean paradigm 363, it constructs the model of the pre-19th century world within a computer and does so by employing the very mathematics used to resolve the 19th century crisis! In such a model, a computational (architectural) object is embedded in the pre-given spatiotemporal setup. The Euclidean simulator—in reference to which the objects are created, manipulated and rendered—provides the only access for the user to the objects. Computer-aided modellers can be regarded as computational tools only in a narrow sense of being computer programs whose infrastructure relies on computation to create geometric objects. However, the modelling process they accommodate for, seldom relies on computation, and consists mostly of manual work. For this reason, the task of modelling is often labour intensive and increases with the size and complexity of the modelled object. Rossignac gave an example of the design of a new aircraft engine within the paradigm of solid modelling, which according to him requires 200 person years. 364 As long as the work of a designer is performed in a manual fashion, which is still a dominant mode of design 365, we cannot regard the tools he/she uses as computational, mathematical, or intelligent, regardless of their sophisticated underlying infrastructure. In terms of modelling, Sutherland’s Sketchpad (1963) can even be regarded as progressive in comparison to the prominent tools from the 1980s, because it allowed the user to draw directly on the screen and also provided some elementary notion of intelligence. For example, if the user wanted to draw a square, he or she could draw it in a rough manner and the software would recognise the intention and transform it to an exact square. 366 computational models in architecture In contrast to the CAD models, we can define computational models in architecture as those that rely on computational procedures as the essential part of the modelling/design process. Beginning this dissertation with an introduction of computation within a larger historical context proves valuable for accessing the general principles and motivations of computational

363 “The main task of computer graphics is to generate and display three-dimensional objects.” Salomon, The Computer Graphics Manual, 429. 364 Rossignac, “Solid and Physical Modeling,” 22. 365 “… although 2D and 3D graphics software proved to be a remarkable departure from pencil and tracing paper and has been adopted almost universally as the predominant, if not exclusive, means of production in architectural practice, it merely represents the commercialization of the simplest and most obvious application of information technology in architectural design: the automation of traditional processes like drawing, modeling, and communicating.” Mitchell, foreword to Architecture’s New Media, xi. 366 Salomon, The Computer Graphics Manual, 10.

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models in architecture and for attempting to categorise them. In respect to the three prominent traditions of interpreting computation, the following categorisation of computational models in architecture are proposed: • Intelligence-based design models, within the tradition of early research of artificial intelligence (AI) • Rule-based design models, within the tradition of transformational grammars and axiomatic systems • Control-based design models, within the tradition of cybernetics and system theory Artificial Intelligence inspired models AI-inspired models sought mainly to exploit the developing field of computer intelligence in order to provide a holistic view to architectural modelling. The early examples of computer ‘intelligence’ in the 1950s included computers playing chess and checkers. 367 These initial experiences raised the question of whether computable machines could be programmed to think on their own. Stories around that question became a great source of inspiration for science fiction. In the 1950 paper “Computing Machinery and Intelligence,” Alan Turing asserted that the question of thinking machines was ill formed. Instead of asking whether machines can think, he suggested we should ask, how well can they perform in the so called “imitation game.” 368 He described a setup in which a computer and a human were having a mediated discussion without being aware of who or what was writing on their opposite side. 369 A decade later, in 1962, Marshall McLuhan wrote Gutenberg Galaxy where he coined the term global village to describe the immediacy of the world that communication technology brought to the whole of mankind. It seemed as if the society was one gigantic electronic brain or a computer. 370 One of the early researchers in AI and one of the inventors of artificial neural networks, Marvin Minsky defined AI as “the science of making machines do things that would require intelligence if done by man,” 371 and in the conclusion of

367 Crevier, AI, 217. 368 Turing, “Computing Machinery and Intelligence,” 433. 369 “We now ask the question, ‘What will happen when a machine takes the part of A (human) in this game? ‘ Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman? These questions replace our original, ‘Can machines think?’” Turing, 435. 370 See the gloss 31: “The new electronic interdependence recreates the world in the image of a global village.” McLuhan, The Gutenberg Galaxy, 31–32. 371 Minsky, preface to Semantic Information Processing, v.

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his 1965 paper “Matter, Minds, Models” described his expectations of the future of intelligent machines: When intelligent machines are constructed, we should not be surprised to find them as confused and as stubborn as are men in their convictions about mind-matter, consciousness, free will, and the like. 372 These ideas greatly inspired computer scientist Nicholas Negroponte to address the question of architectural design amid the promises of early artificial intelligence. In his book, The Architecture Machine (1970), he foresaw the future of design as the collaboration between human designers and fully conscious intelligent machines. 373 He strongly believed that intelligent machines would soon become a reality and create conditions for an evolutionary design process where “a mutual training, resilience, and growth (between human and the machine) can be developed.” 374 He saw the process of communication between man and machine as the main requirement for establishing “computeraided” as opposed to “computerised” architectural design. Thus, in computer-aided design, the machine learns how to adapt to the designer and his taste and at the same time learned some objective truths. 375 Another relevant concept Negroponte introduced was the characterisation of design as the resolution of forces. This idea seems to have greatly inspired architect Greg Lynn to later define his own architectural principles in Animate Form (1999). 376 In comparison to Negroponte’s story, the technical implementation of his ideas appears quite simplistic and he was careful not to expose it. He showed models dealing with problems such as spatial allocation, with defined rules for each spatial typology. 377 Five years later, in Soft Architecture Machines (1975), Negroponte integrated the cybernetic idea of feedback into his work. He decided to completely eradicate architects from the design process, so that design becomes the dialog between the user and the intelligent house. 378 This he described as a respon-

372 Minsky, “Matter, Mind and Models,” 4. 373 “We are talking about a symbiosis that is a cohabitation of two intelligent species.” Negroponte, The Architecture Machine, 7. 374 Negroponte, 3. 375 Negroponte, 23. 376 “…animate design is defined by the co-presence of motion and force at the moment of formal conception.” Lynn, Animate Form, 11. 377 Negroponte, The Architecture Machine, 45. 378 “The general assumption is that in most cases the architect is an unnecessary and cumbersome (and even detrimental) middleman between individual, constantly changing needs and the continuous incorporation of these needs into the built environment. The architect’s primary functions, I propose, will be served well and served best by computers.” Negroponte, Soft Architecture Machines, 1.

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sive architecture, within an intelligent, detached, self-sufficient, sustainable and fully controlled/regulated environment. 379 He illustrated the idea with the project called SEEK from 1970. This project consisted of one Plexiglas box inhabited by gerbils, containing a number of smaller wooden boxes and computer controlled robotic hand which regulated the environment. Fig. 28

Negroponte’s SEEK project. (Negroponte, 1970)

As gerbils would constantly move the boxes, the machine was expected to assist them to succeed in their objectives: Its purpose was to show how a machine handled a mismatch between its model of the world and the real world—in this case five hundred two-inch metal plated cubes. The mismatch was created by a colony of gerbils whose activity constantly disturbed the strictly rectilinear arrangement called for by the machine’s model. 380 What is usually not mentioned with this project, is that most of the gerbils died within this responsive environment. 381

379 Negroponte, 128. 380 Negroponte, 47. 381 Batchen, “DA[R]TA,” 189.

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Another prominent AI-inspired method used in architectural modelling was case-based reasoning (CBR). It was based on an assumption that humans have similar needs, thus their design requirements will be similar too. Therefore, to solve a certain problem, one should be looking for how similar problems in terms of their requirements had been solved before. 382 The biggest prerequisite for the successful application of the case-based design method was the large database of “cases.” Today, in the age of big data and machine learning this would not be considered a problem, but at the time this was the method’s major bottleneck. Before the Internet, there was no adequate infrastructure to serve as the case exchange platform. Thus, the generalisation methods were implemented logically, while today they can be “learned” by computer algorithms. Another difficulty was to formalise the criteria of relevancy of the retrieved cases to the current situation. Kalay argues that “none of these methods, however, has shown much ability to create truly novel solutions,” as they depended on the rule base and adaptation of older solutions. 383 In 1969, Minsky and Papert published a book called Perceptrons: an introduction to computational geometry, which gave rather pessimistic predictions on the future of artificial intelligence methods 384 backed up by the mathematical proofs. The impact of the book drastically reduced the research in artificial intelligence until the 1980s when the interest was revived. 385 Rule-based design models

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Rule-based design models incorporate a set of rules whose application evolves the design of the object. Such models are structured into two distinct levels with dependences between them. Rule-based design methods were among the first to be researched in computational design. Unlike the models characterised by the control paradigm, rule-based models were not implemented into commercial software. Depending upon how the rules are structurally implemented by the models, it is possible to differentiate between at least three approaches to rulebased design: Recursive/grammatical design models Emergent/cellular design models Logical design models 382 Kalay, Architecture’s New Media, 202. 383 Kalay, 202. 384 … also known as “AI winter.” 385 Crevier, AI, 203.

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Recursive/grammatical The design method of the recursive/grammatical models directly resembles the state/rule mechanics of the Turing machine or one of its derivatives. These models draw from a large spectrum of production systems including Semi-Thue processes, Chomsky grammars, graph grammars, shape and attributed grammars. 386 Their dominant mode of operation is reflected in the idea of sequential transformation: A set of geometrical entities is evolved into a new set of entities. These models computationally follow the logicist and axiomatic tradition, especially formalisms of Hilbert’s program, but are pushed towards productivity instead of logical consistency or proof. Recursive/grammatical architectural models stratify a design process into two levels. The first one is the level of the grammar, consisting of the elementary propositions, rules and axioms, which are the starting point of design; the second level is geometry, rendering of the combinatory potential of the rules. The main promise of the method is the efficiency it provides in producing complex forms from a relatively small number of initial conditions. 387 Lindenmayer’s initial research into production systems for the modelling of plants indicated a grammatical path to generating geometric shapes. In 1971, George Stiny, an American design and computation theorist, translated this idea into a system that could transform and manipulate 2D and 3D shapes known as shape grammar. 388 Similar to Chomsky—who sought to encode the complexity of language by means of simple rules— the title of Stiny’s paper, “Shape Grammars and the Generative Specification of Painting and Sculpture,” suggests that he had ambitions to extend rule-based systems outside the domain of architecture. In the paper’s appendix, Stiny submitted his experimental art-works, created by applying his method. Unfortunately, he followed the legacy of Chomsky, completely ignoring the mathematical basis of his procedure, concentrating purely on the mechanics of transformation of geometrical shapes.

386 Müller et al., “Procedural Modeling of Buildings,” 615. 387 Kalay, Architecture’s New Media, 237. 388 Stiny and Gips, “Shape Grammars and the Generative Specification of Painting and Sculpture,” Abstract.

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Fig. 29

An extract from Stiny’s seminal paper on shape grammars. (Stiny and Gips, 1971)

In later work, Mitchell and Stiny appropriated Chomsky’s definition of a language as a set of all possible valid sentences. They outlined a parallel between the idea of a language with the idea of a style. They asserted that if one could describe the geometric rules of an architectural style, then every sentence of that language would be a valid stylistic representation. 389 One of the first examples of shape grammars in architecture was Stiny and Mitchell’s “The Palladian grammar” from 1978. It was a parametric shape grammar used to generate ground floor plans of Palladio’s Villa Malcontenta. 390 Rule 58

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389 “Since the rules of a shape grammar encode descriptions of the shapes that belong to a particular “language, “ they can be used both to describe forms belonging to the corpus of shapes from which they were derived and to generate new forms that belong to the same ‘family’.” Kalay, Architecture’s New Media, 274. 390 “A parametric shape grammar that generates the ground plans of Palladio’s villas is developed as a definition of the Palladian style. The grammar is applied to generate the plan for the Villa Malcontenta.” Stiny and Mitchell, “The Palladian Grammar,” 5.

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Another prominent example was the work of Koning and Eisenberg in 1981, who used parametric shape grammars to generate a variety of instances of houses in Frank Lloyd Wright’s Prairie Style. 391 Fig. 31 11

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One of the historical holy grails of CAAD, automated floor plan generation, is based on the problem of space allocation (also known as automated spatial synthesis or quadratic assignment formulation 392). This approach examines a large number of possible solutions by means of computational combinatorics, aiming to minimise distances between adjacent rooms. Limited by the increase of combinatorial complexity, this approach was proven to be able to deal only with the simplest of spatial arrangements. 393 One of the successful recent implementations of shape grammars in architectural modelling comes from the work of Müller, Wonka, Haegler, Ulmer and Van Gool at the ETH in Zurich, Switzerland. In 2006, they created a sequential shape grammar for the procedural modelling of architecture in computer graphics and named it CGA shape. 394 Unlike L-systems, their grammar system consisted of three sets of rules, applied sequentially, resembling Chomsky’s transformation system: 395 • Crude volumetric model/mass model • Structure of the facade • Details for windows, doors and ornaments

391 “The establishment of a fireplace is the key to the definition of the prairie-style house. Around this fireplace, functionally distinguished Froebelean-type blocks are recursively added and interpenetrated to form the basic compositions from which elaborated prairie-style houses are derived.” Koning and Eizenberg, “The Language of the Prairie,” 295. 392 Kalay, Architecture’s New Media, 241. 393 Kalay, 239. 394 Müller et al., “Procedural Modeling of Buildings,” 614. 395 “While parallel grammars like L-systems are suited to capture growth over time, a sequential application of rules allows for the characterisation of structure i.e. the spatial distribution of features and components. Therefore, CGA Shape is a sequential grammar (similar to Chomsky grammars).” Müller et al., 615.

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The CGA model was never intended for practicing architects, but rather for game designers and special effects engineers who needed to simulate the physical infrastructure of cities. In order to demonstrate the superiority of their method for modelling buildings, they compared it to L-systems: We also use a large set of shape rules not existing in L-systems. Furthermore, the rules governing a biological system do not directly relate to the modelling of buildings. We found that a direct application of L-Systems to architecture overemphasises the idea of growth, a concept that is often counterproductive for the procedural modelling of buildings. 396 In the years that followed, the CGA model would evolve into the most powerful procedural city generator ever made, called CityEngine. The promise was economical efficiency: By using CityEngine and its library of rules, it was possible to “grow” a fully detailed model of a million-inhabitant city in the matter of seconds. 397 Fig. 32

Screen capture from the CityEngine software. (JuhaW, 2012)

Shape grammars were also employed in another example from the ETH in Zurich. In 2010, Michael Hansmeyer, a researcher at the chair of CAAD whose research originated in L-systems, conducted a series of experiments with subdivision surfaces. These are a special case of shape grammars, focusing on the division of the model’s geometric faces. Subdivision rules specify how, and under which conditions one geometrical face can be replaced by a new set of faces. Subdivision processes draw inspiration from the idea of self-similarity, which can be traced to the work of Benoît Mandelbrot in the 1980s, most notably to his book The Fractal Geometry of Nature. 398 Hansmeyer furnished 396 Müller et al., 622. 397 Esri, “Esri CityEngine,” accessed March 1, 2017. http://www.esri.com/software/ cityengine. 398 Mandelbrot, The Fractal Geometry of Nature.

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his subdivision model with elaborate control mechanisms—integrated algorithms that search for interesting solutions—calling it mesh grammars. In 2010, he exhibited a project called Subdivided Columns—A New Order consisting of a number of extraordinarily complex subdivided columns, fabricated out of a large number of 1mm thick slices of cardboard. 399 In 2013, Benjamin Dillenburger joined him on a project of designing and manufacturing the first completely digitally conceived room. The result of this collaboration, which involved a so far unseen level of geometric detail, was referred to as Digital Grotesque. 400 Fig. 33

A detail from Digital Grotesque. (Hansmeyer and Dillenburger, 2017)

Emergent/cellular The design method of emergent/ cellular design models emphasises the interaction of a number of simple, rule-following entities whose interactions within an environment exhibit patterns possessing a global structure. Emergent/cellular architectural models stratify the design process into two levels. The entities of the first level are called agents; they are programmed to follow a set of interaction rules 401 within an environment. 402 The second level is characterised by the emergent phenomena, which are the results of the interaction, usually rendered into geometry. 403 This approach

399 Michael Hansmayer, “Projects: Columns: Info,” accessed March 1, 2017. http://www. michael-hansmeyer.com/projects/columns_info.html. 400 Digital Grotesque, “Concept,” accessed March 1, 2017. https://digital-grotesque.com. 401 “Agents can act together in order to achieve more complex goals than any one agent can achieve on its own. A group of agents acting together was termed agency by Minsky.” Kalay, Architecture’s New Media, 432. 402 “The embedded idea is that you must have a rich environment for any emergent intelligence to happen, and the embodied idea is that you must have a good set of perceptive mechanisms so as to be able to interact with the environment.” Coates, programming. architecture, 83. 403 “Once again we are looking at a series of emergent outcomes which are the result of an extremely simple algorithm. The complexity comes from the many different ways that the expanding cubes intersect. This effect is achieved by simply focusing on the process rather than trying to calculate the intersections in some top-down method.” Coates, 49.

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follows the tradition of von Neumann’s research into cellular automata and Turing’s work on morphogenesis. 404 Often, it is complemented with search and optimisation methods, most notably evolutionary programming. 405 In his 1952 paper named “Chemical basis of morphogenesis,” Turing slowly turned away from the linear conception of his computing machine, towards a non-linear theory of computation. His aim was to show how complex biological processes could be modelled computationally. 406 The focus was on the notion of morphogenesis and how to model it: It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. 407 Turing’s famous differential equations and their numerical interpretation led to the development of reaction-diffusion systems, often exploited in design today for the purpose of simulating natural processes. Unfortunately, many researchers inspired by computation often ignore the historicity of the paradigms they follow. Sometimes, this ignorance gets amplified with successive generations of followers. A very prominent example is Chomsky, who proudly rejected statistics and probability, and then coopted Hilbert’s work on formal systems while ignoring algebra on which it rests upon. Stiny and Mitchell have closely followed Chomsky and further restricted the grammatical approach to operations on geometric shapes. In the next generation, we even have a complete hostility towards mathematics. Coates, a dedicated follower of Chomsky 408 writes: Cellular Automaton model seems to be more natural and considerably simpler. However, the traditional mathematical approach still has greater academic prestige, and is used in preference to the simpler explanation – yet

404 Neumann, Theory of Self-Reproducing Automata; Turing, “The Chemical Basis of Morphogenesis.”; Frazer, An Evolutionary Architecture. 405 Frazer, 9. 406 “The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts.” Turing, “The Chemical Basis of Morphogenesis,” 38. 407 Turing, 38. 408 “I have to admit that the general position of this book is structuralist in nature, partly due to a heavy dependence on the thoughts of Bill Hillier, and of course on the use of artificial language and Chomsky’s generative scheme as a model for form production.” Coates, programming.architecture, 160.

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another example of the dead hand of Victorian mathematics at work! 409 One of the most celebrated rule-based models—cellular automata (CA)—follows the research tradition established by John von Neumann. Frazer describes it as a special class of polyautomata 410, which “lend interesting insights into how complex behaviour can emerge from simple rules.” 411 He also provided a simple explanation of this model: A cellular automaton consists of a regular array of cells in one, two, three or more dimensions. Each cell can have at least two states: in the simplest automata, these might be 0 or 1, true or false, black or white, alive or dead, but in more complex automata more states are possible. Each cell is said to have neighbours which are cells with some specified spatial relationship. In the simplest two-dimensional grid of squares, a cell might be defined as having four edgeneighbours, or eight, including those on the diagonal… 412 Typical explications of CA, such as the one from Steven Wolfram in A New Kind of Science, often lack precision and concentrate on visual effects. 413 From my perspective, cellular automata are conceptually related to Turing’s model of computation and not to Church’s functional model. 414 A cellular automaton takes as an input a valid specification of a Turing machine and renders each of its operations (defined in time) into space. Thus, we can think of cellular automata as the tracing of computer programs. The architectural applications of cellular automata often implemented a (simple) model of the environment. On this account, automata are often described as “natural” or context sensitive. Their main promise is seen in the complexity of forms that can emerge from simple interactions. 415

409 Coates, 165. 410 “Polyautomata theory is a branch of computational theory concerned with a multitude of interconnected automata acting in parallel to form a larger automaton.” Frazer, An Evolutionary Architecture, 51. 411 Frazer, 51. 412 Frazer, 51. 413 See: “How Do Simple Programs Behave?” in Wolfram, A New Kind of Science, 23–39. “Over and over again we will see the same kind of thing: that even though the underlying rules for a system are simple, and even though the system is started from simple initial conditions, the behavior that the system shows can nevertheless be highly complex. And I will argue that it is this basic phenomenon that is ultimately responsible for most of the complexity that we see in nature.” Wolfram, 28. 414 Church’s lambda calculus has fact has the same power as the Turing machine, but is defined by means of functions. 415 Coates, programming.architecture, 41.

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Fig. 34

2-dimensonal cellular automata. (Wolfram Mathworld, 2017)

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The most prominent example of cellular automata is the Game of Life developed by John Conway in the 1960s. It was a two-dimensional, two state cellular automaton with simple rules, but gained its fame from the properties that it exhibited when animated. 416 The destiny of each cell is dependent on the states of the neighbouring cells and their distribution in space at a given time. 417 The Game of Life is a symptomatic example of naturalising computation, characteristic for the field. Its states are described using adjectives such as “alive”, “dead,” and “lonely.” The emergent “life-like” patterns and behaviours were given names such as “pulsar”, “toad”, “glider”, “spaceship,” etc. Fig. 35

John Conway’s Game of Life. (TeamGhostID, 2013)

416 Frazer, An Evolutionary Architecture, 55. 417 “Eight neighbours are considered: a cell will die of loneliness in the next generation if less than two of its neighbours are alive, and die of overcrowding if more than three are present. The optimum number of neighbours for survival is two or three. A dead cell can come to life if it has three live neighbours.” Frazer, 55.

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Virtual Users (VUsers), was an early attempt to adapt the Game of Life model to architecture. 418 Kalay discretised the area of an imported CAD floor plan and populated it with the simulated “users,” each occupying a single field. A virtual user was able to sense its environment—its sensors quantifying social, spatial and psychological awareness—and follow a matching set of rules. Both rules and sensory information defined the relationship of each virtual user to the others within the environment. The intent of the simulation was to “help determine the suitability of a given design to the needs of particular populations, including ones having different physical abilities or disabilities.” 419 Another prominent rule-based model utilised in design was the swarm model, inspired by Craig Reynolds’ flocking algorithm from 1986. 420 Reynolds’ algorithm formulated the rules of interaction between agents called boids 421 such that their behaviour resembled the flocking behaviour of birds. 422 Each boid follows only three simple rules: steering to avoid the crowd, aligning itself towards the average heading of the flock, and moving towards the centre of the mass of its flock-mates. What emerged in the simulation was the organic, life-like movement bringing a new potential of formal expression. In the United Kingdom, such systems were extensively used as design tools, the most prominent examples coming out of the Design Research Laboratory of the Architectural Association School of Architecture. 423 Fig. 36

A swarm-like Softkill project from AADRL. (Stuart-Smith et al., 2013)

418 Kalay, Architecture’s New Media, 382. 419 Kalay, 383. 420 Coates, programming.architecture, 89. 421 The pronunciation of the word “birds” in the New York accent. 422 Coates, 89. 423 AA School, “Intro.”

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In his 1995 book An Evolutionary Architecture, John Frazer described a computational model that aimed to establish a new vision of architecture on the evolutionary and emergent basis. Inspired by the work of von Neumann 424, Frazer’s model investigated “fundamental form-generating processes in architecture, paralleling a wider scientific search for a theory of morphogenesis in the natural world.” 425 In their drift from computation to nature, the proponents of evolutionary architecture distanced themselves from the rational computational approaches, such as Mitchell’s. 426 They positioned themselves as followers of Turing’s legacy of morphogenesis and augmented it with the evolutionary algorithms. A genetic algorithm is an intelligent search and optimisation technique used to simulate Darwin’s model of natural selection. 427 Genetic algorithms consider a population of objects on two distinct levels: that of genotype (a genetic code), and of the phenotype, acting as the candidate solutions of its species. A genetic algorithm implements a fitness function to optimise a particular property or a group of properties considered desirable among the population of objects. The optimisation is consolidated in time intervals called generations. To resemble the process of natural selection, the representation of genotypes is manipulated by using three operators: selection, crossover and mutation. 428 Frazer’s intention was to shift the notion of architectural language (in the sense of encoding) from blueprints to a “genetic language of architecture,” where an architectural concept would be described in a form of “genetic code.” 429 This idea altered the character of his theory from the theory of explanation to the theory of

424 “Von Neumann recognised that life depends upon reaching this critical level of complexity. Life indeed exists on the edge of chaos, and this is the point of departure for our new model of architecture.” Frazer, An Evolutionary Architecture, 20. 425 Frazer, 9. 426 “We do not mean the endless permutational exercises much beloved by computer theorists who can produce, for example, every known Palladian plan (plus a few new ones) after devising a so-called Palladian Syntax.” Frazer, 67. … “From our point of view, there are several problems with this approach. All of these generative systems are essentially combinatorial or configurational, a problem which seems to stem from Aristotle’s description of nature in terms of a kit of parts that can be combined to furnish as many varieties of animals as there are combinations of parts. Fortunately, nature is not actually constrained by the limitations implied by Aristotle.” Frazer, 14. 427 Kalay, Architecture’s New Media, 202–03. 428 “To satisfy the laws of natural evolution, there is no need to have a living organism. All the criteria for success present in a natural evolving system are mirrored in our artificial evolutionary model. Genetic information in the form of computer code is reproduced by the equivalents to cause it to crossover and mutate. Phenotypes in the form of virtual models are developed in simulated environments, performances are compared, and a selection of appropriate genetic code is made and then replicated in a cyclical manner.” Frazer, An Evolutionary Architecture, 98. 429 Frazer, 65.

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generation, which follows Chomsky’s work. 430 Frazer’s approach claimed to be following “nature as the generating force for architectural form,” 431 but in my opinion falls short in its lack of explanation of how the simple algorithms such as cellular automata could possibly be related to the immensely complex generative processes of nature. fig. 37

Genetic algorithm: Ichiro Nagasaka, 1992. (Frazer, 1995)

In the postscript of Frazer’s book, Tim Jachna aims to reinforce the ideas in evolutionary architecture by relating them to a prominent paradigm of the time—cybernetics: An evolutionary architecture will exhibit metabolism. It will enjoy a thermodynamically open relationship with the environment in both a metabolic and a socio-economic sense. It will maintain stability with the environment by negative feedback interactions and promote evolution in its employment of positive feedback. It will conserve information while using the processes of autopoiesis, autocatalysis and emergent behaviour to generate new forms and structures. 432 However, Frazer himself never mentions the work of Wiener. He only refers to Negroponte and his unreasonable expectations of artificial intelligence, which, he says, “didn’t deliver any answers.” 433 In order to avoid the same mistake, he conceived his approach in a pragmatic, rather than a visionary manner. 434 430 431 432 433 434

Frazer, 12. Frazer, 9. Jachna, postscript to An Evolutionary Architecture, 103. Frazer, An Evolutionary Architecture, 17. “Our evolutionary approach is exactly the sort of problem that could be given to an army of clerks the difficulty lies in handing over the rule book. Much of this text concerns the nature of these rules and the possibilities of developing them in such a way that they do not prescribe the result or the process by which they evolve.” Frazer, 18.

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Logic-based The design methodology of logic-based models adopts formal logical reasoning as the principal mode of design. Rather than operate directly with geometry, these models codify the application of design statements or decisions in propositional or predicate form. The first prominent example of logic-based models can be found in Christopher Alexander’s book Notes on the Synthesis of Form (1964). For Alexander, “the ultimate object of design is form,” 435 and its challenge stems from the relation between the form and design requirements. He noted the impossibility of creating a mapping that would allow to evaluate the form according to the needs it should reflect. 436 Instead of concentrating on the operation of “various systems and subsystems” that are generators of form, Alexander defined the idea of a pattern as an independent, abstracted unit that encapsulates a certain bundle of relationships. 437 A pattern “resolves a small system of interacting and conflicting forces, and is independent of all other forces, and of all other possible diagrams.” 438 Thus, the design process consisted of three parts: defining the requirements of design, identifying the patterns that would resolve them, and applying the patterns. Alexander used set theory to represent design problems: The great power and beauty of the set, as an analytical tool for design problems, is that its elements can be as various as they need be, and do not have to be restricted only to requirements which can be expressed in quantifiable form. Thus, in the design of a house, the set M may contain the need for individual solitude, the need for rapid construction, the need for family comfort, the need for easy maintenance, as well as such easily quantifiable requirements as the need for low capital cost and efficiency of operation. Indeed, M may contain any requirement at all. 439 He used diagrams to encode the requirements defined by the sets. 440 To evaluate a design ensemble according to the defined requirements, he employed binary equivalents of Boolean truth values: We shall treat a property of the ensemble (quantifiable or not), as an acceptable misfit variable, provided we can

435 Alexander, Notes on the Synthesis of Form, 15. 436 “But there is no general symbolic connection between the requirements and the form’s description which provide criteria; and so there is no way of testing the form symbolically.” Alexander, 75. 437 Alexander, Preface. 438 Alexander, Preface. 439 Alexander, 79. 440 The task of the designer was to “translate sets of requirements into diagrams which capture their physical implications.” Alexander, 92.

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associate with it an unambiguous way of dividing all possible forms into two classes: those for which we agree that they fit or meet the requirement, which we describe by saying that the variable takes the value 0, and those for which we do not agree, which therefore fail to meet the requirement, and for which the variable is assigned the value 1. 441 In 1977, Alexander wrote A Pattern Language, where he collected 253 patterns for the design of buildings and cities. If correctly followed, a network of patterns should guide and constrain the design. 442 His design vision can be understood as an abstract language model where the organisation of patterns reflects the notions of vocabulary, syntax and grammar. Even though Alexander’s idea of a pattern was an abstract idea, the implementation of patterns and their interaction was in fact not abstract at all. Patterns were defined as concrete practices, and their implementation consisted of successively applying sets of rules within a network. fig. 38

Alexander’s pattern language represented as a graph. (Dillenburger, 2016)

In 1990, William Mitchell, known for his work on Palladian shape grammars, further elaborated his design method in The Logic of Architecture. The main concern of the book was finding the language suited for computer-aided architectural design and showing how such language can be created by means of computation and

441 Alexander, Notes on the Synthesis of Form, 100. 442 Alexander, Ishikawa, and Silverstein, A Pattern Language, Preface, x–xi.

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formal logic. Mitchell picked up Alexander’s distinction between the form and design requirements, but turned it into a pragmatic generative perspective, reminiscent of Chomsky’s work. He transformed Alexander’s idea of the form produced by systems and subsystems in act, into a transformational grammar system. He described the particular constitutions of the generative system as design worlds, which provided “graphic tokens which can be manipulated according to certain grammatical rules.” 443 What is interesting here is to compare how Mitchell implemented the encoding of requirements in contrast to Alexander. Instead of using set theory, Mitchell used first-order predicate calculus. Therefore, to associate the colour, length and material of the Parthenon with variables X, Y, and Z, he used the following predicates: colour ( Parthenon , X ) length ( Parthenon , Y ) material ( Parthenon , Z )

In order to qualify more than one object, he used logical quantifiers, for example: 6 Column (fluted ( Column ))

The equivalent of this statement in language is: for each column, set its property to ‘fluted.’ He also gave an example of possible axioms of his system: A world of wooden blocks, for example, is governed by the axiom that two blocks cannot be in the same place at the same time. 444 Mitchell’s design method employed computation in the form of a transformational grammar, whose combinatory potential defined what he called a design world. Within each world, he encoded design objectives in the form of logical predicates. Thus, to satisfy the design objective requires satisfying the logical predicate. As we have seen in the first chapter, algebraic thinking does not easily coexist among the followers of the logicist tradition. The same can be observed with Mitchell: The trouble with algebras, as universes of design possibilities, is that they usually contain too much. They tend to contain vast numbers of possibilities that have no

443 Mitchell, The Logic of Architecture, x. 444 Mitchell, 54.

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architectural meaning whatsoever, plus possibilities that are meaningful but irrelevant or uninteresting. 445 The last notable example of logic-based models covered here are Expert systems. These models, popular in the 1980s, encoded (design) knowledge directly in form of (design) rules. 446 The idea of an expert system was to put such encapsulated expert knowledge at the disposal of non-experts. In comparison to procedural programs, expert systems utilised “generalised rules and inference, instead of hard-coded stepwise solution procedures.” 447 Thus, the results they provided could not be evaluated with respect to the design problem, but with respect to the validity of employed reasoning. They consisted of situation rules and facts. Situation rules were encoded in the IF-THEN couplets, where an IF encoded the generalised condition used to trigger a THEN part. The set of all rules defined the knowledge base, which was to be supplemented with new knowledge. To reach conclusions by means of rules and facts, expert systems used the mechanism known as the inference engine. 448 Control-based design models Control-based design models are computational models whose unified system of procedures governs the creation and manipulation of spatial form and whose design method allows and endorses controlling this system as a whole. Control-based design models are exclusively a top-down modelling paradigm affirming the concepts of control, automation, regulation, and optimisation, thus following the tradition of cybernetics. Instead of defining rules, this paradigm relies on satisfying logical and geometric conditions of a model, defined by algebraic equalities and inequalities. 449 According to how the control mechanisms are implemented, it is possible to differentiate between a minimum of three approaches to control-based design: • Adaptive models • Parametric models • Generative models Adaptive models The design method of adaptive models directly follows Wiener’s legacy by relying on feedback chains to automate and optimise model’s operation. Kalay describes the 445 Mitchell, 131. 446 Kalay, Architecture’s New Media, 267. 447 Kalay, 268. 448 Kalay, 269. 449 Golovanov, Geometric Modeling, 260.

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concept of automation as feedback-dependent: It is a system “where the output of a machine is linked to its input and compared against some intended performance measures.” 450 Examples of adaptive models are common in everyday architecture. Adaptive mechanisms are embedded in building services, including “control, regulation, and supervision of electrical, mechanical, and climatic control equipment.” 451 Such models monitor streams of values produced by the system (measured by sensors), and when required, their actuators respond by adjusting the system so that the future measurements stay within the desired limits. Adaptive models of greater complexity make a step further than responding by anticipating the events that could happen in the future. Kalay defined this kind of adaptability as model-based: Similarly, a house equipped with such a model-based adaptability system could optimise household operations, for example by turning on the water heater an hour before the washer is scheduled to be used, thus ensuring that there is plenty of hot water available by the time it is needed and avoiding wasting energy when hot water is not needed. 452 As an example of total environmental adaptability, Kalay points to Negroponte’s Soft Architecture Machines (1975). The University of Colorado’s Adaptive Control of Home Environments (ACHE) project can serve as the real-world illustration of Negroponte’s ideas. The project combined machine learning with feedbackbased automation to achieve total environmental adaptability, in order to anticipate inhabitants’ needs and to save energy. 453 This was achieved by training a reinforcement learning model which gradually learned the users’ behaviours thus freeing them from having to manually control their environment. Parametric models In parametric modelling, the associative relations between geometrical/topological elements 454 are specified by the equations, whose variables serve as the model’s control parameters. The parametric design method encourages the creation of new, aggregated parameters, thus simplifying the creation of efficient parametrised libraries of normalised components. A geometric model of an object usually accommodates a large number of diverse geometric elements. In the case of boundary

450 Kalay, Architecture’s New Media, 448. 451 Kalay, 448. 452 Kalay, 450. 453 Kalay, 451. 454 … which depending on the modeling schema could be points, vertices, edges, curves, faces, surfaces, shells or volumes.

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representation schema, such a model will store a number of vertices, edges, loops, faces and shells. 455 Each of these diverse elements has a characteristic data structure. On the most elementary level, parametric modelling exposes these data structures by representing them with variables called parameters. 456 On the next level, parametric modelling allows to create new, arbitrary parameters. These new parameters can replace, modify or link the existing ones (of arbitrary geometric objects) to algebraic equations or inequalities. By doing so, parameters set the objects in a specific relation to each other. 457 The result of parametric modelling is an ensemble of geometric elements governed by a network of parameters. The structure of the network takes the role of a grammar (or a prototype), while the parameters define the particular rendering (or a type 458) of that grammar. In a complex model, a change of value of a single parameter may ripple through the entire network and affect many objects at once. Parametric modelling puts this mechanism at the disposal of the user to create and control the geometric model. Now, let us look at the parameters from the perspective of geometric objects. The conditions that parameters, equations, and inequalities impose on a geometric object are known as geometric constraints or variational relations. 459 They are implemented as objects containing algebraic equations, inequalities, and a list of parameters they affect. 460 The crucial part to consider is that one and the same parameter can partake in many geometric constraints. In this respect, geometric constraints “model the relations between the elements of the model creation and control of such models.” 461 One of the roles of a geometric modeller is to satisfy geometric constraints, which is satisfied “if the algebraic equations described by the object are satisfied. ” 462 Another role of a geometric modeller is to keep the geometric model in equilibrium. 463 This is achieved by satisfying all of its geometric constraints. 464 Editing a model—either by trans-

455 Rossignac, “Solid and Physical Modeling,” 18. Because of that, boundary representation model is easier to parametrise than CSG model. 456 Golovanov, Geometric Modeling, 259. 457 For example: multiple objects can be located on the same axis or lay on the same plane; a size of a certain object can be dependent on the size of another object, etc. Golovanov, 12, 259. 458 “Specifying a set of values of the control parameters, we obtain a set of parts of a certain type.” Golovanov, 259. 459 Golovanov, 259. 460 Golovanov, 260. 461 Golovanov, 259. 462 Golovanov, 260. 463 Golovanov, 260. 464 “A constraint solver (130) adjusts the model to meet all the constraints simultaneously.” Rossignac, “Solid and Physical Modeling,” 18.

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forming or by the addition or removal of elements—disturbs the equilibrium of constraints. To return the system back to equilibrium, a system of equations is formed of the constraints. 465 The parameters of the model that represent it in the state of equilibrium are taken as the unknown variables of a new system of equations. 466 When this system is solved, the model is rebuilt with new parameter values, which restores equilibrium. 467 Parametric modellers were first developed in the 1970s and extended the possibilities of solid modellers. 468 In 1977, French aerospace company Dassault Aviation began developing a software, CATIA, which set a standard for all parametric modellers at the time. In the 1980s and 1990s the CAD/CAM market expanded with the development of MicroStation, Pro Engineer and SolidWorks. The arrival of the fourth version of CATIA had the greatest implications on large-scale architectural modelling. In a video interview for Dassault Systèmes, American architect Frank Gehry, greatly inspired by the power of parametric CAD modellers used in the automotive and aerospace industry, conveyed his vision of the future of architectural modelling: Dassault have built Boeing 777 with no paper. That’s what we want. 469 He used CATIA v4 to design and develop a project for the Guggenheim museum in Bilbao (1997). The future building was completely contained in a parametrically defined model and each of its elements referenced within a global chain of dependencies and constraints. Such geometric description was accurate, exhaustive and for the first time, able to be directly used for manufacturing. 470 In the following years, Dassault Systèmes and Gehry Technologies began to collaborate on a new version of the CATIA software, tailored for architecture. In 1999 they announced the Digital Project, which was a fully parametric architectural modeller containing parametrically defined building components. It allowed the architects to gradually evolve their

465 Golovanov, Geometric Modeling, 260. 466 Golovanov, 260. 467 Golovanov, 260. 468 “Some design constraints may be expressed by substituting the parameters of the primitives or of the transformations by symbolic parameter expressions. This approach was first demonstrated in the 1970s with the PADL-2 solid modeling system (66) and is now in widespread use.” Rossignac, “Solid and Physical Modeling,” 17. 469 “Frank Gehry on Dassault Systémes and Architecture,” YouTube video. 470 “The common denominator for all the participants was the CATIA 3D model, which was used to connect and inform all the parts of the project, as well as for virtual walkthroughs, interference checks, quantities take-offs, steel detailing, cutting the components of the steel ribs, dimensions, setting the concrete formwork and embeds, and defining survey points.” Kalay, Architecture’s New Media, 446.

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models from simple two-dimensional sketches to an elaborate, fully-detailed three-dimensional model. Fig. 39

Frank Gehry’s CAD drawing from 1993.

Design possibilities of parametric modelling also affected architectural theory. In 2008, Patrick Schumacher wrote the “Parametricist Manifesto,” in which he celebrated the efficiency of the new tools and their potential to radically economise architecture. 471 Parametric modellers often incorporate the technology known as building information modelling (BIM). For that reason, they are often (mis)represented as BIM software. The idea of BIM was conceived in the 1970s, in parallel to the first CAD software that established the geometrical approach to architectural modelling. Another kind of approach, of which BIM is a successor, was concerned with adequately describing buildings for the purposes of the construction industry 472, of which one of their pioneers was Charles Eastman, whose well-known paper “An Outline of the Building Description System” appeared in 1974. His main concerns were the cost of the design, construction and building operation; his main complaint was that the models relied too much on graphical descriptions. 473 In the prototype he presented, drawing elements were grouped together and the groups nested in a hierarchical system. These new 471 Bühlmann, Hovestadt, and Moosavi, Introduction to Coding as Literacy—Metalithikum IV, 13. 472 Kalay, Architecture’s New Media, 67. 473 Eastman et al., “An Outline of the Building Description System,” 5.

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aggregates were named, measured and described with metadata. Such “semantic ontology,” he believed, would be a much better description for the industry, than simple drawings. 474 The development of solid modellers, whose focus on physical fidelity improved measuring of geometric objects, additionally empowered Eastman’s conception. His work also coincided with the development of computer databases 475, which made the task of storing and querying building descriptions easy. The research subject of this field in the following years was the design of semantic ontologies. The important question was how to name, group and characterise architectural objects such that they are interoperable between different models. Autodesk’s Industry Foundation Classes initiative (IFC) from 1994 was one of the attempts to make a consensus on this difficult (if not impossible) task. The IFC model was “an industrydeveloped product data model for the design and full lifecycle of buildings.” 476 Today, it is a commonly used collaboration format in projects that integrate BIM. Therefore, BIM is not a modelling system, but a semantic ontology applicable to any modelling schema that allows its elements to be measured and described. Since the turn of the 21st century, all major architectural modelling software packages have introduced parametric modelling and have incorporated BIM. A notable example of a prominent simple to use BIM-focused software is Autodesk Revit, released in 2000. On the other side there is a less simple, but feature-prominent software called Generative Components, released in 2003. Generative models Generative modelling is a historybased modelling where geometric objects are evolved by freely applying stacks of transformation algorithms to a starting geometry. The major advantage of the method is flexibility. Unlike a pure grammatical approach, the transformations need not be predefined, but can be freely added, removed or combined. Transformation algorithm stacks are directly linked to a geometric object, and the only requirement of their protocols is the compatibility of their inputs and outputs. If implemented well, a change in a single step of the construction will result in consistent changes within the whole construction. Generative modellers are characteristic for polygonal modelling, of

474 “In a similar vein, because the building description is now in a machine readable form, any type of quantitative analysis could be directly coupled to the system. All data preparation for such analyses would be automatic, greatly reducing their cost.” Eastman, 6. 475 The “SQL” language, one of the computer languages for querying databases, was introduced in the same year as his paper (1974). 476 Eastman et al., BIM Handbook, 111.

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which the first successful software included Autodesk’s 3D Studio Max(1990), Cinema 4D (1991) and Maya (1998). 477 Generative modellers are the main modelling tools in computer graphics. To achieve realistic depiction, these programs always include physical simulations including kinematics, skeletons, dynamics, particle systems, etc. The dynamic aspect of these tools greatly inspired American architect Greg Lynn to define his own architectural theory and program. In his book Animate Form (1999), he invited architects to abandon the safety of Cartesian static organisations and to embrace dynamism by inviting the ideas of virtual force and motion into the environment. 478 Virtual movement of such forces, he explained, “allows form to occupy a multiplicity of possible positions continuously with the same form.” 479 To unlock the world of forces and dynamics he proposed to use calculus. 480 To embrace his perspective, he offered a (surprisingly) easy solution: All of the necessary instruments could be found in modern animation software and their topological landscapes: 481 Underlying all of the contemporary animation software is a mathematics of the infinitely small interval which simulates actual motion and time through keyframing. These transformations can be linearly morphed or they can involve nonlinear interactions through dynamics. 482 The geometry and the mathematics that Leibniz invented to describe this interactive, combinatorial, and multiplicitous gravity remain as the foundations for topology and calculus upon which contemporary animation technology is based. There can be little doubt that the advent of computer-aided visualization has allowed architects to explore calculus-based forms for the first time. 483 As an example of a new generation of artefacts made possible by the generative paradigms, Lynn introduced dynamical design

477 Salomon, The Computer Graphics Manual, 26–27. 478 “An object defined as a vector whose trajectory is relative to other objects, forces, fields and flows, defines form within an active space of force and motion. This shift from a passive space of static coordinates to an active space of interactions implies a move from autonomous purity to contextual specificity.” Lynn, Animate Form, 11. 479 Lynn, 10. 480 Lynn, Folding in Architecture, 16. 481 “Other topological landscapes include isomorphic polysurfaces (or blobs), skeletons (or inverse kinematics networks), warps, forces, and particles.” Lynn, Animate Form, 30. 482 Lynn, Folding in Architecture, 23. 483 Lynn, 16.

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elements called blobs 484 and exemplified their use in design throughout the rest of his book. Fig. 40

Greg Lynn’s blob architecture: Embryological House: Size “A” eggs. (Lynn, 1999)

state-of-the-art and its limits From the models examined so far, we can distinguish between two different contexts in which they have evolved. On one side we have the computational design models embedded in a commercial software like AutoCAD, Maya and CATIA. On the other side are the computational models applied to individual research or design projects. At the turn of the 21st century, the Internet disrupted the stability of these contexts. It started with the extension of existing tools with so-called scripting. The ability to programmatically take charge of the modelling within the commercial software had been first introduced with AutoLISP in 1980s. AutoLISP was a programmatic extension of AutoCAD software implemented in the functional programming language from 1958 called LISP. 485 In 1997, Autodesk introduced MaxScript for 3D Studio Max R2 and in 1998, Alias Systems Corporation introduced the MEL scripting language for Maya software. Today, the majority of software modelling platforms include scripting abilities. 486 The possibility of collaborative, location-independent work on the Internet has spawned open-source software development communities, brought together by their common interests. Notable products of such collaborative development include software like Blender (2003) and FreeCAD (2002). The open-source movement also encouraged the development of modelling libraries for programming languages, such as 484 “A blob is defined with a centre, a surface area, a mass relative to other objects, and a field of influence. The field of influence defines a relational zone within which the blob will fuse with, or be inflected by, other blobs.” Lynn, 30. 485 Kalay, Architecture’s New Media, 53. 486 Rhinoscript, Rhino.Python, SketchUp Ruby, etc.

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Open Cascade (1999) and Toxiclibs (2010). Finally, new programming languages and their dialects were created for the purposes of graphic and spatial design. Prominent examples include Processing (2001), OpenFrameworks (2005), Grasshopper (2007), Cinder (2010), and their web ports, including Processing.js (2008) and WebGL (2011). The Internet enabled not only the easy access to these tools, but also a means to collaboratively create and aggregate their technical documentation. Soon, this included a plethora of tutorials, available to anyone interested in learning. However, the biggest difficulty with these developments was the requirement from users to learn how to program computers, whether to actually write the code or to manipulate the already written code by means of graphical programming interfaces. Seemingly without warning, all of the computational models of the past, with their infrastructure in geometric modelling and computer graphics became readily available, encapsulated in code, documented, discussed, and taught in schools. One of the most prominent universities that has embraced computational modelling from the start was the Architectural Association School of Architecture (AA) in London, some of whose models we have already covered. 487 Since computing was thought to be an advanced topic, architectural education saw the rise of post-graduate programs aimed to introduce computer technology to architects. In 1997, the AA created a new graduate programme led by Bret Steele and Patrick Schumacher called Design Research Lab, with the goal of “systematic exploration of new design tools, systems and discourses, targeting design innovations in architecture and urbanism.” 488 A similar postgraduate programme, with the interest in exploring the relation between computers and architecture in the virtual had been already established in Switzerland in 1992, by Professor Gerhard Schmitt. The program took a new direction in 2000 when Ludger Hovestadt replaced Schmitt as the professor of the chair for ComputerAided Architectural Design (CAAD). Hovestadt introduced his research under the slogan “back to reality.” 489 Over the next ten years, the research at the chair of CAAD comprised more than a hundred experiments in exploring the application-orientated potential of information technology, an overview of which appears in the book Beyond the Grid (2009). This long introduction to the elusive relationship between architecture and digital computers helps in delineating the current scope of computational paradigms used in architectural modelling. Sadly, most of the models used in architectural praxis today are still not computational

487 Most notably: Frazer, An Evolutionary Architecture. 488 AA School, “History.” 489 Hovestadt, Beyond the Grid, 12.

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in nature. 490 The majority of those that are computational follow the traditions of early research in artificial intelligence, top-down, control paradigm of cybernetic, or bottom-up, rule-based paradigm of formal systems. These paradigms were further developed and hybridised within the advanced architectural education and research institutes, resulting in formally exciting and sometimes unimaginable artefacts. Internet technologies have democratised the access to the fruits of this collective effort, which resulted from the development of new tools, methods and a new generation of architects/programmers. Here, I would like to mention a potential risk that comes unseen with the cheerful optimism of programmable contemporary design tools. In many such tools, the ability to program often takes place within an already existing model, whose ontology is hidden from the user for the sake of simplicity. This can empower a designer to quickly create results, but it also stops her from comprehending the bigger picture. In this sense, knowing how to program, without understanding the general abstract nature of models can even be counterproductive! Finally, I must mention that in terms of representation, all of these models have simply appropriated the model of space and time introduced by computer graphics without reflecting on its potential impact. As computer graphics and geometric modelling heavily depend on mathematics (differential geometry, topology, calculus, etc.) this can give a false sense of mathematical depth to computational modelling used in architecture, depending above all on powerful computers. The next chapter will try to show that computational paradigms employed in the constitution of geometric modelling, CAD, and from them derived computational models, represent only a small part of the spectrum of what is possible and actually used in information technology today. In this respect, I do not think it is an exaggeration to say that computational models in architecture have not conceptually advanced much since their early beginnings. The chair of CAAD and its associates experienced this conceptual deadlock early on and have taken its overarching consequences to architecture and society seriously. In 2009, they left the generally favoured pragmatic route of “doing,” and shifted the perspective towards questioning what are computers actually good for and what computation is all about. These questions could simply not be answered within

490 “Most generative and evaluative software that have been developed over the past five decades failed to gain a foothold in architectural practice, hence to add value to professional design practices and its products. As a result, architectural design solutions are still crafted manually, much the same way they have been over the past 500 years, except that the drawings and models that represent them can now be edited more easily and communicated more expeditiously among the members of the design team.” Kalay, Architecture’s New Media, xi.

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the old established perspective. The philosophical turn started with a question: “What’s next?” This research is an attempt to sketch one of the possible answers to this question. III towards a new vision of architectonics Let us quickly jump to the present day and concentrate on a single fact: this document is composed by means of a technical device, whose central processing unit is a piece of silicone, 3 × 3 centimetres in size and whose clock runs at the frequency of 2,800,000,000 times per second. Today, there are more than 6,000,000,000 devices of the same kind, whose characteristics may be described within the same order of magnitude. Any of these devices bear the potential to communicate with any other such device, regardless of the time or the place. Only a few generations before us, the numbers that characterise our computers were just a theoretical possibility, having nothing to do with our human capabilities. Yet today, in the age of social and logistic networks, nanotechnologies and self-driving cars, they are taken for granted so much so that they do not even make for an interesting discussion. In the trilogy The Information Age: Economy, Society and Culture (1996–1998), Manuel Castells gave an exceedingly broad overview of the transition from the industrial to the informational society. However, in the thousands of written pages, there is no mention of the mathematics of the 20th century nor quantum physics, the very developments that according to physicist Kakalios made our informational society possible. 491 In order to orientate ourselves and our interests in the complexity of the present moment, I believe that these concerns should be taken into consideration. from certainty to probability A formal account for ignorance If we were not ignorant there would be no probability, there could only be certainty. But our ignorance cannot be absolute, for then there would be no longer any probability at all. Thus the problems of probability may be classed according to the greater or less depth of this ignorance. 492

491 Kakalios, Introduction to The Amazing Story of Quantum Mechanics, xii-xiv. Kakalios argues that the field of solid-state physics which provided us with the technology of transistors among many others, would not be possible without quantum mechanics. Without transistors, he says: “We would consequently live in a relatively computer-free world.” Kakalios, xiii. 492 The calculus of probabilities. Poincaré, Science and Hypothesis, 189.

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In the introduction of his 1985 book on quantum electrodynamics, Richard Feynman, as if he was announcing the most shocking discovery to the general audience, asserted that the most advanced physics of the time cannot exactly predict any event but only calculate its probability. 493 A reason for opening the book like this cannot be unintentional. Although the concepts of probability theory were part of every school’s curriculum, Feynman did not assume that the reader understood the full extent of it. If we compare the computational models in architecture with those commonly used in Information Technology, we can also safely assume that architects belong to that group too. It was only in the 16th century that science started taking account of that which we cannot know: the future. The theory of probability emerged from gambling halls and insurance brokerages to become a coherent branch of mathematics. The foundational event was the publication of Jacob Bernoulli’s treatise The Art of Conjecturing (1713). 494 Bernoulli was interested in accurately estimating probabilities of events, based on the number of their occurrences. One of the major achievements in his Ars Conjectandi (as it appeared in Latin) was the refinement of the idea of expectation. The manner in which Bernoulli introduced the idea is very interesting: It can be seen from what we have said that we are not using the word expectation in its ordinary sense, according to which we are commonly said to expect or to hope for what is best of all, though worse things can happen to us. Here account is taken of the extent to which our hope of getting the best is tempered and diminished by fear of getting something worse. So by its “value” we always mean something intermediate between the best we hope for and the worst we fear. 495 He conducted an experiment that involved 5000 pebbles stashed in an urn: 3000 white and 2000 black. The ratio of the pebbles was supposed to be unknown to the observer, which is why the pebbles were hidden from him/her. Bernoulli offered an interesting way to estimate their ratio. The experiment involved repeating a single procedure: drawing a pebble, marking if it was black or white, and returning it to the urn. He observed that as the number of trials increased, the ratio of white to black started to converge to the actual ratio of three to two. 496 This became known as the (weak) law of large numbers. As an implication of the law, he drew a philosophical conclusion that if the principle was to be

493 “… Does this mean that physics, a science of great exactitude, has been reduced to calculating only the probability of an event, and not predicting exactly what will happen? Yes.” Feynman, QED, 19. 494 Originally published as Ars Conjectandi. 495 Bernoulli, The Art of Conjecturing, Together with Letter to a Friend on Sets in Tennis Court, Part I: 134. 496 Bernoulli, Part IV: 328.

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extended ad infinitum, it would show that the whole world is governed by precise ratios and a constant law of change. 497 Soon, it was discovered that not only the probability of an independent event converges to the expected value, but that the probability of the variation away from the average follows a certain distribution. It was named the binomial distribution. Its graph resembled the shape of a bell, thus becoming known as the “bell curve.” 498 Whenever the variation of a large number of random trials was observed in nature, its outcomes seemed to be following the pattern of the binomial distribution. This introduced the belief that the average fate of these events might somehow be naturally predetermined, which was later formalised as the central limit theorem. In 1902, Russian mathematician and theologist Pavel Nekrasov, published a paper that scrutinised the law of large numbers in the context of an old theological debate about free-will and predestination. 499 As a strong opponent of free will, Nekrasov did not like the implications of the law of large numbers. In order to “save” free will, he made a claim that the law of large numbers only applies to independent events, with no causal links between them. 500 Another Russian mathematician Andrey Markov read his paper and concluded that Nekrasov’s mathematical work was “an abuse of mathematics.” 501 To prove him wrong, he extended Bernoulli’s results by proving that the law of large numbers applies perfectly well to systems of dependent variables if they meet certain criteria. 502 Markov’s mechanisation of entropic cuts Probability theory recognises independent events and those that are dependent on other events. An outcome of a flip of a coin does not depend on the previous throw, which is why we regard it as an independent event. The same is true with a roll of a dice. In a game of Monopoly, for example, the roll of a die will determine how many steps the token will advance on the board. But if we wish to predict where the token will land next, we need to know its current position on the board. 503 This is an example of dependent events. Markov first addressed dependent variables his 1906 paper. 504 Since the prediction in this case depends on the previous outcome, a model that handles dependent variables requires a means to store it. To achieve this, Markov created a model consisting of two states, whose dependency on each other was described by

497 “… for whatever God has done, God has, by that very deed, also determined at the same time.” Bernoulli, 329. 498 “Origin of Markov chains | Journey into information theory | Computer Science | Khan Academy,” YouTube video. 499 Seneta, “Markov and the Birth of Chain Dependence Theory,” 257. 500 Hayes, “First Links in the Markov Chain,” 94. 501 Seneta, “Markov and the Birth of Chain Dependence Theory,” 257. 502 Seneta, 94–95. 503 Hayes, “First Links in the Markov Chain,” 92. 504 Seneta, “Markov and the Birth of Chain Dependence Theory,” 259–61.

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four probability values. Two values defined the probabilities of transitioning between one state to another, and the other two probabilities that the state will not change. Such construction can be recreated in an experiment involving two sources of random variables, for example, two containers with black and white pebbles (like those Bernoulli used in his experiment). Since both containers contain the two types of pebbles in different distributions, states of the model are defined to reflect the possibility of drawing each kind. In our case, we will have only two states: s1—black previously drawn, and s2—white previously drawn. Depending on the current state (first state is chosen at random, either s1 or s2), we will draw a pebble from the respective container. Depending on whether we draw black or white, we will either remain in the current state or transition to another. This procedure can be repeated indefinitely, transitioning in four possible ways [s1–s1], [s1–s2] [s2–s1] [s2–s2]. Markov made an assumption that if both states were reachable 505, the frequency of each state would converge to a fixed average value, as well as their ratio. 506 Such a model became known as the Markov chain. Turing’s 1937 paper addressed Hilbert’s Entscheidungsproblem; Markov’s 1906 paper addressed Nekrasov’s claim that independence is the necessary condition for the law of large numbers. Both papers introduced an ingenious mathematical construction that could be implemented mechanically and that proved to be universally applicable. To illustrate the potential of Markov’s idea, we will reproduce, but in the opposite direction, his 1912 experiment from the paper “An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains.” This experiment assumes that the object we would like to model unfolds itself in time or sequentially, by means of which it can be partitioned. The second assumption is that the discretisation yields at least two parts that are different from each other. For the sake of simplicity, we can assume that our object is represented symbolically, by the following sequence of ones and zeroes, read from left to right: 0 0 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 11 0 11 0 1

The classical way of modelling this sequence would be to calculate the probability of ones and zeroes appearing within the sequence, the first step being to simply count the number of ones and zeros within the total number of symbols. 0 0 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 11 0 11 0 1

505 Probability greater than zero and less than one, meaning at least one white and one black pebble in each container. 506 Hayes, “First Links in the Markov Chain,” 95.

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[0 1]

0:17 1:14 31

Now that these values are known, it is easy to calculate the probability of each class of symbols appearing independently in the sequence, by dividing the number of parts within each class with the total number of symbols: 17 ≈ 0.548 ≈ 55% 31 14 P(1) = ≈ 0.452 ≈ 45% 31 P(0) =

As we have stated in the beginning, our sequence of numbers is not random. It is a physical trace of an unfolding of the object, described symbolically. 507 The attempted statistical treatment gave us very little information about the object. Modelling this sequence by Markov chains requires a different strategy. First, we need to introduce the notion of history, which is done by cutting the sequence in such a way that every element is joined with his successor. Then, we need to determine how many different successions are possible if this sequence occurs. In the case of two different symbols, there are only four such possibilities: 55%

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Then, we need to determine how these pairs of symbols are distributed within the four possible classes of successions. This is done by counting how many times each sequence of two actually appears in the entire sequence. 507 This unfolding can be in time, but not necessarily spatio-temporal.

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≈ 0.353 ≈ 35% ≈ 0.647 ≈ 65% ≈ 0.769 ≈ 77% ≈ 0.231 ≈ 23%

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Fig. 41

Markov chain represented diagrammatically and as a table of probabilities.

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What is truly remarkable about the resulting chain of probabilities is that it enables us to construct a machine that will mechanically produce a sequence of symbols that can imitate the original sequence and have the same statistical

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properties! We could construct a machine capable of doing this in a similar way as Bernoulli, only with two urns. The first urn would contain 353 black pebbles (representing zeroes) and 647 white pebbles (representing ones). The second urn would contain 769 black pebbles and 231 white pebbles. In his 1913 paper “An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains,” Markov showcased the potential of his model with an unusual example: the famous poem by Pushkin. 508 He discretised the poem into 20,000 characters, in order to estimate the extent to which Pushkin’s text violated the principle of independence of the characters. 509 Figure 42 compares the distribution of twenty thousand characters from “Eugene Onegin” 510 (coloured) with the random distribution of letters in a sequence of the same length (grey). As we can see, the distributions differ drastically. ...wasto oyou ngto havebeenbli ghte d...

He was too young to have been blighted by the cold world’s corrupt finesse; his soul still blossomed out, and lighted at a friend’s word, a girl’s caress. In heart’s affairs, a sweet beginner,

20,000 letters from Eugene Onegin v 7,788 c 12,212

he fed on hope’s deceptive dinner; the world’s éclat, its thunder-roll, still captivated his young soul. He sweetened up with fancy’s icing the uncertainties within his heart; for him, the objective on life’s chart was still mysterious and enticing— something to rack his brains about, suspecting wonders would come out.

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probability 0.389 v 0.611 c v c v 0.175 0.825 c 0.526 0.474 v

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The Markov chain is one of the first mathematical and computational models that took advantage of self-reference. By cutting the entropic universe of all messages into parts and putting them into proportion 511 it allowed to characterise any unfolding in probabilistic terms. According to this, it can be seen as the pre-specific representation of concrete objects. 512 Forty years later, Claude Shannon found interest in the model but eventually discarded it as it was too computationally expensive to be pragmatically useful. 513 Sixty years later, this was no longer a problem. In 1996, 508 Markov, “An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains,” 591. 509 “He counted 1,104 vowel-vowel pairs and was able to deduce that there were 3,827 double consonants; the remaining 15,069 pairs must consist of a vowel and a consonant in one order or the other. With these numbers in hand, Markov could estimate to what extent Pushkin’s text violates the principle of independence.” Hayes, “First Links in the Markov Chain,” 95. 510 Based on the translation, by Charles H. Johnson, published in 1977. Hayes, 95. 511 “The world of numeric specification, the infinity of the chronological order is cut into two, the parts are subscribed and arranged in, as we suggest, probability values.” Hovestadt, “Elements of Digital Architecture,” 100. 512 Moosavi, “Pre-Specific Modeling,” 41–45. 513 Moosavi, 86.

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fig. 42

Part of Markov’s experiment with an English translation of Eugene Onegin. (Hayes, 2013)

Larry Page and Sergey Brin created the PageRank algorithm, which introduced an objective measure of popularity of a web page. The algorithm that was to become the foundation of Google’s search engine and revolutionise the Internet, was in fact a gigantic Markov chain. 514 The sequence we have used to demonstrate the calculation of a Markov chain involved only two states. The PageRank is implemented such that each state represents a web page, and each individual transition a single hyperlink. 515 From that we can infer that the computation of PageRank in 2015 required computation of probabilities for approximately 50 billion states. 516 Markov’s idea from 1906, combined with enormous computing power, allows Google to answer more than one billion questions per day, from people around the globe, in 181 countries and in 146 languages, in less than one second per question. 517 Probability of a quantum event At approximately the same time, when the big battles of logic and formal systems were fought, something quite “illogical” started to happen in physics. Einstein introduced his theories of special (1905) and general relativity (1916) and showed that space and time are in fact inseparable from each other. Newton’s laws, that were confidently used to describe planetary motion, failed to describe the interactions of elementary particles, especially the motion of electrons going around the nucleus. In 1926, a new theory was developed to explain the behaviour of electrons. From the standpoint of classical physics, it was a rather counter-intuitive theory, known as quantum mechanics. 518 Pretty soon, this theory became the foundation for all chemistry. In 1929, quantum theory was able to integrate the theory of electricity and magnetism and thus referred to as quantum electrodynamics. 519 One of the many counter-intuitive ideas quantum theory proposed was that energy was not continuous, but came in discrete packages called quanta. Moreover, light could be described both as a wave, and a particle stream of quanta. 520 Unlike classical mechanics, quantum theory fully relies on probability theory to explain nature, which allows Feynman to remark that: “Nature permits us to calculate only probabilities. Yet science has not collapsed.” 521 However, what is astonishing is not the probability itself, but rather the philosophical implications of how it is calculated. Richard

514 Hayes, “First Links in the Markov Chain,” 92. 515 Hayes, “First Links in the Markov Chain,” 92. 516 Bosch, Bogers, and de Kunder, “Estimating Search Engine Index Size Variability: a 9-Year Longitudinal Study,” 848. The estimation showed that in 2015 Google’s index contained around 50 billion pages. 517 Google, “About Search.” The numbers are from 2013. 518 Feynman, QED, 5. 519 Feynman, 6. 520 Feynman, 14. 521 Feynman, 19.

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Feynman’s 1985 book QED: The Strange Theory of Light and Matter was an attempt to bring the peculiarities of the calculus of quantum physics to the reader. To calculate the probability of light reflecting from two surfaces of a glass sheet, one can try adding the probabilities of single surface reflections. Oddly, the result will not match the actual observations. Feynman showed that depending on the thickness of the glass, the overall reflection will in fact vary from 0–16% in an oscillating sine wave pattern. From the standpoint of classical physics and its probabilistic calculus, this result simply did not make sense. Light source

A 0 to 16 100

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By means of an analogous but simplified model, Feynman has demonstrated how the correct values are calculated in quantum electrodynamics. 522 First, he changed the interpretation of Figure 43. The paths drawn from the light source to the multiplier were not any longer taken as two light rays, but as two possible ways a photon could travel from the light source to point A. He represented these two events with two arrows, technically known as probability amplitudes. The length of each arrow is determined probabilistically as if the light bounced from a single surface. In that case, 4 out of 100 photons are reflected from the surface, which corresponds to a probability of 0.04 (4%). According to the rules, the “probability of an event is equal to the square of the length of the arrow,” 523 which gives us the value of 0.2 for the length of both arrows. fig. 44

Arrows representing different reflection probabilities ranging from 0 to 16%. (Feynman, 1990)

4% 0.2 1% 0.1 0%

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To determine the direction of the arrows, he introduced an imaginary stopwatch timing the photon: When a photon leaves the source, we start the stopwatch. As long as the photon moves, the stopwatch hand turns (about 36,000 times per inch for red light); when the photon ends up at the photomultiplier, we stop the watch. The hand ends up pointing in a certain direction. That is the direction we will draw the arrow. 524 Finally, when the photon bounces off the front surface of the glass, the direction of the arrow is flipped. The final arrow is obtained by adding the two arrows, which corresponds to vector addition. Finally, the squaring of the length of the final arrow gives the probability of the event. 525 fig. 45

Composing a number of arrows into the final arrow. (Feynman, 1990)

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The reflection of light from two surfaces can be seen as an interplay of two probability amplitudes, whose directions in respect to each other can either increase or decrease their combined value. This interplay which can lead to either reinforcement or cancellation of two or more amplitudes is known as interference. 526 In an example where light is reflected from the surface of a mirror, Feynman introduced a case of an infinite number of possible events. 527 He subdivided the surface of the mirror in a finite number of patches and considered every possible path that could start from the light source, bounce from the mirror, and end up in the photomultiplier, including the cases that physics considers impossible.

524 Feynman, 27. 525 “If there are multiple surfaces, there will be one arrow for each event that can happen, and the arrows will be combined in a ‘final arrow’.” Feynman, 25. “square of that arrow would represent the probability of the event.” Feynman, 26. 526 Feynman, 22, Figure 5. 527 Feynman, 40.

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Something remarkable shows up in the interplay of the arrows. The probability amplitudes of those impossible and unintuitive reflections simply cancel each other out! What remains corresponds to those patches where the incidence angle of the path matches the angle of reflection, which is what we already know from optics. At precisely this place where “the time is least,” 528 arrows are pointing in the same direction. The probability amplitude corresponding to the sum of all paths illustrates the quantum interpretation of the principle of least action from classical mechanics: The least action is found where the interference is minimised. However, the crux of this example is that all the impossible and non-intuitive ways in which an event can happen must be part of the calculation in order to obtain values corresponding to the actual real-world observations! What does not seem to exist in reality for classical physics, indeed contributes, in quantum physics, to account for that reality. When nothing seems to happen, probability amplitudes are dancing in circles. The Markov model showed the potential of probabilistic characterisation in contemporary computational models. However, the affirmation

528 “To summarise, where the time is least is also where the time for the nearby paths is nearly the same; that’s where the little arrows point in nearly the same direction and add up to substantial length”. Feynman, 45.

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of the three fundamental ideas of the quantum paradigm: That light is a photon; that matter has a wavelength nature associated with its motion; and that everything, light and matter, has an intrinsic angular momentum or spin that can only have discrete values, made computers what they are today, and introduced a host of new technologies. 529 Their products include transistors (microchips), lasers, electronic microscopes, magnetic resonance imaging, LED lights, etc. Superposition and modal logic In his 1996 book, Necessity or Contingency, French philosopher Jules Vuillemin, gave a historical and philosophical account of the famous problem of future contingents, known as the master argument. 530 The argument made by Diodorus Cronus around 280 BC deals with necessity and contingency, therefore with the possibility of freedom. The argument involves three statements: 1 Every true proposition about the past is necessary. 2 The impossible does not logically follow from the possible. 3 What neither is presently true nor will be so is possible. 531 Diodorus had noticed that the asserted truth of any pair of propositions would be in contradiction with the remaining proposition. He attempted to resolve the problem by denying the third axiom and giving a sui generis definition of contingency. 532 In his book Vuillemin thoroughly investigated the possibilities of resolving the argument throughout history. 533 My particular interest in this issue concerns an inherent contradiction from the perspective of probability theory and quantum physics. Vuillemin recreated the argument in probabilistic terms by an analogy with a “steam of projectiles shot from a cannon too inaccurate to control their angular dispersion and subjected to passing through a screen pierced by two holes in the manner of Young’s experiment and then reaching a detector that absorbs and counts them.” 534 He showed that when the law of large numbers is applied to the experiment, “there is some notion of contingency that is not incompatible with the Diodorean premises and that leaves the principles of logic intact.” 535 Hence, the Diodorean solution, it implies, would be pertinent to the classical notion of probability. By making a deliberate simplification, we could say that Vuillemin’s construction points to the limit of scientific paradigms on the basis of their ability to productively employ probabilistic characterisation. This characterisation, he shows, is inseparable from the logical reasoning employed. He demonstrated this by

529 Kakalios, The Amazing Story of Quantum Mechanics, 9. 530 Vuillemin, Introduction to Necessity or Contingency, xi. 531 Vuillemin, 3. 532 Vuillemin, 257. 533 Vuillemin, xii. 534 Vuillemin, 257. 535 Vuillemin, 257.

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reconstructing the experiment in the quantum paradigm. If two projectiles are regarded as photons, it is not their probabilities that are added, as we have seen with Feynman, but their probability amplitudes: For the probability amplitude, which is generally a complex quantity, does not figure among the elements of reality. To obtain a probability we must multiply two conjugated probability amplitudes. This means that, when we attribute that amplitude to a system, it is attributed neither as an actual property or magnitude nor as an actual disposition or propensity to having such property or magnitude, but as purely virtual disposition or propensity to having it. The second-order potentiality, as it were, thus put into play is no longer the measure of an ignorance that might have some chance of being only provisional. It is physical. It describes nature. 536 During their flight, photons are in a state of superposition, and this very state invalidates Diodorus’s arguments “in changing the sense of its premises.” 537 In relation to the first two, the third axiom becomes “strictly speaking undecidable.” 538 Within the quantum paradigm, the Diodorean solution loses its cogency, and physics can do without it. 539 This throws a particularly important light on Gödel’s paper and its implications on computation. It affirms my position that computers are not about limits of formal systems, but about our own capacity to think and make models, including those that, from the classical perspective, appear to be paradoxical. from structure to structure-ability Spatio-temporality in its own terms Geometric modelling and computer graphics provided an infrastructure for computational models in architecture, most notably their common conception of space. This implementation was realised by means of the Cartesian paradigm directed towards emulating the Euclidean notion of empty generic space. 540 Two pillars of modern physics, theory of relativity and quantum physics, could not be realised within such a conception of space. 541 German mathematician Bernhard Riemann, introduced a radically different conception of spatiality that paved the way for the physics of the 20th century. 542 536 Vuillemin, 264–65. 537 Vuillemin, 263. 538 Vuillemin, 264. 539 Vuillemin, 259. 540 “Since the Euclidean space can be thought of as physically empty, in physical terms it then is the vacuum, or mathematically, the substrate of the vacuum.” Jost, historical introduction to On the Hypotheses Which Lie at the Bases of Geometry, 22. “The Cartesian space therefore is actually ideally suited for the description of the vacuum, at least if it properly captures the topological and dimensional properties of the vacuum, as was implicitly assumed at that time.” Jost, 22. 541 Zafiris, Natural Communication. 542 Jost, historical introduction to On the Hypotheses Which Lie at the Bases of Geometry, 1.

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In his 1854 habilitation speech “On the Hypotheses Which Lie at the Bases of Geometry,” Riemann envisioned the notion of spatiality through the general notion of magnitude. He defined it as an n-ply (multiply extended) magnitude. 543 Euclid’s propositions of geometry, he noted, could not be derived from such a notion, because the characterisation of space by such propositions, could only be deduced from experience, and its validity can only rest on observation. 544 There can, on the contrary, exist multiple systems that would suffice in defining the metric properties of space, as we have seen with Hilbert. 545 What Riemann introduced was a distinction that separated the notions of what would later be known as the topological 546 and the metric structure of space. Riemann first accounted for the very possibility of magnitude relations, arguing for their reliance on the existence of an “antecedent general notion which admits of different specialisations.” 547 The possibility of a continuous path between these specialisations he called a continuous manifoldness and its individual specialisations points. If the path between them is not continuous, they would form a discrete manifoldness whose individual specialisations can be considered elements. 548 He derived his conception of quantification from the notion of quanta, which involve “definite portions of a manifoldness, distinguished by a mark or by a boundary.” 549 The discrete magnitudes are quantified by counting and the continuous magnitudes by means of measuring 550, for which he provided the following definition: Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case also we can only determine the more or less and not the how much. 551 For the first time, the notion of magnitude needed not be instituted on the basis of some external, absolute standard of measure, but could be defined from inside out, within the regions of manifoldness. 552 This appears to be the point where Riemann’s work laid down the groundwork for relativity theory.

543 Riemann, On the Hypotheses Which Lie at the Bases of Geometry, 32–34. 544 Riemann, 31. 545 “The idea is that if the metric properties of the space do not necessarily follow from its structure, then the space can carry several possible metrics, and the mathematician then can specify any such hypothetical relations and examine the resulting structures and distinguish them with regard to their characteristics.” Jost, historical introduction to On the Hypotheses Which Lie at the Bases of Geometry, 46. 546 Dealing with qualitative aspects of position. 547 Riemann, On the Hypotheses Which Lie at the Bases of Geometry, 32. 548 Riemann, 32. 549 Riemann, 32. 550 Riemann, 32. 551 Riemann, 32. 552 Riemann, 32.

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Riemann then described how triply extended magnitudes 553 could be derived by a composition of variabilities of simply and doubly extended magnitudes. He demonstrated how any path within the manifoldness could be indexed by a path within a manifoldness of lower dimensionality. He thereby observed the true character of manifoldness in the property “that the determination of position in it may be reduced to n determinations of magnitude.” 554 As the construction of a manifoldness determined only the qualitative aspect of position 555, in the second part of his habilitation Riemann aimed at showing how to assign measure relations to a manifoldness. 556 This required that a “quantity should be independent of position,” which was possible to achieve in many ways. 557 To assign metrical-relations to manifoldness, Riemann stratified his model on two different levels. One level in respect to the other was constructed on an infinitesimal scale, on which Riemann’s manifoldness is locally Euclidean. 558 He expressed the notion of curvature within the manifoldness as the local deviation from the Euclidean model, which is normalised in such a way that the Euclidean space has zero curvature. On that account, manifoldness locally carries a linear structure, known as a vector space. 559 The introduction of two different levels to describe manifoldness, required Riemann to establish a model that could translate between them. Riemann established one direction of communication by the construction of a tangent space, which corresponds to a real vector space that can be attached to a point. That very point needs to contain all the possible “directions” in which one can tangentially pass through it. Such a tangent space is constructed around each point of manifoldness and it translates it into a local Euclidean picture, in which angles and distances can be measured in a classical Pythagorean way. He accomplished the second direction of communication by means of the integration of infinitesimals, which allowed him to return from the infinitesimal to the topological scale. This process involves ‘gluing together’ an infinite number of infinitely small local Euclidean patches. The integration yields a special class of curves (least action paths in space) that can describe a manifoldness globally, known as geodesics. 560

553 We can think of it as a generalisation of three-dimensionality. 554 Riemann, 34. 555 “The manifold structure refers only to the neighbourhood structure and to the relative positions, i.e. to the qualitative aspects.” Jost, presentation of the text On the Hypotheses Which Lie at the Bases of Geometry, 44. 556 “Riemann thus recognizes that in order to measure lengths and angles, an additional structure is required which is of a quantitative nature.” Jost, 43. 557 It this hypothesis is correct, it comes with a consequence that “length of lines is independent of their position, and consequently every line is measurable by means of every other.” Riemann, On the Hypotheses Which Lie at the Bases of Geometry, 34. 558 Jost, presentation of the text On the Hypotheses Which Lie at the Bases of Geometry, 21. 559 Jost, 24. 560 Larger and a more detailed exposition of the topic can be found in Zafiris, Natural Communication.

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In his doctoral dissertation, Riemann introduced the notion of a surface into the context of complex analysis, a branch of mathematics investigating functions of complex numbers. The complex surface model became known as a Riemann surface. 561 Such surfaces sometimes include localities where the notion of a distance cannot be defined, even on an infinitesimal scale. For that reason, the model involves a gluing procedure which is not based on metrical compatibility. This introduces a novel idea of mapping between different levels of the model, known as a covering space. A covering space characterises a non-trivial deformation property by means of an unfolding. 562 The circulation of a path on the surface of a ring can serve as an example of a non-trivial deformation. If two paths are defined on a plane or a disc, there is no obstacle to prevent one path from being deformed into another. This is not the case when the paths are moving on the surface of a ring, whose hole is an inherent obstacle to any path on its surface. On the level of the ring, there is no way to account for the obstacle, as this latter is not part of the world in which the path lives. Riemann proposed to characterise such a path by indexing its movement using another path created in a space considered to cover the ring. In the case of a ring, such a space will be a spiral. Every time the path makes a full turn around the obstacle (the hole), the spiral generates a new dimension which can be indexed by an integer. In this way, every non-trivial property can be made to unfold a new dimensionality by means of a covering space. The spiral itself is not embedded in any pre-given space-time, as each of its turns requires a new modelling of time. 563 fig. 47

Illustration of a covering space of a ring.

561 Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. 562 Zafiris, Natural Communication. 563 Zafiris.

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An object risen from the foam Riemann’s novel characterisation of spatiality, which involved localisation and different modes of mapping between the global and the local, found a fertile ground in quantum physics. There, for example, one level of observing quantum phenomena would be on the scale of electron interactions. This level comes with an intrinsic limitation to the very act of observation itself. Yet on another scale, these interactions are given physical form and can be observed as interference patterns. Riemann’s ideas provided means to think of the relation between these different levels in terms of communication. This development also challenged the established conceptions of what is an object. The set-theoretic paradigm of the 19th century reinforced the intuitive assumption that objects are sharply defined entities, embedded in space and indexed by time. This view could not be sustained in the quantum paradigm. Objects in the quantum paradigm are not sharply defined; there is no space in which they can be located; and no casual relations could explain their behaviour. 564 The etymology of the name of the Greek goddess Aphrodite 565, reveals to us that she has risen from the foam. 566 We can think about the quantum objects in the same way as they emerge from indistinguishability. 567 Contemporary models of communication demonstrate that it is possible to operate within such an objectivity. Unfolding architectonics of the Internet Since the beginning of the 21st century, with the development of search engines and social media, the Internet has challenged us to reconsider what computers are good for. Unlike the articulations of computation that we are using to model architecture, the Internet cannot be adequately considered as a tool. The Internet, with the index as its principle operating entity abstracts from the structural paradigm of tools and machines, and propels the very concept of a structure to the infinite. This allows it to outperform the immense efficiency of machines and at the same time preserve the richness and diversification of the communication it medialises. 568 One very materialistic, necessity-driven (and in my opinion inadequate) way of looking at the structure of the Internet is to see it as a vast network of computers whose core routers are interconnected via data backbones, ultimately resting upon millions and millions of cables laid in space. From a more abstract standpoint, we can observe a very different Internet. It is a place where things are being born, emerging, growing, expanding,

564 Zafiris. 565 derived from aphrós (ἀφρός) “sea-foam.” 566 Cyrino, Aphrodite, 14. 567 Zafiris. 568 “The development of the Internet, and even more so the development of the World Wide Web, have changed the concept of computers from machines that calculate to machines that communicate.” Kalay, Architecture’s New Media, 36.

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transforming and dying all the time and without any particular reason. This form of “life” need not reflect any underlying physical infrastructure, although of course the infrastructure makes it possible in the first place. Let us consider this more abstract vision of the Internet from a Riemannian perspective of unfolding. This perspective shows that the structure of the Internet is not predefined, nor embedded in some kind of space-time, but rather grows from the inside-out, unfolding space(s) and time(s). We can refer to such an unfolding architectonics as structure-ability. Fig. 48

Reddit place, 2017. (https://www.reddit. com/r/place/)

from computation to communication The following table is a summary of what was presented in this overview. The proposed distinctions prove to be very helpful in defining the expectations of models that can be used to challenge computational models in architecture, and in setting up operative concepts required for that purpose. The strategy, in that respect, is to look for models that are conceptually closer to the ideas on the right side of the table, than those on the left.

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Modern world

Quantum /  information world

Space

Spatio-temporal embedding

Unfolding space and time

Structure

Top-down and bottom-up

Structure-ability

Reference

Global structure

Localisation principle

Relations

Casual relations and determinism

Probability of an event

Variational principle

Principle of least action

Path-integral formulation

Role of computers

Computing machinery

Infrastructure for communication

From this chapter, we adopt the mathematics of probability 569, the idea of structure-ability and the quantum-characterisation of an object as the elements that can help us to present a new vision of architectonics. My hypothesis is that they will bring greater potency, contingency and freedom of articulation to our models, in comparison to what we as architects currently have at our disposal. What seems to set apart the aspiring models of this chapter from the previous ones is whether their reference assumes global view, or differentiates between global and local perspectives. The particularly interesting models in this respect do not operate within any absolute objectivity, such as space and time, but rather concentrate on what can be exchanged between different localities, or structural levels. Making these levels “talk” to each other requires formalising their means of communication in a purely abstract, yet operative way. We can characterise such models as communication models. In the next chapter, two communication models will be presented in detail. The first one is the natural communication model, created by mathematician and quantum physicist Elias Zafiris. The second model comes from language, and it is described in Louis Hjelmslev’s book Prolegomena to the Theory of Language (1940). Seemingly very different, both models show remarkable similarities in the way they approach modelling.

569 Both Markov’s operational treatment of probability and philosophical implications of the idea of probability amplitudes.

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ARCHITECTONICS OF COMMUNICATION HOW DIFFERENT NATURES COMMUNICATE

II

I natural communication model The natural communication model, conceived by the mathematician and physicist Elias Zafiris, is a mathematical framework that provides an abstract skeleton for representing and solving a wide range of scientific and technical problems. 570 Most computer modelling, including in the field of computeraided architectural design, traditionally approaches modelling from a set-theoretical point of view. 571 Zafiris aims to challenge the modelling paradigm founded upon set theory with one based on mathematical category theory. The main obstacle of such an ambition stems from the fact that category theory, as an abstract part of pure mathematics, is a recent development, and not yet established in technical applications. 572 This chapter introduces the natural communication model and investigates the possibilities of applying it to architecture. Due to its complexity, the model will not be presented in its full scope, but only in its most important notions as well as those that are closely related to the problem at hand. This dissertation insists that the natural communication model provides a promising avenue to computationally address architectural questions, which cannot be adequately addressed within the current computational paradigms. anatomy of the model What is a model? The purpose of a model, for Zafiris, is to reduce complexity and create new information. The model needs to satisfy two elementary requirements, to be able to both encode and decode information, which thereby characterises any such model as a cryptographic model. In order to be encoded and decoded, that which is to be modelled must be stratified into at least two different levels (A and B in Figure 49). Zafiris notes that the stratification can accommodate the modelling of a wide range of phenomena from a more abstract perspective: The levels of the model can be seen as the levels of hierarchy, levels of analysis or some local pointers. 573

570 Zafiris, Natural Communication. 571 Moosavi, “Pre-Specific Modeling,” 41. 572 “Categories, functors, natural transformations, limits and colimits appeared almost out of nowhere in a paper by Eilenberg & MacLane (1945) entitled “General Theory of Natural Equivalences.” We say “almost,” because their earlier paper (1942) contains specific functors and natural transformations at work, limited to groups.” Marquis, “Category Theory.” 573 Zafiris, Natural Communication.

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Fig. 49

Cryptographic model connecting levels A and B by means of encoding and decoding.

decoding

encoding

Level A

Level B

In the context of different strata, encoding and decoding serve as communication bridges. In the simplest case, a model contains encoding and decoding bridges, which are necessary for establishing communication between the two levels. The quality of a model, according to Zafiris, directly depends on the quality of the bridges. This quality is manifested by the property of naturality, which defines the ability of a bridge to reflect properties of a level as a whole, rather than of any individual choice of objects. For Zafiris, the creation of the bridges is the creative part of the modelling process (in the sense of Dedekind 574), and its most crucial task. In his book, Philosophy of Mathematics and Natural Science, Herman Weyl restricts the possible domain of scientific investigation to an isomorphic mapping. He finds science “quite indifferent as to the ‘essence’ of its object.” 575 Zafiris agrees with Weyl and adds that in nature we do not have direct access to the things we would like to model. For us, they are always encoded in geometric structures. In this respect, a code would correspond to the semantic elements of modelling 576, both geometrical and topological. Therefore, every problem has a unique matching geometry or topology. The fact that we can characterise a structure as geometric, indicates an embodiment of a particular cipher. Zafiris describes ciphers as the algebraic (algorithmic or syntactic) elements of the analysis. Ciphers formalise processes of reasoning and computation by being conceptually isomorphic to what we can observe in nature. 577

574 „Meine Hauptantwort auf die im Titel dieser Schrift gestellte Frage lautet: die Zahlen sind freie Schöpfungen des menschlichen Geistes, sie dienen als ein Mittel, um die Verschiedenheit der Dinge leichter und schärfer aufzufassen.” Author’s translation: “My answer to the problems propounded in the title of this paper is this: Numbers are free creations of the human mind, they serve as a means to regard the diversity of things easier and sharper.” Dedekind, Was sind und was sollen die Zahlen?, iii. 575 Weyl, Philosophy of Mathematics and Natural Science, 25. 576 Zafiris, Natural Communication. 577 “When a system of ‘meaningless’ symbols has patterns in it that accurately track, or mirror, various phenomena in the world, then that tracking or mirroring imbues the symbols with some degree of meaning—indeed, such tracking or mirroring is no less and no more than what meaning is.” Preface to Hofstadter, Gödel, Escher, Bach, P-3.

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For example, the natural communication model uses an encoding bridge to encode geometrical or topological information into an algebraic structure. On this new level, the problem, or an obstacle which was unsolvable on the original level can be resolved on the level of syntax. Then, by means of a decoding bridge (finding the inverse procedure) the solution can be decoded to the original domain, where it can be interpreted. 578 Motivic key: Cryptographic unit To formalise such procedures, Zafiris introduces the notion of a motivic key as a relational unit, or a recurrent monad possessing identity. The anatomy of the key relates two different levels W and T by means of the encoding bridge S+1 and the decoding bridge S-1, as shown in Figure 50. 579 W

Fig. 50

Motivic key.

S +1

S -1

T

The direction of the arrows in the diagram indicates the inverse relationship of bridges S+1 and S-1. Depending on which level we decide to take as the reference, we obtain a corresponding algebraic structure. If we take W as the level of the original problem (or an obstacle), then we can say that T is a conjugate to W under the encoding S+1 and decoding S-1. This can be written as: 35%

W = S T S –1

Conjugation has the reflection property, meaning that if we can define level W as a conjugate to T, the opposite is also possible: W = S T S –1 & T = S –1 W S

578 Zafiris, Natural Communication. 579 Zafiris.

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One of the basic properties of the motivic key is that of extension, for which two basic types are allowed: extension of the key in depth (stacking) and extension of the key in length. 580 If two keys share a common level, such that W is the level conjugate to the level T, and T is conjugate to the level K, both keys can be described in terms of their original level as: W = S T S –1 T = K R K –1

The fact that the level T takes part in both keys allows Zafiris to extend the first key in depth and express both keys in terms of the level W. This operation is also known as stacking, and the newly obtained key can be formulated as: W = SK R K –1 S –1 W = ( SK ) R ( SK ) –1

and illustrated as follows: Fig. 51

W

S +1

Extension of the motivic key in depth.

S -1

T K +1

K -1

R

Both the diagram and the equation show that the newly obtained key preserves the key’s original form. 581

580 Zafiris. 581 Zafiris.

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If in one key, W is a level conjugate to the level T, and in another, level A is conjugate to the level B under the same encoding and decoding bridges S+1 and S-1, Zafiris expresses both keys in terms of their original level: W = S T S –1 A = S B S –1

This particular operation is called extending the key in length. The result of this operation is called a product key, which can be formulated in terms of a product of the two original levels as: WA = S TB S –1

and diagrammed as follows: W

Fig. 52

Extension of the motivic key in length, resulting with a product key.

A

S -1

S +1

T

B

As both the diagram and the equation show, the product key too, preserves the original form of the key. Example: Encoding a geometrical problem into an algebraic structure To illustrate how a topological problem can be encoded in an algebraic structure, Zafiris gives the example of homotopy classes. In topology, homotopy describes a deformation between two functions such that one can be continuously deformed into another. 582

582 “If two loops have the same base point, they are called homotopic if one can be continuously deformed to the other, with all the intermediate paths living in X and beginning and ending at the given base point.” Gowers, “Homotopy Groups,” 221.

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Fig. 53

Uninterrupted deformation of a curve into another in 16 successive steps.

Analogous to the example of covering spaces, Zafiris examines homotopic properties of the paths on the surface of the ring 583, described here in Figure 54. On the left drawing, there two points A and B and two paths, passing between them. These two paths can be deformed into each other just like in Figure 53. On the right drawing, between the same two points passes a red path, but unlike the blue or green one, it winds around the hole in the middle. It is obvious from the drawing that the red path cannot be deformed to match the others. Every attempt to deform it necessarily fails, as the path cannot cross the surface of the hole. 584 Fig. 54

Curve deformation on the surface of a ring.

A

B

A

B

583 They can be seen as continuous functions, computational paths, or curves in general. 584 Zafiris, Natural Communication.

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In this example, the hole represents a structural invariant of the problem of deformation. This invariant property is manifested in the fact that if we have a path that winds around the hole n times, and another path that winds around the hole m times, those two paths cannot be deformed into each other if n!m, they do not wind around the hole the same number of times. How can this invariant be qualified in algebraic terms? Zafiris shows that the simplest way would be to simply count the number of times the path winds around the hole and index a path with this number. Thus, the minimal characterisation of a deformation property of a path on the ring requires only integers. This procedure encodes the geometric problem of path deformations in an algebraic structure called the ring of integers. 585 The result of this is twofold: The complexity of the paths in question is reduced to the level of integers, and each integer obtains geometric semantics indicating the number of times a path winds around the obstacle. 586 In the chapter “An experiment: Communication and natures of architectural representation,” by making an analogy with this example, it will be shown how geometric information of a floor plan can be encoded in an algebraic structure defined with the ring of integers and the field of real numbers. relation between processes: partition spectra The natural communication model challenges the established modelling paradigms that adopt the set-theoretical assumptions, in which objects are seen as positive, sharply defined and naturally distinguishable entities. 587 As a quantum physicist, Zafiris finds this assumption unnatural, and adds that no clearly defined objects can be discriminated on the quantum level, as the whole observable domain is in a “fuzzy” state. 588 In his natural communication model, he derives the notion of an object spectrally on the basis of a partition procedure applicable to the modelling of quantum phenomena. Before introducing the partition paradigm, the assumptions and the implications of the set-theoretic point of view will be described, which will help in identifying the differences between the two perspectives.

585 “A ring, like a group or a field is an algebraic structure that satisfies certain axioms… In general, a ring is a set R with two binary operations, denoted by “+” and “×”, which satisfies all the field axioms apart from the one that says that nonzero elements have multiplicative inverses.” Gowers, “Rings, Ideals, and Modules,” 284. 586 Zafiris, Natural Communication. 587 “… majority of current computational urban modeling approaches are based on the ideas of Abstract Universals that can be seen as a Set Theoretical Modeling Approach.” Moosavi, “Pre-Specific Modeling,” 10–11. 588 Zafiris, Natural Communication.

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Set-theoretic vs partition ontology Set theory is the field of mathematics that studies abstract sets and their properties, commonly used as a foundational system for mathematics. 589 German mathematician Georg Cantor introduced set theory in his 1874 paper “On a Property of the Collection of All Algebraic Numbers.” The elementary notion of the set-theoretic paradigm is that of a set, a mathematical object used to define a collection of other objects. Cantor established the concept by proving that there are more real numbers than algebraic numbers. 590 In order to compare sets, he introduced the property of cardinality, which addresses a set according to its size. The property of ordinality allowed him to account for the order of elements within a set. These are the only properties used to characterise a set. 591 They convey the basic assumption of the set-theoretic point of view: It is analytically possible to precisely distinguish all the elements within a set. 592 Modelling within the set-theoretical paradigm consists of creating structures on top of sets. 593 Such structures allow sets to be nested within each other and form a hierarchical structure known as a von Neumann universe, formalised in the Zermelo—Fraenkel set theory. 594 The creation of the sets proceeds from the notion of an empty set, into which the elements are introduced by the notion of membership. At any stage of this procedure, the sets are totally formed objects. The partition point of view assumes completely different initial conditions. It begins with the assumption that no objects can be fundamentally distinguished. In order to discriminate anything from this state and create information, the partition paradigm builds models that can act as communication bridges. The application of bridges to an object yields a partition spectrum, which serves as the basis for defining objects. The spectrum can be further refined until the satisfactory approximation of an object is reached. Only at this point can that which is actually distinguished be represented as a set. Therefore, sets are not

589 Bagaria, “Set Theory,” 616. 590 Cantor, “On a Property of the Class of All Real Algebraic Numbers,” 258–62. 591 “This reflects two different ways in which we can think about the set {1, 2, … , n}. Sometimes all we care about is its size. Then, if we have a set X in one-to-one correspondence with {1, 2, … , n}, we conclude that X has cardinality n. But sometimes we also take note of the natural ordering on the set {1, 2, … , n}, in which case we observe that our one-to-one correspondence provides us with an ordering on X too. If we adopt the first point of view, then we are regarding n as a cardinal, and if we adopt the second, then we are regarding it as an ordinal.” Bagaria, “Set Theory,” 616. 592 Zafiris, Natural Communication. 593 “The basic idea behind the axioms of ZFC is that there is a “universe of all sets” that we would like to understand, and the axioms give us the tools we need to build sets out of other sets. In usual mathematical practice we take sets of integers, sets of real numbers, sets of functions, etc., but also sets of sets (such as sets of open sets in a topological space [III.92]), sets of sets of sets (such as sets of open covers), and so on.” Bagaria, “Set Theory,” 619. 594 Bagaria, III:249, IV:619.

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discarded in the partition paradigm. They are used to store distinctions obtained in the final step of the modelling process. 595 In his doctoral dissertation, Moosavi introduces three types of models: deterministic (natural), specific (rational) and pre-specific 596 (complex) models. He adopts the notion of abstract universals 597 to demonstrate that deterministic and specific modelling approaches subscribe to the set-theoretic paradigm. For Zafiris, the main implication of the set-theoretic point of view to modelling is that systems and subsystems are conceptualised as sets and subsets. Thus, set-theoretical models include relations and create their algebraic structures on top of this conceptualisation. Both Zafiris and Moosavi conclude that the approaches based on set-theoretical assumptions have an inherent limit in modelling complex phenomena. To go beyond the limits of set theory, they propose to abandon its assumptions altogether and adopt the mathematics of category theory. 598 In computer science, computation is traditionally described by means of an effective procedure, introduced by Church and Turing in 1936. 599 An effective procedure assumes a class of problems inherent to the procedure. Thus, a procedure can be effective in respect to one class of problems, but ineffective in respect to another class. Each procedure consists of an input and an output, relating them functionally. By means of this simple structure, multiple procedures can be joined into a chain, in order to extend their functionality. This chain of procedures is usually described as an algorithm. In a running computer program, the functionality of an algorithm can be assessed only by means of its inputs and outputs. In this regard, any computational procedure can be seen as a black box, as illustrated in Figure 55. Fig. 55

Single level linear computation chain.

INPUT

BLACK BOX

OUTPUT

595 Zafiris, Natural Communication. 596 Term pre-specific was coined by Bühlmann for characterising an emerging kind of information-based design: “A thing is pre-specific in the sense that we consider relevant here: it stands for something that has the paradoxical feature of being simultaneously indetermined and specific in its particularity.” Bühlmann, “Pseudopodia,” 55. 597 “Given a relation n, an entity uF is said to be a universal for the property F (with respect to n) if it satisfies the following universality condition: for any x, x n uF if and only if F(x).” Ellerman, “Category Theory and Concrete Universals,” 409–410. “This condition is called Universality and it means that the universal is the essence of that property the universal is referring to… Therefore, any entity that satisfies the conditions of universality and uniqueness for a certain property is a universal for that property. Now, if a universal is self-participating, it is called a concrete universal and if it does not have self-participation property it is an abstract universal.” Moosavi, “PreSpecific Modeling,” 39–40. 598 Zafiris, Natural Communication. 599 Church, “An Unsolvable Problem of Elementary Number Theory,” 356–63; Turing, “On Computable Numbers, with an Application to the Entscheidungsproblem,” 230–31.

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The notion of computation in the natural communication model involves at least two computational paths, both implementing effective procedures within their respective reference plane. Inputs and outputs of these models are connected by means of communication bridges that are inverse to each other, as illustrated in Figure 56. This particular schema will be used as the framework for conducting an experiment in the last chapter of this work. Fig. 56

INPUT

BLACK BOX

OUTPUT

INPUT

BLACK BOX

OUTPUT

Stratified computation involving communication bridges.

Equivalence and partition: From bridges to spectra To derive the notion of partition from the elementary notions, Zafiris presents the natural communication model in terms of communication between different processes. In this case, processes W and T are conjugate to each other under the encoding bridge S+1 and decoding bridge S-1: 35%

W = S T S –1

Zafiris shows that in the case of conjugation, the relation between processes W and T is an equivalence relation in the class of all processes between different levels. Equivalence relation is defined by three properties: reflexivity—process W is conjugate to itself; symmetry—W can be described in terms of T and vice versa; and transitivity—keys can be extended in depth and in length. With the equivalence relation, Zafiris defines the notion of partition, which allows him to derive the notion of a partition spectrum. 600 A partition spectrum involves clustering within the class of all conjugate processes, according to the bridges that connect them. The processes are divided into different cells (represented by different colours in Figure 57), such that the entities belonging to the same cell are conjugate to each other by means of the same bridges. Entities that are in different cells of the partition spectrum are conjugate to each other by means of different bridges. The cells of the partition spectrum are 600 Zafiris.

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called equivalence classes. They partition the class of all processes into disjoint subclasses. The following table illustrates the stage of a modelling process where the application of bridges to an indistinguishable object creates the partition spectrum. The equivalence classes within the partition spectrum serve as an informational basis for approximating the object of our investigation. Fig. 57

Illustration of a partition spectrum with different cells resembling equivalence classes.

In the chapter “An instrument for communication: self-organizing model,” it will be demonstrated how to computationally implement a procedure that creates a partition spectrum from a set of data. It is important to note that a distinction within an equivalence class cannot be interpreted as an element of a set. The notion of a class does not presuppose that corresponding entities are distinguishable with an absolute precision as in the set. Such entities can be interpreted as objectively indistinguishable, as indistinct entities, or as types. 601 However, they do bear the potential of being distinguished by further partitioning, thus are potentially representable by the set. Virtual and actual distinctions: Refining the spectrum Zafiris shows that the entities contained within the equivalence classes have two basic properties: distinction and differentiation. 602 Distinction implies that different equivalence classes (cells) can be recognised within a partition spectrum. Thus, the elementary distinction within the partition spectrum is that types within different equivalent classes are different from each other. Formally, a distinction within a partition spectrum relates a pair of processes (W, W’), belonging to different equivalence classes, as illustrated in Figure 58.

601 Zafiris. 602 Zafiris.

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Fig. 58

Spectral distinctions relating a pair of processes belonging to different equivalence classes.

W W'

Differentiation is a virtual distinction between entities that belong to the same equivalence class (cell). It implies that equivalence classes can be further refined by the repeated application of the key. By refining the partition spectrum, what was a virtual distinction becomes an actual distinction. Theoretically, the process of refining the partition spectrum can be repeated up to a point when everything about an object is distinguished, thus everything that remains to characterise it are elements of a set. However, this is not an ideal outcome of modelling, as it comes with a price of breaking the symmetry of the object, which will be explained in the subchapter on information. 603 Partition codes and ciphers There are three pairs of codes and ciphers that characterise the partition perspective. The code of symmetry defines the relation between the encoding and decoding bridges. If the communication bridges are exact inverses to each other, the cipher of symmetry is a group. 604 If the bridges are not exact inverses to each other, but inverses with different localities, the corresponding cipher is a groupoid. Finally, if communication bridges are conceptual inverses, the cipher is an adjunction. An adjunction is the most complex notion of inversion in mathematics. According to Zafiris, an adjunction signifies an “amphidromous dependence of the involved, information descriptive languages in communication.” 605 The code of duality defines the relation between sets and partitions. In set-theory, partitioning starts with an empty set and continues by adding elements and creating a hierarchy of sets until a desired approximation is reached. In the partition paradigm, the process starts from the indiscrete partition (the black circle in Figure 59). It follows with an application of 603 Zafiris. 604 Gowers, “Some Fundamental Mathematical Definitions,” 19–20. Group is a basic algebraic structure used to describe symmetries and invariants. 605 Zafiris, “Complex Systems from the Perspective of Category Theory: I. Functioning of the Adjunction Concept,” 157.

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communication bridges which yields a partition spectrum containing equivalence classes (the cloudy gray circles in the middle). The classes contain virtual distinctions. Further refining an equivalence class leads to creating new virtual distinctions and actualises the existing ones. The actual distinctions can be represented by a set (the circle on the bottom). The cipher which manifests duality between sets and partitions is called a category. 606 fig. 59

Illustration of a partition procedure.

Indiscrete partition

No distinctions

S -1

S +1

Maximum symmetry

Spectrum refinement W

Partition spectrum

W'

Discrete partition No symmetry

S -1

S +1

Partition spectrum

Virtual distinctions Equivalence classes

Everything is distinguished

The code of order among partitions is manifested in the process of actualising the virtual distinctions, by partially ordering partitions for that purpose. Its corresponding cipher is a partial order. Although any comparison must assume some notion of order, a partition paradigm does not assume an absolute, but a partial order, and establishes it by refining the partition

606 Zafiris, Natural Communication.

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spectrum. 607 The initial state in which nothing can be distinguished—the indistinct (indiscrete) partition—represents the state of maximum symmetry. In the partitioning process, the symmetry breaks, making it possible to distinguish entities belonging to different equivalence classes. In the case of a discrete partition, where everything is distinguished, all symmetry is destroyed. What breaks the symmetry is the process of creating the actual distinctions, which involves breaking the bridges of a model. 608 iNforMaTioN uNiT Zafiris also shows how the notion of information differs between the partition and set-theoretic points of view. 609 The latter’s notion of information can be demonstrated with an example of a set whose elements are totally distinguished, and a subset within it. The larger set is symbolised with 0, and its subset with 1. For each element of the larger set, there are only two options in relation to its membership in the subset: the element is either a member of the subset which corresponds to 1, or not, which corresponds to 0. fig. 60

Information unit within the paradigm of set-theory.

0

1

Boolean logic is the simplest type of logic that corresponds to this construction. If an element belongs to the subset, the proposition stating that the element is a member of 1 is a TRUE proposition, represented with the symbol 1. If the element does not belong to the subset, the proposition stating that the element is a member of 1 is a FALSE proposition, represented with the symbol 0. The true proposition, symbolised with 1, is interpreted as the unit of information. Thus, a true proposition yields one bit of information, which is the equivalent to locating an element of a set within its subset. This illustrates the elementary method of quantification within the digital paradigm. 610 607 608 609 610

Zafiris. Zafiris. Zafiris. Zafiris.

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The partition perspective, however, introduces the notion of information on top of a partition spectrum. The information unit is given by the actual distinction within the spectrum. In the partition perspective, an ordered pair of processes (Q,R) belonging to the different equivalence classes defines a partition information unit. The information obtained cannot any longer be represented by Boolean logic. 611 Fig. 61

Information unit within the partition paradigm.

Q

R

The virtual distinction within an equivalence class implies the conjunction between two distinct levels of a model W and T. These levels are actually distinguished only if the bridges can be established within each level. When this happens, the bridges that were constructed to actualise the distinction are no longer needed and are discarded. 612 (Figure 62) W

Fig. 62

Breaking of the communication bridges occurs when the distinctions become actualised.

S +1

S -1

�

T

The partition perspective requires two entities to obtain one information unit. The state of an indiscrete partition, which is the state of maximal

611 Zafiris, Natural Communication. 612 Zafiris.

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symmetry, thereby yields no information. The state opposite to it, that of a discrete partition where all symmetry has been broken, yields the maximal amount of information. The process of actualising distinctions breaks the symmetry and creates information, which renders symmetry and information as inverse concepts. 613 framework of category theory Category theory, the state of the art in pure mathematics, reaches the highest level of abstraction in characterising mathematical domains and their relations. In characterising a domain, it goes a step further from describing the common properties of its elements: It abstracts the nature of the elements and focuses on their connectivity as their structural property. 614 Corry and MacLane’s, and many other category theory textbooks, introduce the topic starting with the notion of the category, followed by functors (morphisms), natural transformations and finally, adjunctions. 615 Genetic process of unfolding categories According to Zafiris, the emphasis on this particular sequence detaches category theory from any possibility of application. He proposes a genetic process of arriving at categories 616, which starts with the notion of the model containing communication bridges. He prefers to introduce the elements of category theory, in a reversal of the textbook presentation, in the following order: 1 Adjunctions 2 Naturality 3 Functors 4 Categories He introduces the sequence with the notion of a model. To call something a model implies the establishment of communication bridges between different domains, through which the problem or an obstacle can be resolved.

613 Zafiris. 614 “As a counterpart to the set-theoretical analytic championed by Cantor’s heirs, category theory no longer dissects objects from within and analyses them in terms of their elements, but goes on to elaborate synthetic approaches by which objects are studied through their external behavior, in correlation with their ambient milieu.” Zalamea, Synthetic Philosophy of Contemporary Mathematics, 121. 615 Note the page numbers: “First we describe categories directly by means of axioms, without using any set theory, and calling them ‘metacategories’”. MacLane, Categories for the Working Mathematician, 7; “A functor is a morphism of categories.” MacLane, 13.; “… natural transformation is a function which assigns to each object c of C an arrow … of B in such a way that every arrow f: c$c’ in C yields a diagram… which is commutative.” MacLane, 16.; “Let A and X be categories. An adjunction from X to A is a triple : X$A, where F and G are functors.” MacLane, 78. 616 Zafiris, Natural Communication. Zafiris argues that genetic process that unfolds to category theory started with Thales and natural philosophy, continued with Galois—the birth of the universal algebra, and finally crystallised with unfolding covering spaces in algebraic topology.

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The notion of a functor formalises the notion of a single communication bridge. A bridge can only be called a functor if it enables two categories to communicate such that the associativity under composition is preserved. A pair of functors that form a bi-directional connection between two structural levels of a model are technically known as adjoint functors. A good communication bridge should be able to transform objects from one domain into another regardless of any particular choice of objects that it is presented with, as long as it doesn’t change their structure. The property of preserving structural properties is called naturality 617, and a transformation that preserves them is called a natural transformation. Functors play yet another role in the relational constitution of an object: formalising the idea of a level constituted by functors requires the notion of a category. 618 Elements of category theory A category is an algebraic structure consisting of objects and arrows forming a network. It can be seen as a closed universe of discourse in which an arrow from one object to another has to preserve the structure given by the whole category. 619 A category is characterised by the transitivity or associative property of a composition of arrows and the existence of identity arrows. Thus, the notion of a structure is not contained in the constitution of the elements within the category, but in the constitution of the network. Figure 63 is a schematic representation of a category composed from objects X, Y, Z by means of morphisms f, g, and f%g. Fig. 63

f

X

Schematic representation of a category. (Denoir, 2006)

Y g

fog

Z

An object in set theory is characterised externally by the properties assigned to it. An object in category theory is characterised by all of its relations with the other objects within the same category. Metaphorically speaking, an object is characterised in terms of its social environment, instead of some pre-given externally defined properties. On that account, Zafiris describes categories as structural species. 620 An example (Figure 64) illustrates a category containing objects A, B, C, D, E, connected by arrows. By definition, a functor maps one category to another, such that the composition of the objects is preserved.

617 Zafiris. 618 Zafiris. 619 Zafiris. 620 Zafiris.

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Fig. 64

B

An example of a category composing out of five objects.

C A

composition

D E

To illustrate the idea of natural transformation, Zafiris gives an example in which one category (network) can be mapped to another by more than one functor. 621 Figure 65 shows the functor F mapping objects A, B, and C to the category F (A), F (B), F (C). Another functor G can map the same object to another category G (A), G (B), G (C). F B functors

functor

Fig. 65

F B G

C F

C F A G

F A

D

F G

D E

E

If we attend to objects A and B, the mapping can be symbolised as: F ( A)

F(B)

G ( A)

G(B)

Thus, a natural transformation makes a connection between two functors while preserving the object’s composition:

621 Zafiris.

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A single category can be mapped by a single functor (left) but also more than one (right).

F ( A)

F(B)

G ( A)

G(B)

It is possible to traverse the network from F (A) to F (B), but also from G (A) to G (B). In such a setup, the characterisation is independent of the objects themselves. It is not important what A and B represent, if they are composed in a certain way. As such, the transformation can be characterised as natural. 622 The next step in the process includes an inversion that makes it possible to traverse a path from F (A) " G (A) " G (B) " F (B): F ( A)

F(B)

G ( A)

G(B)

This is a bi-directional schema referred to as a natural isomorphism. 623 Bridges, which are the most fundamental concept in the natural communication model, have their counterpart in category theory and are technically known as adjunct functors. The adjunction corresponds to a conceptual inversion that can be used to represent a pair of functors mapping two domains in opposite directions. Thus, an adjunction is a bi-directional connection represented by two arrows. The adjunction between the functors F and G is symbolised as: A

F G

B

By constructing an adjunction, two levels of a model can be represented as a conjugate to each other, and their relation can be described as communication. 624 After clarifying adjunctions, Zafiris reintroduces the original modelling scenario in newly established mathematical terms. Thus, the motivic key stratifies the model into two categories W and T by means of the adjoint functors S+1 and S-1. In a general modelling scenario (see Figure 50), there is a complex problem or an obstacle that cannot be resolved within the domain

622 Zafiris. 623 Zafiris. 624 Zafiris.

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represented by the category W. Thus, the information represented in this domain needs to be encoded into a different domain represented by the category T, using the encoding bridge represented by the functor S+1. Let us assume that when represented in a different domain, a problem or an obstacle can be resolved algorithmically or on the level of syntax. This, however, is not yet sufficient. The resolution described on the level T does not automatically bring any meaning back to the original level. In order to obtain the semantics of the resolution, the information that resolves the problem/obstacle on the level T needs to be decoded back to the original domain. This is accomplished by the decoding bridge represented by the functor S-1. Communication bridges S+1 and S-1, in this case, will not be exact inverses to each other, but adjoints. 625 Objects and identity in category theory An object in category theory is fully characterised by how every other object relates to it. A Hom (or classifying functor) is obtained by investigating the mappings between all the objects within a category. This functor is represented by a category of sets, where sets are used to store relations that describe the structure of the category. 626 The notion of an identity in mathematics is closely related to the notion of an inversion. If a mathematical object is multiplied by its inverse, their product is defined as the identity unit. When multiplied, two inverse numerical entities will annihilate each other, and their products will be 1. S · S –1 = 1

In the understanding of communication developed here, the notion of a unit does not correspond to the classical mathematical notion of identity by which inverses cancel each other out. Zafiris introduces the term homotopic identity, which instead formulates identity based on structural invariance. 627 Identity so defined requires a notion of inversion that does not necessarily annihilate the levels that are inverse to each other. This inversion is exactly what the adjunction brings, by means of self-reference inherent to category theory. If the bridges between levels are adjoint, the identity of the process is not fixed but fluid. Zafiris metaphorically calls this an oscillating identity, which sounds like a promising idea to be pursued in the domain of architectural modelling.

625 Zafiris provides an interpretation of the adjunction with the example of translation: “In general, the content of the information is not possible to remain completely invariant translating from one language to another and back, in any information exchange mechanism.” Zafiris, “Complex Systems from the Perspective of Category Theory: I. Functioning of the Adjunction Concept,” 156. 626 Zafiris, Natural Communication. 627 Zafiris.

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One of the examples given by Zafiris was the problem of similarity of topological spaces. In topological spaces, the notion of similarity is often non-intuitive and sometimes impossible to formalise. Nevertheless, by encoding it in terms of algebraic invariants (ciphers), it is possible to solve the problem in an analytical or algorithmic way without difficulties inherent to the spatial order. Such a solution then needs to be decoded back into its original domain, where its semantics can be extracted. 628 The intention of this work is to apply the conceptual framework of natural communication in the domain of architecture. By means of communication bridges between different domains, the resolution of a problem unsolvable within architecture will be presented as the resolution of a matching problem in its conjugate symbolic domain. The solution can then be decoded back to architecture. II glossematics prolegomena to a theory of language Louis Hjelmslev (1899–1965) was a Danish linguist whose ideas formed the basis of the Copenhagen school of linguistics. His major ambition was to provide a self-sufficient, scientific basis for the theory of language. In order to do so, he found it necessary to rethink the very notions of theory and science. In his most important work, Prolegomena to the theory of Language (1943), he set a path for such a theory. 629 The only reference he explicitly recognised as productive for establishing such theory was the work of the Swiss Ferdinand de Saussure whose ideas we have already introduced. Prolegomena is characterised by a very peculiar use of terminology. Hjelmslev took terms and concepts already established in different disciplines and redefined them formally, building a formal system upon these definitions. In order to read Hjelmslev consistently, one must attend closely to his attitude and not fall into the temptation of referring to the usual meanings of familiar terms he uses. Hjelmslev found it difficult to accept how the science of his time looked upon language. He described it as a fascination that never lead to the serious investigation of language in itself, but rather as a means to a knowledge. He described how the interpretation of language, gestures and the physics of language (sounds, writing, organs of speech), were used in an attempt to understand human psyche and thoughts. 630 628 Zafiris. 629 Hjelmslev, Prolegomena to a Theory of Language, 6. 630 In general problem with linguistics was that “we are studying the physical and psychological, psychological and logical, sociological and historical precipitations of language, not language itself.” Hjelmslev, 5.

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For him, the difficulties inherent to language could be seen as a kind of intrinsic blindness. By virtue of being an integral part of our existence, language is easy to overlook, making it often naturalised and taken for granted. This is of major importance for Hjelmslev, as he concludes that our blindness for language means that our knowledge is actually limited by our language. He described the current usage of the language as a mean to transcend knowledge. However, the idea that fascinated him the most was to establish a path towards imminent knowledge, which was possible if we would finally become aware of the language itself. If Hjelmslev’s ambition is taken to be this extensive (for which I would argue in favour), by providing an immanent understanding of language, Hjelmslev wishes to recreate the whole knowledge of the world on the linguistic basis. 631 This work will also try to demonstrate that Hjelmslev actually anticipated a novel way of thinking about models and computation, which was far ahead of his time. In part, his theory becomes quite similar to the model of natural communication. Such parallels between two seemingly different models will be investigated in the following chapter. redefining the notion of theory Axioms and principles To illustrate the main assumption of his theory, Hjelmslev introduced metaphors of constancy and fluctuation. 632 On the most general level, he saw constancy as something that was never-changing, and fluctuations as the very opposite. From the perspective of physics, gravity could be taken as an example of a constant that satisfies Hjelmslev’s argument. In the same respect, fluctuations could be seen as the movement of bodies in a space-time affected by it. By generalising on the basis of constancy and fluctuations, he introduced his major hypothesis and offered it in a form of an axiom: A priori it would seem to be a generally valid thesis that for every process, there is a corresponding system, by which the process can be analysed and described by means of a limited number of premises. 633 This proposition could be illustrated by the relation of symbols X and Y, standing for an arbitrary pair of phenomena:

631 “Linguistics must attempt to grasp language, not as a conglomerate of non-linguistic (physical, physiological, logical, sociological) phenomena, but as a self-sufficient totality, a structure sui generis.” Hjelmslev, 6. 632 “Linguistic theory … while continually taking account of the fluctuations and changes of speech, necessarily refuses to grant exclusive significances to changes; it must seek a constancy … that makes language a language, whatever language it may be, and that makes a particular language identical with itself in all its various manifestations.” Hjelmslev, 8. 633 Hjelmslev, 9.

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SYSTEM

Fig. 66

Hjelmslev’s elementary distinction serving as an axiom.

X constancy

PROCESS Most general distinction in the world

Y fluctuation

The illustration shows a relation between symbols, where X stands for something that exhibits constancy and Y for something that exhibits fluctuation or variability. Hjelmslev’s intention was to formalise this state of affairs and test it as the basis for a linguistic theory. From a mathematical perspective, X and Y could be seen as exhibiting invariant and variable properties. In physics, Y could stand for observables 634, as it is closely related to the notion of data. The system/process distinction is the fundamental distinction Hjelmslev made in order to give an account of reality. It is important to keep in mind that whenever Hjelmslev assumes the process or a system perspective as a basis for an explanation, it actually means that he suggests that we look at the phenomena from the standpoint of a unilateral dependence. For Hjelmslev, constancy and fluctuation were exactly what was at stake with the distinction between science and humanities. He argues how the “humanistic tradition denies a priori the existence of the constancy and the legitimacy of seeking it,” 635 since humanistic, as opposed to natural phenomena, are non-recurrent. Hjelmslev claimed that a main standpoint of classical history was that humanistic phenomena are not formally interpreted, only described. He noticed the same with language from the perspective of philology. 636 All of this, he noted, led to the rejection of the possibility of establishing a system. Science, on the contrary, attempts to generalise phenomena. The ability to generalise makes it possible to describe parts of reality with models that are used to answer the question of how a phenomenon occurs. However, if this kind of reasoning was applied in the study of language, it only leads to numerous exceptions. Regularities in different languages are not as intuitive in terms of representation, which makes the problem impossible to grasp. In this respect, Hjelmslev and Zafiris face the same problem and offer, in response, a novel perspective for looking at the problem. This perspective leads to the stratification of a problem on different domains.

634 “Moreover, the usual underlying assumption on the basis of physical theories postulates that our form of observation is represented by coefficients in a number field, which is usually taken to be the field of real numbers.” Zafiris, “The Nature of Local/Global Distinctions, Group Actions and Phases,” 174. 635 Hjelmslev, Prolegomena to a Theory of Language, 8. 636 “Philology—the study of language and its texts as a means to literary and historical insight”. Hjelmslev, 5.

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Hjelmslev introduced another important observation that he wished to utilise as a principle. He noted that if we were able to recognise some entities in a certain process as appearing in a certain (local) order, we could learn something about them purely on the basis of that ordering, without taking their nature into consideration. This observation was mathematically formalised in the beginning of the 20th century by the Russian mathematician Andrey Markov as what later became known as Markov chains. He also noted that even if we do not characterise elements in any other way, there is always a possibility of conducting an analysis according to the possibilities of combination. 637 This would permit the establishment of a relational method between elements, even if the nature of the elements remained unknown. In mathematics, similar abstraction is also encapsulated by the notion of the category where objects and arrows are used to map the objects according to their relations without considering their nature. The principles Hjelmslev mentioned are heavily used in computational modelling today. Models like PageRank, bag of words, sentiment analysis, etc., are good examples of the potency of such thinking. Only by counting the words and measuring the probability of their appearance in sentences, Google is able to translate texts without a language model. Such models, Moosavi refers to as data-driven. 638 In regard to the setup he introduced, Hjelmslev’s expectations were that language was indeed an object that could affirm his thesis that “a process has an underlying system.” 639 To set up the theory, Hjelmslev was faced with the same problem that had been puzzling Hilbert: how to give an account of his theory within itself. Unlike Frege and Russell, Hjelmslev was not afraid of selfreference, which was the main stumbling point of the set-theory. 640 In order to account for his theory within the theory itself, he put special attention to the architectonics of the system of definitions. He needed to introduce only those premises “that are necessarily required by its object.” 641 Like any scientific theory, a scientific theory of language needs to be verifiable, thus “a theory must be capable of yielding, in all its

637 “It must be assumed that any process can be analysed into a limited number of elements recurring in various combinations. Then, on the basis of this analysis, it should be possible to order these elements into classes according to their possibilities of combination.” Hjelmslev, 9. 638 Moosavi, “Pre-Specific Modeling,” 23. 639 Hjelmslev, Prolegomena to a Theory of Language, 10. 640 “Problems with set theory had emerged in the form of paradoxes, the most famous due to Russell [VI.71]: if S is the set of all sets that do not contain themselves, then it is not possible for S to be in S, nor can it not be in S. Zermelo’s axiomatics sought to avoid this difficulty, in part by avoiding the definition of set.” Archibald, “The Development of Rigor in Mathematical Analysis,” 128. 641 Hjelmslev, Prolegomena to a Theory of Language, 10.

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applications, results that agree with so-called (actual or presumed) empirical data.” 642 The main principle of the theory was concerned with its relation to experience. Hjelmslev set it above all other principles, as it marked a clear cut with all previous undertakings of linguistic philosophy. He formulated it as the empirical principle: The description shall be free of contradiction (self-consistent), exhaustive, and as simple as possible. The requirement of freedom from contradiction takes precedence over the requirement of exhaustive description. The requirement of exhaustive description takes precedence over the requirement of simplicity. 643 This precedence determines the relative level of importance of each requirement. Therefore, being free from contradiction is more important than being described in a simple way. Later it will be shown how all three requirements, in the specific order of importance, work together as a strategy for finding an optimal/generic solution, similar to Wikipedia. PRECEDENCE

Fig. 67

Precedence of the requirements of the empirical principle.

FREEDOM OF CONTRADICTION EXHAUSTIVE DESCRIPTION SIMPLICITY

Ascent After the empirical principle, Hjelmslev made another inversion which dealt with the direction of reasoning, or as he called it ascent. To explain it, he contrasted his mode of reasoning with that of previous linguistics. First, he described a familiar type of ascent called inductivism. He defined it as the gradual ascent from something particular towards something general and subscribed it to the linguistic reasoning of his contemporaries. Such reasoning is a progression from a component to a class and a synthetic, generalising movement. Fig. 68

Hjelmslev’s description of inductivism.

SOMETHING PARTICULAR (more limited)

SOMETHING GENERAL (less limited)

COMPONENT

CLASS

642 Hjelmslev, 11. 643 Hjelmslev, 11.

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He gave an example of inductive reasoning characteristic of the study of language at that time: … it ascends, in its formation of concepts, from the individual sounds to the phonemes (classes of sounds), from the individual phonemes to the categories of phonemes, from the various individual meanings to the general or basic meanings, and from these to the categories of meanings. 644 Hjelmslev found this reasoning deficient and his point of view can be backed up, if we refer back to the chapter on natural communication. In light of the partition paradigm, it becomes evident that the mode of reasoning Hjelmslev opposes is characteristic of set theory. Assumptions about its elements appear to be exactly the same for phonetics and set theory. The individual sounds that phonetics takes as the departure point fully correspond to the idea of a finite set. All the elements in this set are fully defined, which assumes that each element can be clearly recognised from another according to their intrinsic properties. The problem with this perspective is that the sounds that the phonetic takes as the basis of description actually never exist in an isolation. Therefore, the process of distinguishing them was already assumed behind the scenes and made in a completely arbitrary fashion from a linguistic standpoint. The set-theoretical point of view is at the core of the international phonetic alphabet (IPA), which is a system that may be very useful in practice, but according to Hjelmslev, cannot serve as the basis of linguistic analysis. For him, such thinking necessarily leads “to the abstraction of concepts which are then taken to be real—and then they become not generalisable beyond a single language in an individual stage.” 645 Conceived in an inductive manner, generalisations can only be appropriate in a very limited scope, as they depend on the artificial choice of objects that are taken to be real. This is exactly what is at stake with the concept of naturality in the natural communication model. In language this manifests itself such that generalisations can be made only for a single language in a single stage. But even in a single language, there will be a large number of cases where such a theory will yield exceptions. 646 Hjelmslev concluded that induction was not compatible with his empirical principle, namely with the principle of selfconsistency, and he proceeded by inverting the mode of reasoning. The inversion of the ascent was done in the same manner as Zafiris does with the partition paradigm. Where Zafiris speaks of an indiscrete partition—which is further refined into sets of virtual and actual distinctions—Hjelmslev speaks of analysing (later: partitioning,

644 Hjelmslev, 12. 645 Hjelmslev, 12. 646 “…induction leads from fluctuation, not to constancy, but to accident.” Hjelmslev, 12.

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articulating) the text in its full integrity. 647 Both approaches start with an unanalysed/unknown whole, whose substance needs to be distinguished by means of a process that yields certain inventories of entities. For both, the integrity of the unanalysed whole is something that must be assumed in the beginning. Hjelmslev suggests that the only way to find constancy or “to order a system to the process of that text” is to analyse into components and continue doing so until the analysis is exhausted. Such a progression from class to component he defined as deduction. 648 Fig. 69

INDUCTION

Differences between induction and deduction.

COMPONENT

CLASS Synthetic, generalising movement

DEDUCTION COMPONENT

CLASS Analytic, specifying movement

Theory and the world Another inversion that Hjelmslev made regarded the relation between the theory and its object. For him, this was necessary because the concept of theory already contains assumptions about its relation to the world. In the prevailing sense of the word theory at the time, “the influence between the theory and its object are unidirectional: The objects determines and affects the theory, not vice versa.” 649 In science, theory is related to the idea of modelling (some aspect of reality), and the idea of verification: comparing the results with empirical data. To redefine the term theory, Hjelmslev needed to redefine how and to what extent theory should reflect the world. He defined it in a two-fold way by means of two principles that are somewhat in opposition to each other, namely arbitrariness and appropriateness. From the standpoint of arbitrariness: A theory, in our sense, is in itself independent of any experience. In itself, it says nothing at all about the possibility of its application and relation to empirical data. It includes no existence postulate. It constitutes what has been called a purely deductive system. 650

647 “If the linguistic investigator is given anything, it is the as yet unanalysed text in its undivided and absolute integrity.” Hjelmslev, 12. 648 Notion of deduction comes from axiomatic systems. The notion of deduction introduced by Hjelmslev has the meaning of aphaeresis—quite different meaning from the standard conception of deduction by means of implication. If we take the notion of deduction to the partition paradigm, it corresponds well to the process of refining the spectrum and creating distinctions. 649 Hjelmslev, Prolegomena to a Theory of Language, 13. 650 Hjelmslev, 14.

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From the standpoint of appropriateness: A theory introduces certain premises concerning which the theoretician knows from preceding experience that they fulfil the conditions for application to certain empirical data. 651 This supports his point that the structure of language is arbitrary and that the linguistic theoretician should know which model works under which local conditions. The theoretician recognises certain patterns in data and incorporates them in the theory under the conditions corresponding to that data. From the standpoint of arbitrariness, to create a theorem means to provide a function in a conditional form; if the condition is fulfilled, the truth of a given proposition follows. Yet, from the standpoint of appropriateness, establishing a theorem is an act that involves more than just applying logic. It is a creative, intellectual act in the same sense in which making bridges is for Zafiris. The logical apparatus, however, remains responsible for the verification of the theorems. 652 From these two principles, it follows that the theory defines its object and at the same time an object defines the theory. Gilles Deleuze, highly influenced by Hjelmslev’s work, appropriated this gesture and called it a double articulation. 653 From the standpoint of calculation (arbitrariness), the theory defines the object and from the standpoint of empirical knowledge (appropriateness), the object defines the theory. Theory aims at providing a procedural method by means of which objects of a premised nature can be described self-consistently and exhaustively. 654 From this follows that the theory aims at providing a procedure for describing an object. Such description is what we refer to as knowledge or comprehension. However, Hjelmslev’s notion of a formal system does not correspond to a purely logical, but rather procedural algebraic system. From the standpoints of arbitrariness and appropriateness it necessarily follows that it is impossible to establish a single absolute model that would entirely contain a theory of language. Hjelmslev’s model consists of a network of arbitrary, localised models he calls a system. In contemporary mathematics, such gluing that allows operating on multiplicities without reducing them, is known as a sheaf. 655 Zafiris 651 Hjelmslev, 14. 652 “On the basis of a theory and its theorems we may construct hypotheses (including socalled laws), the fate of which, contrary to that of the theory itself, depends exclusively on verification.” Hjelmslev, 14. 653 Deleuze and Guattari, “10,000 B.C.: The Geology of Morals,” 45. In category theory, double articulation could be described as functorial bidirectional correspondence. Zafiris, Natural Communication. 654 Hjelmslev, Prolegomena to a Theory of Language, 15. 655 “A sheaf is a type of mathematical object that allows for the global gluing of whatever proves to be coherently transferable in the local.” Zalamea, Synthetic Philosophy of Contemporary Mathematics, 162.

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uses sheaves to represent complex objects through local topological gluing. 656 Premises introduced by the linguistic theoretician are of a local character. 657 Therefore, they only apply to those objects that lie within the boundary within which the applicability of a premise yields results that agree with empirical data. Objects within such a boundary are said to be of the same premised nature. 658 In other words, a premise can be seen as a system corresponding to some topological locality. A system of premises, then, is expected to expand these boundaries by means of gluing. Hjelmslev gave an example of Danish texts. If one could construct a system of premises that generalise the Danish language, it should in fact be possible to comprehend all Danish texts, those existing and non-existing. One can get a better insight into Hjelmslev’s vision of his formal system, by comparing it with a simpler, yet powerful formal system of regular expressions. The system was formalised by Stephen Kleene 659 in the 1950s and implemented computationally on the Unix platform by Ken Thompson in 1968. 660 The basic idea was that, by using a sequence of characters, we can define a search pattern that can locate (match) and replace specific character strings in a text. 661 To illustrate Hjelmslev’s ideas, it can be helpful to examine regular expressions from the standpoint of arbitrariness, appropriateness and a premised nature. From the standpoint of arbitrariness, regular expressions are simply a formal system. Every operation they perform on a text is executed mechanically by following a set of search-replace patterns. From the standpoint of appropriateness, the only possible way of achieving meaningful results when working with regular expressions is to creatively experiment by applying arbitrary search-replace structures until a satisfactory solution is reached. It is impossible to generalise any rules that would be universally useful, but it is

656 “We stress the point that the transition from locally defined properties to global consequences happens via a compatible family of elements over a covering system of the complex object. In this perspective a covering system on a complex object can be viewed as providing a decomposition of that object into simpler objects.” Zafiris, “Complex Systems From the Perspective of Category Theory: I. Functioning of the Adjunction Concept,” 150. “The above general scheme accomplishes the task of comprehending entirely the complex object through covering families of well-known local objects pasted together appropriately, in case there exists an isomorphism between the operationally or theoretically specified structure representing a complex system and the sheaf of compatible local viewpoints imposed upon it.” Zafiris, 150. 657 One must keep in mind that this refers to topological, not metrical notion of locality. 658 “But at the same time, a theory is not only meant to provide us with the means of knowing one define object. It must be so organised as to enable us to know all conceivable objects of the same premised nature as the one under consideration.” Hjelmslev, Prolegomena to a Theory of Language, 15. 659 Kozen, “Kleene Algebra and Regular Expressions,” 55. Kleene proved algebraically that every finite automation has an equivalent regular expression. 660 Wikipedia, s.v. “Regular Expression,” last modified September 1, 2017, 18:59, https:// en.wikipedia.org/wiki/Regular_expression. 661 Thompson, “Programming Techniques: Regular Expression Search Algorithm,” 419.

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evident that there can be many rules which will work for a specific purpose. Therefore, a regular expression created by an experienced user is often useful for a certain local case and not at all useful for another. In that respect, objects of a premised nature would be those which fall within the boundary within which a certain regular expression yields satisfactory results. This can be illustrated with an example of using regular expressions to extract text from movie subtitle files. As shown below (Figure 70), subtitles contain a lot of meta-information used to match dialogs to a specific time. 1⁋ 00:05:13,328 --> 00:05:16,991⁋ Darling, don't be difficult.⁋ Let's take our sweet...⁋ ⁋ 2⁋ 00:05:17,098 --> 00:05:20,363⁋ Iovely children on an outing.⁋ ⁋ 3⁋ 00:05:31,813 --> 00:05:35,408⁋ You can't do this.⁋ I know my rights.⁋ ⁋ 4⁋ 00:05:35,483 --> 00:05:38,111⁋ I have the authority to close⁋ you down, and I'm doing just that.⁋ ⁋ 5⁋ 00:05:38,186 --> 00:05:42,714⁋ This exhibit degrades everybody who⁋ sees it and the creature himself.⁋ ⁋ 6⁋ 00:05:42,791 --> 00:05:46,056⁋ He is a freak!⁋ How else will he live?⁋ ⁋ 7⁋ 00:05:46,127 --> 00:05:49,392⁋ Freaks are one thing.⁋ This is entirely different.⁋ ⁋ 8⁋ 00:05:49,464 --> 00:05:52,797⁋ This is monstrous⁋ and should not be allowed.⁋ ⁋ 9⁋ 00:05:52,867 --> 00:05:56,462⁋ These officers will see to it⁋ you're on your way soon. Good day!⁋ ⁋

10⁋ 00:05:57,972 --> 00:06:01,373⁋ Move along please.⁋ ⁋ 11⁋ 00:06:04,112 --> 00:06:06,046⁋ Hold it there, sir.⁋ ⁋ 12⁋ 00:06:06,114 --> 00:06:08,139⁋ Come this way.⁋ ⁋ 13⁋ 00:06:19,828 --> 00:06:21,762⁋ On the move again...⁋ ⁋ 14⁋ 00:06:26,835 --> 00:06:28,268⁋ my treasure.⁋ ⁋ 15⁋ 00:07:06,574 --> 00:07:08,872⁋ We'll be seeing a lot more⁋ of these machine accidents...⁋ ⁋ 16⁋ 00:07:08,943 --> 00:07:10,740⁋ - Mr. Hodges.⁋ - Yes, sir.⁋ ⁋ 17⁋ 00:07:14,983 --> 00:07:19,317⁋ Abominable things, these machines.⁋ You can't reason with them.⁋ ⁋ 18⁋ 00:07:20,622 --> 00:07:22,385⁋ What a mess!⁋ ⁋ 19⁋ 00:07:26,161 --> 00:07:28,789⁋ Pull on the rope.⁋ 20⁋ 00:07:29,931 --> 00:07:31,762⁋ Irons, please.⁋

Figure 71 presents a collection of twelve regular expressions, where those marked by a are search patterns, and those marked by b are replace patterns. Parentheses are used to indicate which parts of the selected text

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Fig. 70

A subtitle extract from David Lynch’s Elephant Man.

should be stored into a variable represented by the symbol $. Colours indicate which selected part matches which variable.

Fig. 71 01a ^\d+\n[\d:, ->]+ 07a (\w)$\n([A-Z]) Move along please.⁋ ⁋ A collection of 07b $1 $2 01b 11⁋ 12 regular 00:06:04,112 --> 00:06:06,046⁋ expression rules 08a ,$\n([A-Za-z]) 02a Hold it there, sir.⁋ and replacements. (\b\w\b$)\n([a-z]+) 08b , $1 ⁋ 02b $1 $2 12⁋ 09a 00:06:06,114 --> 00:06:08,139⁋ ([\w?.!])$\n([\w]) 03a Come this way.⁋ 09b $1 $2 (\.\.\.)$\n+([a-z]) ⁋ 03b $2 13⁋ 10a \.\.\. 00:06:19,828 --> 00:06:21,762⁋ 04a ,$\n*([a-z]) 10b . On the move again...⁋ 04b , $1 ⁋ 11a 14⁋ 05a (\w)$\n([a-z]) (^[^\.!?]+)([!?.]) 00:06:26,835 --> 00:06:28,268⁋ 05b $1 $2 11b $1$2\n my treasure.⁋ ⁋ 12a \n\n 06a ^15⁋ 12b 06b 00:07:06,574 --> 00:07:08,872⁋ We'll be seeing a lot more⁋ of these machine accidents...⁋ By applying regular expressions, it was possible to extract clean text ⁋ 16⁋ from several hundreds of subtitle files in a matter of seconds with a 00:07:08,943 --> 00:07:10,740⁋ satisfactory quality of extraction (Figure 72). The applied method was - Mr. Hodges.⁋ - Yes, sir.⁋ completely arbitrary and created on the fly. ⁋ 17⁋ 00:07:14,983 Fig. 07a 72 --> 00:07:19,317⁋ (\w)$\n([A-Z]) Darling, don’t be difficult. Abominable things, these machines.⁋ Result of the Let’s take our sweet lovely children on an outing. 07b $1 $2 You can't reason with application of them.⁋ You can’t do this. ⁋ regular 08aexpressions ,$\n([A-Za-z]) I know my rights. 18⁋ from08b fig. 71 ,on$1 I have the authority to close you down, and I’m doing just that. 00:07:20,622 --> 00:07:22,385⁋ the subtitles from This exhibit degrades everybody who sees it and the creature himself. What a mess!⁋ fig. 70. He is a freak! 09a ⁋ How else will he live? ([\w?.!])$\n([\w]) 19⁋ Freaks are one thing. 09b $1 $2 00:07:26,161 --> 00:07:28,789⁋ This is entirely different. Pull on the rope.⁋ This is monstrous and should not be allowed. 10a \.\.\. These officers will see to it you’re on your way soon. 10b . 20⁋ Good day! 00:07:29,931 --> 00:07:31,762⁋ Irons, please.⁋ Move along please. 11a Hold it there, sir. (^[^\.!?]+)([!?.]) Come this way. 11b $1$2\n On the move again my treasure. We’ll be seeing a lot more of these machine accidents… 12a \n\n Mr. Hodges. 12b Yes, sir. Abominable things, these machines. You can’t reason with them. What a mess! Pull on the rope.

The creative act of creating regular expressions would, in Hjelmslev’s terms, correspond to appropriateness. The arbitrary way of reaching the result and formality of the system that implements searching and

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replacing would correspond to arbitrariness. 662 Objects of a premised nature would correspond to any movie subtitle in a “SubRip” text file format 663, analysed with the aforementioned chain of 12 regular expressions. Rejecting a single absolute model 664 in a linguistic theory led to the strategy of gluing multiple local models. Hjelmslev took great interest in establishing a stock of local conditions found in individual objects (texts) that could be reused in analysing other texts. The introduction of a stock also implies that his system of premises is not fixed and cannot be exhausted. It was conceived as an arbitrary system which should attain stability with an increased usage. A similar process can be observed today with the notion of big data. The more data available, the more the quality of data driven models increases. 665 In Hjelmslev’s theory this means that on a more abstract level, but never explicitly, some very systemic knowledge about the language in general will be reflected in the system of premises. If the localised models taking part in Hjelmslev’s theory are created collaboratively, and are in fact arbitrary as he suggests, how it is possible to establish grounds for comparison and evaluation of different local models? The empirical principle provides a solution to that problem: Solutions are first tested on the basis of self-consistency, then on the basis of providing an exhaustive description, and finally on the basis of simplicity. The simplicity principle is used to decide a “winner” among the already correct solutions by implying that the solution which yields the simplest explanation and offers the simplest procedure should be taken as the correct solution. 666 Therefore, the empirical principle is a testing mechanism for linguistic theory as well as its optimisation strategy. To assure the quality of a model, Hjelmslev utilises the simplicity principle. To assure the quality of communication bridges, Zafiris utilises natural transformations. skeleton for a new theory: system of definitions Hjelmslev aimed to create a reasoning apparatus and a body of concepts that will meet the requirements he had previously specified. He named it a system of definitions. The system was to contain as few implicit premises as possible while defining as much as possible. Everything that was defined needed to rest on the premised definitions, and everything he would need to define later should rest on what was already defined.

662 Similar or better result could also be made in less steps, possibly even in one, but the important part is that it is impossible to know or predict beforehand how to construct such an expression. 663 Matroška, “SRT Subtitles.” 664 Moosavi, “Pre-Specific Modeling,” 43. In the Moosavi’s classification these would correspondent to deterministic (natural) and specific (rational) models. 665 Halevy, Norvig, and Pereira, “The Unreasonable Effectiveness of Data,” 9. 666 “Any concretely developed linguistic theory hopes to be precisely that definitive one.” Hjelmslev, Prolegomena to a Theory of Language, 19.

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Principle of analysis For Hjelmslev, a deductive progression was the necessary requirement for achieving a self-consistent and exhaustive description of a text. Such a progression he formally defined as an analysis 667, which starts with a text, and in a deductive progression, defines components and components of components. In the natural communication model, such progression corresponds to the partitioning process. Fig. 73

Deductive progression.

DEDUCTIVE PROGRESSION

TEXT

COMPONENTS

COMPONENTS

COMPONENTS OF COMPONENTS

Since division could be done in any number of ways, we need to choose a basis for the division and a dividing strategy that will produce a topological description of a text. The second empirical principle ensured that the basis for analysis of linguistic texts was not universal 668 and that it could vary from text to text. Thus, the mode of progress should have been chosen in a way that yields the most exhaustive description. Hjelmslev suggested that the important thing about the division was not the obtained parts, but the mutual dependences that should permit him to give account for the obtained parts. 669 This means that division makes sense only if the relations can be discriminated within a whole. In Zafiris’s partition paradigm this would correspond to the distinction of different equivalence classes according to the bridges that apply to them. 667 “Analysis—a deductive progression from class to component and component of component (practical definition).” Hjelmslev, 21. 668 “An operation with a given result we shall call universal if it is asserted that the operation can be performed on any object whatsoever; its resultants we shall call universals. On the other hand, an operation with a given result we shall call particular, and its resultants particulars, if it is asserted that the operation can be performed on a certain object but not on any other object.” Hjelmslev, 40. 669 Hjelmslev, 22.

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Finally, Hjelmslev asserts that objects are in fact nothing more or less than a sum total of their dependencies. 670 An object is defined by the sum-total of all the relations of its components. To put it in architectural terms, we can say that objects are in fact nothing more than their own architectonics! If this is to be taken seriously, it could lead to redefining the notion of the architectural profession. However, it would also require a different notion of literacy to be acquired in architectural education. Architectonics of relations To further develop the assertion that “totality does not consist of things but of relationships,” 671 Hjelmslev extended and formalised the notion of a relationship, introducing a new set of definitions for relationships, independent of their function in language. He began by introducing three generic types of relationships: interdependence, determination and constellation. Fig. 74

THEORY

Three general types of dependences.

INTERDEPENDENCE A

B Mutual dependence, in which one term pressuposes the other and vice versa

DETERMINATION A

B Unilateral dependence, in which one term pressuposes the other but not vice versa

CONSTELLATION A

B Dependence, in which two terms are compatible but neither presupposes the other

670 “Both the object under examination and its parts have existence only by virtue of these dependences. The whole of the object under examination can be defined only by their sum total; and each of its parts can be defined only by the dependences joining it to other coordinated parts, to the whole, and to its parts of the next degree, and by them sum of the dependences that these parts of the next degree contract with each other.” Hjelmslev, 23. 671 Hjelmslev, 23.

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He further refined them into two classes: one within a system, and one within a process. For the process, he defined the notions of solidarity, selection and combination. Fig. 75

PROCESS

Dependences within a process.

SOLIDARITY A

B Interdependence between terms in a process

SELECTION A

B Determination between terms in a process

COMBINATION A

B Constellation within a process

For the system, he defined the notions of complementarity, specification and autonomy. SYSTEM

Fig. 76

Dependences within a system.

COMPLEMENTARITY A

B Interdependence between terms in a system

SPECIFICATION A

B Determination between terms in a system

AUTONOMY A

B Constellation within a system

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These three sets of terms allowed him to take a different point of view on the same object according to the assumed perspective of a theory, of a system or of a process. Furthermore, an introduction of these highly abstracted notions of dependence allowed him to widen the field of linguistics by concentrating on relations at any stage of the procedure, rather than to see the objects as positive entities. From the perspective of category theory, these dependences could be seen as functors. Hjelmslev used the word terminal 672 to refer to the endpoints of a relationship. Therefore, analysis was about registering dependences between certain terminals, or parts of text. These parts have existence precisely by virtue of these dependences and only by virtue of them, meaning that there must be a relationship criterion for addressing something as a part of something else. The existence of the whole and its parts suggested that the whole must be larger than its parts. 673 Thus, the terminals were said to enter the whole. Fig. 77

WHOLE (TEXT)

Parts obtained in an analysis can be said to enter the whole.

PARTITION

DISCOVERED OBJECTS (PARTS)

...

DISCOVERED OBJECTS (PARTS)

discovered objects are laying within the whole

By recognising the relation between individual parts as different from the relation between a part and a whole, Hjelmslev defined two families of dependences: The first characterised the dependences between

672 Mac Lane, Categories for the Working Mathematician, 20. It corresponds to the notion of the terminal object in category theory. 673 Parts are therefore discovered in a whole, and lay within it.

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the parts, or between the wholes; the second characterised the dependences between the whole and its parts. Here, the difference cannot simply be depicted with arrows in one dimension. Figure 78 shows an object depicted both as a whole and as a collection of parts. Because of the assumption that parts belong to the whole, a dependence between the whole and its parts is principally different than the dependence between the parts themselves, or between two wholes. He called the factor that makes these distinctions the uniformity of the dependence. 674 In the context of the natural communication model, uniformity of the dependence corresponds to the notion of sheaf-theoretic uniformity. 675 Fig. 78

Uniformity of the dependence.

Elements of the analysis From the standpoint of these dependences, Hjelmslev defined analysis as the “description of an object by the uniform dependences of other objects on it and on each other.” 676 An object that is subjected to the analysis he named a class and the other objects that are registered by a particular analysis as uniformly dependent on the class and on each other, he named components of the class. Fig. 79

CLASS (subjected to an analysis) DEDUCTION

Analysis as a description of an object by all of its uniform dependences.

UNIFORM DEPENDENCES ON THE CLASS

CLASSES

UNIFORM DEPENDENCES ON EACH OTHER

674 “Coordinate parts, which proceed from an individual analysis of a whole, depend in a uniform fashion on that whole.” Hjelmslev, Prolegomena to a Theory of Language, 28. 675 Zafiris, “Generalized Topological Covering Systems on Quantum Events’ Structures,” 1487–88. 676 Hjelmslev, Prolegomena to a Theory of Language, 29, 15, 22, 28. description—that which leads to the knowledge or comprehension. object—object of premised nature. what you want to know about. dependences—how objects relate. uniformity—form of dependences which do not change.

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Hjelmslev defined hierarchy as a class of classes and made a distinction between two sorts of hierarchies: processes and systems. Classes within a linguistic process he named chains, and components of a chain he named parts. Classes within a linguistic system he named paradigms, and components of a paradigm he named members. 677 PROCESS

THEORY

SYSTEM

CHAIN

CLASS

PARADIGM

PARTS

COMPONENTS

MEMBERS

Fig. 80

Class and component within the system/process perspectives.

Corresponding to the distinction between parts and members, he defined partition as the analysis of a process and articulation as the analysis of a system. Thus, the analysis consists of both partitioning the text and articulating the system. PROCESS

THEORY

SYSTEM

ANALYSIS

PARTITION

ARTICULATION

The first step of the analysis is to undertake a partition of a textual process. The analysis starts from an unanalysed text that is a chain. Then, the basis of analysis is selected: for example, the partition. The partitioning process defines and registers parts. This minimal chain of events 677 Hjelmslev, 29.

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Fig. 81

Analysis from the system/process perspective.

is called a single minimal partition, or single operation. In the next step, the obtained parts may again be seen as chains and the procedure can be repeated for as many times before it can no longer be achieved. Fig. 82

CHAIN

Elements of an analysis.

partition

SINGLE MINIMAL PARTITION

BASIS OF ANALYSIS PARTS

CHAINS SINGLE MINIMAL PARTITION

partition

PARTITION (ANALYSIS)

PARTS BASIS OF ANALYSIS

CHAINS

partition

SINGLE MINIMAL PARTITION

PARTS BASIS OF ANALYSIS

The description of the given object cannot be exhausted by a single partition, because every partition must choose a specific basis of analysis. This basis can vary, leading to different descriptions. This multiplicity of analyses from different standpoints he called the analysis complex: a class of analyses of one and the same class according to the selected bases of analysis. 678 An operation in which the determination 679 is taken as the basis of the analysis Hjelmslev defined as a procedure. 680 A procedure can, among other things, either consist of analyses and be a deduction or, on the other hand, consist of syntheses and be an induction. If a procedure consists of both an analysis and a synthesis, the relationship between them will always be a determination, in which the synthesis premises the analysis but not vice-versa. This means that the text had to be created 678 Hjelmslev, 30. 679 “Unilateral dependence in which the one term presupposes the other but not vice versa.” Hjelmslev, 24. 680 “Procedure is a class of operations with mutual determination.” Hjelmslev, 31.

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(synthesised) before its existence can be affirmed. Once obtained, we can only see it as an unanalysed whole because the system that was used to create it in the first place is unknown to us. PROCEDURE

DETERMINATION

Relation between deduction and induction.

INDUCTION

DETERMINATION

DETERMINATION

SYNTHESIS

ANALYSIS

DEDUCTION

Fig. 83

Operation 681 is one of the concepts which implements self-reflection. It was defined as something that leads to knowledge/comprehension by using the theory that has established that very definition. Thus, we can describe an operation as that which leads to knowledge or comprehension and at the same time guarantees that it will be free of contradiction, exhaustive and as simple as possible. If the analysis is seen as the repetition of the single minimal operation, then components (parts, members) can be seen as the derivates 682 of the class. By the degree of derivates, Hjelmslev refers to the “number of the classes through which they are dependent on their lowest common class.” 683 Any class at any stage, except for the last, possess its own relative hierarchy of derivates. Fig. 84

CLASS (LOWEST COMMON CLASS)

Derivates.

1st DEGREE DERIVATES (COMPONENTS)

2nd DEGREE DERIVATES (COMPONENTS OF 1st DEGREE DERIVATIVES)

3rd DEGREE DERIVATES (COMPONENTS OF 2nd DEGREE DERIVATIVES)

681 “Operation is a description that is in agreement with the empirical principle.” Hjelmslev, 31. 682 “By the derivates of a class we shall understand its components and components-ofcomponents within one and the same deduction.” Hjelmslev, 33. 683 Hjelmslev, 33.

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The idea of derivates bears similarity to the refinement of the spectrum in Zafiris’s partition paradigm. Each time the spectrum is refined it yields new distinctions that can be seen as the derivates of the previous refinement, as well as the derivates of the indiscrete partition from which the analysis starts. Functions and functives To formalise the notion of dependence, Hjelmslev introduced the notion of a function, as a dependence that fulfils the conditions for an analysis. Thus, there is a function between a class and its components. Terminals, that are the end points of the dependence, he named functives. A functive is at the same time terminal of a function and an object that has a function to another object. Accordingly, a functive is said to contract its function. Fig. 85

A and B are functives, contracting a function between them.

A

B

Hjelmslev’s term functive, and the mathematical term functor, seem to be closely related. A functive, which is the terminal of a relationship, can also be seen as the terminal of a functor, which is a mathematical generalisation of a relationship. Functives can also be functions, since there can be a function between functions. This just means that there can exist a function between, for example, two dependences.

Fig. 86

Functions can also take place of functives in a dependence.

?

?

?

?

To separate the functives that are not functions, Hjelmslev used the term entity. 684 It is important to notice that the term function is quite differently defined as in mathematics, where functions are operations applied on the sets. In mathematics one term is a function of another, meaning that for a single input, there only can be a single output. When applying function f to the input x, we obtain output y.

684 “A functive that is not a function we shall call an entity.” Hjelmslev, 33.

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MATHEMATICS

GLOSSEMATICS

Fig. 87

output y of the function f to the input x

y = f(x)

A

B

A

B

A

B

Function within mathematics and function within glossematics.

functive A has a function to the functive B

For Hjelmslev, the function implies that the functive A is related to the functive B. They have a dependence that fulfils conditions for an analysis. This also marks the difference between a set-theoretical and categorical approach. Dependences can also be described from the standpoint of the functives that contract them. We can start from a functive and examine what its presence (existence) means for the existence of another functive. In that respect: Constant is a functive whose presence is a necessary condition for the presence of the functive to which it has function; by a variable we shall understand a functive whose presence is not a necessary condition for the presence of the functive to which it has function. 685 In other words, constants are necessary, while variables are contingent functives. Fig. 88

A and B are constants.

A

B

From this perspective, interdependence can be seen as a function between two constants. Determination is a function between a constant and a variable. Constellation is a function between two variables. He defined functions containing at least one constant as cohesions. Functions which employed the two identical classes of terms—only constants or only variables—he named reciprocities. Fig. 89

INTERDEPENDENCE A

COHESIONS

Cohesions and reciprocities.

B

function between two constants DETERMINATION A

RECIPROCITIES

B

function between a constant and a variable CONSTELLATION A

B

function between two variables

685 Hjelmslev, 35.

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Determinations could be regarded from the standpoints of a constant or of a variable. In the most general way, one can say that a constant is determined and, a variable, a determining functive. Thus, the constant is determined by the variable and the variable determines it. Within the process, one can say that a constant is selected and, a variable, a selecting functive. Thus, the constant is selected by the variable and the variable selects it. Within the system, one can say that a constant is specified and, a variable, a specifying functive. Thus, the constant is specified by the variable and the variable specifies it. Fig. 90

Determination, selection, and specification from the functival perspective.

THEORY

PROCESS

SYSTEM

A

B

A

B

A

B

DETERMINED

DETERMINING

SELECTED

SELECTING

SPECIFIED

SPECIFYING

A is determined by B B determines A

A is selected by B B selects A

A is specified by B B specifies A

Relation and correlation Some functions involve functives that enter the process, while others involve functives that enter the system. Hjelmslev described them respectively as both-and functions and either-or functions according to their relation to the logical and linguistic terms conjunction and disjunction. Both-and functions are characterised by the coexistence of functives that enter the process (text), and either-or functions by their alteration. PROCESS

SYSTEM

“both-and” function

“either-or” function

CONJUNCTION

DISJUNCTION

coexistence between the functives entering therein

alteration between the functives entering therein

Fig. 91

Hjelmslev’s distinction between conjunction and disjunction.

Hjelmslev’s notions of conjunction and disjunction have some similarity to the corresponding terms in logic. When analysing a text, the exact same entities enter both the system and the process. Therefore, the exact same entities enter both the text and language. This can be seen with the example of two chains: pet and man. Here, an alternation between p and m enters the system. Meaning that it defines a potential scope for creating chains such as pet or pan by the virtue of disjunction (alternation).

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Fig. 92

p e t

Hjelmslev’s example of a conjunction and a disjunction.

m a n DISJUNCTION:

p↔m, e↔a, t↔n paradigms which enter into the linguistic system

CONJUNCTION:

pet, pen, pat, pan, met, men, mat, man chains that enter into the linguistic process (text)

From the point of view of the process, p is a part and from the point of view of the system, p is a member. Therefore, all functions of language enter into both process and system, thus contract both conjunction, or coexistence, and disjunction or alternation. Their definition will depend on the point of view we take. p e t

Fig. 93

Conjunction and disjunction within the same object.

m a n PROCESS:

p as a component (derivate) of pet enters into a process (conjunction)

SYSTEM:

p as a component (derivate) of p↔m enters into a system (disjunction)

Since all of these terms imply a lot of different meanings, Hjelmslev decides to name them relation and correlation. In respect to the previous example, we can say that p, e, and t contract relation, while p and m contract correlation. 686 Thus, p, e and t taken together are relates, while p and m are correlates. p e t

RELATION

p e t

p e t

RELATES

p e t

— m a n

m a n CORRELATION

m a n

m a n CORRELATES

This implies that the system is a correlational hierarchy, while the process is a relational hierarchy, where hierarchy is simply a class of classes.

686 “We shall thus understand by correlation the either-or function, and by relation the both-and function.” Hjelmslev, 38.

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Fig. 94

Relates and correlates.

In a process, there is always a coexistence of functives present, while in a system, there is an alteration. Thus, all functives enter both processes and systems and contract both relations and correlations. Hjelmslev is now able to evolve his definition system that started with the system-process distinction. Instead of speaking of a semiotic process and a semiotic system, he can now speak of a paradigmatic, and a syntagmatic. SYSTEM

Fig. 95

Extension of the system/process distinction.

PROCESS The most general distinction in the world

LANGUAGE

TEXT

… when talking about language in the ordinary sense of the word

PARADIGMATIC

SYNTAGMATIC

… when talking about semiotic process and semiotic system

The process/system distinction can be represented more clearly from the standpoint of constants and variables. It becomes clear that the process determines the system, which in fact means that the existence of a system is a necessary condition for the existence of the process. Therefore, text cannot exist without a language laying behind it, but it is possible to have a language without a text constructed in that language. In that case, language is possible but textual process is virtual: it is possible but not actually realised. 687 The notion of virtuality exists in Zafiris’s partition paradigm as well. The distinctions that are obtained by means of equivalence classes he calls virtual distinctions. Finally, from the perspective of their affiliation with either process of a system, relationships can be seen as relations and correlations. Fig. 96

Relation and correlation.

RELATION (CONNEXION)

CORRELATION (EQUIVALENCE)

PROCESS

SYSTEM

A

B

A

B

A

B

SOLIDARITY SELECTION COMBINATION

A

B

A

B

A

B

COMPLEMENTARITY SPECIFICATION AUTONOMY

687 “On this basis we call a class realised if it can be taken as the object of particular analysis, and virtual if this is not the case.” Hjelmslev, 40.

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Elements inventory: Theory database One of the potential problems Hjelmslev faced with hierarchies was the alignment of functives belonging to different levels of abstraction. Some entities can, at the same time, be viewed as a sentence, clause, or a word. 688 According to what was registered and how that relates to other functives within the chain, there will be consequences of registering any specific functive at a certain stage. For that reason, Hjelmslev introduced a rule of transference: For this purpose, we must introduce a special rule of transference, which serves to prevent a given entity from being further analysed at a too early stage of the procedure and which ensures that certain entities under given conditions are transferred unanalysed from stage to stage, while entities of the same degree are subjected to analysis. 689 Hjelmslev introduced the concept of an inventory and joined it to each stage of the procedure. The inventory gathers entities existing on the same level, which employ the same relations. 690 Hjelmslev noticed that the size of various inventories can vary, and usually decreases as the procedure advances. This is because there are many more possible combinations on the level of clauses and sentences than on the level of words. Moreover, there are also many more words than syllables, and many more syllables than phonemes. It is important to consider that even though most of the inventories will be unrestricted, it is still possible to keep track of them by means of inventories. Google utilises this principle by making an inventory of words taking part within bi-grams, tri-grams, four-grams, etc., for the purpose of translation. Nonetheless, for Hjelmslev, it is still of crucial importance to end up with the restricted inventories at some point. Without restricted inventories, he concluded, “linguistic theory could not hope to reach its goal, which is to make possible a simple and exhaustive description of the system behind the text.” 691 relations between the language and the world Sign function To challenge Saussure, Hjelmslev needed to address the very important linguistic notion of a sign within his theory. The traditional, vague conception of the term sign is described as “a sign for something else.” According to this definition, Hjelmslev tried to challenge the traditional idea that a language is a system of signs. He discovered that this assertion is true not only on the level of words and word groups,

688 “an entity can sometimes be of the same extension as an entity of another degree.” Hjelmslev, 43. 689 Hjelmslev, 41. 690 “In each single partition we shall be able to make an inventory of the entities that have the same relations, i.e., that can occupy the same “position” in the chain.” Hjelmslev, 41. 691 Hjelmslev, 42.

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but also within one word. As an example, he showed how the word inactivates contains five signs and described their contextual meaning. inactivates

Fig. 97

Hjelmslev shows how the word “inactivates” consists of five different signs.

in act iv ate s 4 5 2 3 1

From this follows that meaning is a purely contextual notion. None of the elements in the word inactivates have an independent existence, as they are all defined relatively to each other. If all the unrestricted inventories seem to contain signs, the question is, what happens when we reach restricted inventories? 692 With the example of the word inactivates, Hjelmslev showed that further analysis would lead to a discovery of two different objects: one being a sign and other being a phoneme. For Hjelmslev this was an indicator that we should not try to break language into signs, because the same object can be two or more things at the same time, depending on the perspective taken. This is why he suggested that content and expression should be analysed separately, each of them with their own restricted inventories. These two inventories would not be necessarily matching each other, even though they are two faces of the same substance-purport. In the natural communication model, this would correspond to the idea of adjunction, where communication bridges would not be exact, but conceptual inverses. If a language is to be an adequate system of signs, it must always permit construction of new signs/words/roots. The language should also be practical to use, easy to learn, and possible to master. Hjelmslev’s strategy to achieve this contained a proposition that signs must be constructed out of a restricted number of entities that are in fact not signs, but whose combinatory potential allows creating an unlimited number of signs: Such non-signs as enter into a sign system as parts of signs we shall here call figuræ. 693 From the perspective of figuræ, each word was not only a sign for something else but also a substance or “matter” put together. For him, this was essential for the structure of any language. 694 From this perspective, languages cannot be described as purely sign systems. Rather, languages are systems of figuræ that can be used to construct signs.

692 For example, what happens when we continue partitioning words more and more and when we end up with phonemes? 693 Hjelmslev, 46. 694 Hjelmslev, 47.

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As an alternative to Saussure’s system of signs, Hjelmslev introduced a sign function. It defines a solidarity between two entities, an expression and a content, which renders them as mutually dependent constants. 695 CONTENT

EXPRESSION

Fig. 98

Sign function is a solidarity between expression and a content.

SIGN FUNCTION

The proposition that there cannot be content without an expression, and vice versa 696, bears similarity to Zafiris’s view that everything observable is encoded in geometric structures. Hjelmslev advised that without the sign function, it would in fact be impossible to delimit signs from each other and to provide an exhaustive description. On that account, a sign function is what permits us to discriminate signs from each other. Unlike process and system, which are related by a determination, content and expression are interdependent, which grants them dual identity. This bears a similarity with the mathematical notion of conjugation. It is not possible, or even valid, to say which conjugate domain is the “true” code of the phenomena, because all of them are different encodings of the same inaccessible thing. Formation of the substance To make a parallel with the natural communication model, the sign function can also be seen from the perspective of the motivic key. This allows us to introduce a new term, the sign key, involving sign function and contracting functives. In that case, the first dimension of the key is introduced by the sign function, which establishes the solidarity between content and expression (Figure 111). Hjelmslev adds that both content and expression are only apparent as substances. 697 This is the second dimension of the sign function that describes how both content and expression are formed into observables. Metaphorically speaking, form can be 695 “Thus there is also solidarity between the sign function and its two functives, expression and content. There will never be a sign function without the simultaneous presence of both these functives; and an expression and its content, or a content and its expression, will never appear together without the sign function’s also being present between them.” Hjelmslev, 48. 696 “there can be no content without an expression, or expressionless content; neither can there be an expression without a content, or content-less expression.” Hjelmslev, 48. 697 Zafiris would say: encoded in geometric structures.

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thought of as a mould and a substance—that which is moulded. Form is always given to the substance and without it, substance is inaccessible to us. Object-oriented programming languages use a similar but less abstract distinction between a class, which is thought of as a blueprint, and an object, which is an actual instance of that class. 698 For Zafiris, the process of formalisation appears when two categories are connected by adjoint functors, where one category can be seen as giving form to another. 699 If language is always accessible as a formed substance (i.e., text, speech, thoughts…), then what can be said about the unformed substance? Hjelmslev noticed that whatever this substance might be, it would in any case be a factor which is common to all languages. Such a factor can only be defined by having a function to the structural principle of the language (expression and content), and to all the other factors that make languages different from each other. This factor he named a purport. 700 The basic idea of a purport is illustrated in Figure 99. The figure contains five sentences (which are in fact expression-substances) in different languages, which all share the same meaning: Fig. 99

“I don’t know” in five different languages.

jeg véd det ikke

Danish

I do not know

English

je ne sais pas

French

en tiedä

Finnish

naluvara

Eskimo–Aleut

These sentences look nothing alike and seem to be differently constructed. Words, word groups, and parts of the words within these different languages, stratify and relate the world they tend

698 “Object-oriented programming is a method of implementation in which programs are organized as cooperative collections of objects, each of which represents an instance of some class, and whose classes are all members of a hierarchy of classes united via inheritance relationships.” Booch, Object-Oriented Analysis and Design with Applications, 41. 699 Zafiris, Natural Communication. For example, one level could be the level of atoms, which is observational universe where the movement of electrons cannot be grasped; second level would be microscopic scale, where the interference patterns could be observed. Functors established between two levels could be seen as giving form to the not-directly accessible scale. 700 “…a principle that is naturally common qua principle to all languages, but one whose execution is peculiar to each individual language.” Hjelmslev, Prolegomena to a Theory of Language, 50.

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to describe in a completely different way. 701 From the standpoint of the purport, the important point is not how they are constructed (ciphered), but rather that they convey the same meaning (geometry/code), by means of which they can be (algebraically) related to each other. Each language lays down its own boundaries within the amorphous “thought-mass” and stresses different factors in it in different arrangements, puts the centres of gravity in different places and gives them different emphases. 702 In relation to form and substance, purport has the existence only in being the substance for one form or another. If not formed, it has no scientific existence. 703 What Weyl describes when saying that we do not have a direct access to nature, Hjelmslev formulates by saying that we do not have direct access to the unformed substance. Therefore, this “unknown” can only be characterised by its external functions. The open question remains whether this linguistic purport and non-linguistic purport are actually the same thing, which Hjelmslev does not attempt to answer until the end of the book. Figure 100 illustrates the principle of formation. It starts with a form which is instituted by the sign function as a solidarity between content and expression. It is independent of purport and stands in an arbitrary relation to it. Since it governs the process of formation, it is the very thing that makes possible the distinction of substances from each other and that makes them observable as a substance.

701 Hjelmslev, 51. For example, in English and in French you have the word I. But in Finish you have the word which means not-I. And in Eskimo, this is a verb meaning “notknowing-am-I-it” derived from word ignorance with the suffix for the first-person subject and third-person object. 702 Hjelmslev, 52. 703 Hjelmslev, 52.

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fig. 100

Illustration of the formation principle. FORM

PURPORT

SUBSTANCE

The process of formation takes place both in the content and expression and can be seen from both perspectives of a process and of a system. This indicates that the sign key is viewed from the perspective of determination, in which the system determines the process. fig. 101

CONTENT IN ITS PROCESS

Different perspectives on the process of formation.

CONTENT FORM

EXPRESSION FORM

CONTENT PURPORT

EXPRESSION PURPORT

CONTENT SUBSTANCE

EXPRESSION SUBSTANCE

SYSTEM OF A CONTENT

176

EXPRESSION IN ITS PROCESS

CONTENT FORM

CONTENT PURPORT

SYSTEM OF AN EXPRESSION

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EXPRESSION PURPORT

CONTENT SUBSTANCE

SYSTEM OF A CONTENT

EXPRESSION SUBSTANCE

SYSTEM OF AN EXPRESSION

CONTENT FORM

EXPRESSION FORM

CONTENT PURPORT

EXPRESSION PURPORT

CONTENT SUBSTANCE

EXPRESSION SUBSTANCE

coNTENT iN iTS ProcESS Hjelmslev describes the notion of a content first and starts from the process perspective. He takes the process/system perspective to demonstrate how processes of content are determined by the systems of content. On the basis of the formation principle within the process, Hjelmslev recognises content-form, which is independent of, and stands in arbitrary relation to, the purport and forms it into a content-substance. The process perspective entails that content can be seen as an effect or manifestation of some unrevealed system behind it. fig. 102

Content form and content substance from the perspective of the process.

CONTENT FORM

CONTENT PURPORT

CONTENT SUBSTANCE

Hjelmslev gives an example of a thought chain. The formation principle implies that the only way a thought can have a scientific existence is as a formed substance. This assumes that it is possible to recognise it from other thoughts or a lack of them. An example of such a thought could be that which is expressed as “je ne sais pas,” or “I don’t know.” The form of thoughts would then

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be a specific projection of the purport, or what makes it possible to recognise such a thought-substance amid the others. The process perspective suggests that thought must be an effect of some kind of a system that makes thinking articulable. 704 Therefore, the existence of thoughts presupposes a system of thinking (or an organ of thinking) that incorporates reality in some way. SySTEM of a coNTENT After investigating the content in its process, Hjelmslev illustrates the system of a content. Similarly, on the basis of the formation principle, within the system, Hjelmslev recognises content-form which is independent of, and stands in arbitrary relation to, the purport and forms it into a content-substance. The system perspective entails that the existence of such a content is a necessary condition for the existence of certain kinds of processes. fig. 103

Content form and content substance from the perspective of the system.

CONTENT FORM

CONTENT PURPORT

CONTENT SUBSTANCE

Hjelmslev provides different paradigms 705 as examples: Paradigms of colour are specific articulations within the colour spectrum, which shows the possibilities of purport formation. 706 These paradigms exclusively depend on the organs of sight and the interpretation of their activity. A person with “normal” vision should be able to recognise that blue is something different than red. Once this distinction is made, these colours exist as paradigms of content-substance. Hjelmslev shows how each language sets the boundaries on the purport arbitrarily, as shown 704 …for example brain activity. 705 “Paradigm—class within a semiotic system.” Hjelmslev, 29, 53. 706 “Behind the paradigms that are furnished in the various languages by the designations of color, we can, by subtracting the differences, disclose such an amorphous continuum, the color spectrum, on which each language arbitrarily sets its boundaries.” Hjelmslev, 52.

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on the illustration below. By looking the word “glas” in Welsh, it is clear that it is not possible to translate it literally to English. 707 Therefore, in every translation there is a loss of information which needs to be compensated by creating new information. gwyrdd green blue

glas

Fig. 104

Image of a color spectrum and expressions made on top of it in English and Welsh.

gray llwyd brown

Paradigms of morphemes are specific articulations of the conception of identity from the perspective of quantification. Concepts in different languages can be differentiated from each other, on the basis of being a single thing (singular) or a collection (plural). Purport, whose possibilities of formation we see represented within the spectrum of all things in relation to their number, allows arbitrary characterisation in this respect. Therefore, some languages differentiate three, even four, different zones of morphemes according to the quantity of things that a concept refers to. Fig. 105

træ skov

Baum

arbre

Holz

bois

Wald

Morpheme zones for numbers in Welsh, German, and French. (adapted from Hjelmslev, p.54)

forêt

Paradigms of time are specific articulations within the spectrum of temporality. Tenses in different languages set their own boundaries in a form of relative timelines on the purport. Thus, certain languages that are structured around time, like English, consist of a lot of tenses. For the native English speaker, differentiating the time of an event in relation to other events is a very important aspect of communication. In contrast to it, some languages, like Japanese, integrate time in a more linear fashion. This implies that language seems to be actually shaping our conception of time. 707 due to the two different projections on a purport.

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SySTEM of aN ExPrESSioN Expression is presented from the system perspective. Hjelmslev takes the process/system perspective to demonstrate how processes of expression are determined by the systems of expression. On the basis of the formation principle within the system, Hjelmslev recognises expression-form, which is independent of, and stands in arbitrary relation to, the purport and forms it into an expression-substance. The system perspective entails that the existence of such an expression system is a necessary condition for the existence of certain kind of processes. fig. 106

Expression form and expression substance from the perspective of the system.

EXPRESSION FORM

EXPRESSION PURPORT

EXPRESSION SUBSTANCE

Hjelmslev gives an example with the paradigm of phonetic zones: Paradigms of phonetic zones are specific articulations of the vocalic continuum permitted by the anatomy of the vocal tract, in which different classes of sounds-expressions can be produced. 708 From the perspective of a sound spectrum, these articulations are arbitrary. However, they are not at all arbitrary from the perspective of our hearing organs, which are sensitive to the frequencies of their oscillation. Without ears, it would not be possible to discriminate shifting frequencies of the mechanical wave produced in the mouth. It is also feasible to imagine different hearing organs, which would develop alternative sensibilities to these oscillations and indicate that the substance has taken a different form. Languages take advantage of the ability of the vocal tract to produce a variety of sounds and to arbitrarily set up the boundaries on the spectrum. This is why every language is equipped with a different set of sounds.

708 International Phonetic Association, “Full IPA Chart.” International Phonetic Alphabet recognizes 11 such zones: Bilabial, Labiodental, Dental, Alveolar, Postalveolar, Retroflex, Palatal, Velar, Uvular, Pharyngeal and Glottal.

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fig. 107

The human vocal tract can be divided into different zones according to how and where different sounds are produced in speech. (Gray, 1918)

ExPrESSioN iN iTS ProcESS After investigating the system of expression, Hjelmslev illustrates expression in its process. Similarly, on the basis of the formation principle within the process, Hjelmslev recognises expression-form, which is independent of, and stands in arbitrary relation to, the purport and forms it into an expression-substance. The process perspective entails that expressions can be seen as an effect or manifestation of some unrevealed system that makes them possible. fig. 108

Expression form and expression substance from the perspective of the process.

EXPRESSION FORM

EXPRESSION PURPORT

EXPRESSION SUBSTANCE

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Purely by virtue of the cohesion between the system and the process, the specific formation of the system in a given language inevitably involves effects in the process. One and the same expression-purport may be formed differently in different languages. 709 However, most of the expression-substances are closely related to the specific languages in which their pronunciation is trained. This corresponds to the idea of a mother-tongue. Fluency, a characteristic of a mother tongue, means that a speaker has totally mastered, thus naturalised the boundaries of expression that the language sets in purport. When learning a new language, one tries to reproduce its expressions, although they often come substantiated by means of a different form that does not match the substance of learner’s mother-tongue. This is what is usually understood as speaking with an accent. 710 Hjelmslev gives two examples of this: Chains of the different expression-purport where the content-purport is the same are illustrated by the pronunciation of the word Berlin in different languages. As Figure 109 shows, the phonetic transcription varies very much even though the content-substance (the city of Berlin) is the same. Fig. 109

Different pronunciations of the city name Berlin.

[bə'lɪn]

English

[bɛ 'liːn]

German

[bœ ’li?n]

Danish

[beiuinu]

Japanese

['bɛrlin]

Hungarian

Chains of the same expression-purport but a different contentpurport are exemplified with the different writings of the same sound chain. Hjelmslev gives example with the words got, Gott, and godt in English, German and Danish. Fig. 110

Three words pronounced the same but standing for a different content.

got

simple past of get

English

Gott

God

German

godt

good

Danish

709 Hjelmslev, Prolegomena to a Theory of Language, 56. 710 “speaking with an accent consists in forming a perceived expression-purport according to predispositions suggested by functional facts in the speaker’s mother tongue.” Hjelmslev, 56–57.

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Anatomy of a sign To complete the definition of the sign, Hjelmslev combines both the sign-function and the formation principle. He concludes that the sign is a sign for both contentsubstance and expression-substance. This implies that the sign is a unity of both content-form, which we recognise by finding it projected into the content-substance, as well as the expression-form, which we recognise by finding it projected onto the expressionsubstance. 711 A sign unites the form of every content-substance (object) that can be classified under the same expression-form by any possible symbolisation or pronunciation. Figure 111 provides an illustration inspired by Hjelmslev’s description. 712 It is important to keep in mind that such a diagram can be seen from both a process and a system perspective. Fig. 111

FORM

CONTENT

Diagram showing the relations between the notions of form, substance, content and expression.

EXPRESSION

SUBSTANCE

Finally, Hjelmslev defines a sign as a unit consisting of contentform and expression-form established by the solidarity of the sign function. 713 The sign must thereby be used such that it shows clearly the relation between the sign and the designatum.

711 “That a sign is a sign for something means that the content form of a sign can subsume that something as content-substance.” Hjelmslev, 57. 712 “The sign is a two-sided entity, with a Janus-like perspective in two directions, and with effect in two respects: “outwards” toward the expression-substance and “inwards” toward the content-substance.” Hjelmslev, 58. 713 Hjelmslev, 58.

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The notion of a sign that relates four different entities corresponds well with Zafiris’s notion of the motivic key, which relates two conjugate domains by means of two cryptographic bridges. W

FORM

Fig. 112

Diagrams of Hjelmslev’s sign function and Zafiris’s motivic key.

CONTENT

EXPRESSION

S +1

S -1

T

SUBSTANCE

In this respect, content and expression could be seen as two different domains giving form to each other. The process of formalisation could be seen as a form of communication. Content / Expression planes For Hjelmslev, the first stage of analysis must be the analysis into expression-form and content-form, as they are the parts offering the greatest extension. 714 If the text (process) is divided into content and expression, Hjelmslev refers to the results of the division as an expression line and a content line. The same gesture in the system yields respectively an expression side and a content side. 715 In order to provide three sets of terms, he refers to the content line and the content side as a content plane. In the same manner, the expression line and the expression side are referred to as an expression plane. Fig. 113

Content and expression plane from the process/ system perspectives.

THEORY

CONTENT PLANE

EXPRESSION PLANE

SYSTEM

CONTENT SIDE

EXPRESSION SIDE

PROCESS

CONTENT LINE

EXPRESSION LINE

714 “When this principle is carried through, it will appear that any text must always be analysed in the first stage into two and only two parts, whose minimal number guarantees their maximal extension: namely, the expression line and the content line, which have mutual solidarity through the sign function.” Hjelmslev, 59. 715 Hjelmslev, 59.

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generalisation and inventorying processes Towards restricted inventories The analysis starts from the division of the text into two planes: content and expression. The procedure continues with the partition and articulation of the planes in a number of stages. At a certain point in the analysis, Hjelmslev expects to reach entities which are basic to the system, out of which all others can be constructed. The number of these entities should be as low as possible. To allow for this generalisation, he introduced two principles, which he deduced from the simplicity principle: the principle of economy and the principle of reduction. The principle of economy: The description is made through a procedure. 716 The procedure shall be so arranged that the result is the simplest possible, and shall be suspended if it does not lead to further simplification. 717 The principle of reduction: Each operation in the procedure shall be continued or repeated until the description is exhausted, and shall at each stage lead to the registration of the lowest possible number of objects. 718 Objects that are inventoried at each stage of the procedure he calls elements. Hjelmslev’s inventories bear resemblance to Zafiris’s idea of sets of virtual and actual distinctions, while his elements correspond to Zafiris’s entities or types (eidos). In order to account for the frequent appearance of the same kind of objects in language, Hjelmslev requires a procedure in which at least two elements are reduced to one. For him this was attainable because in language often “we have -one and the same- sentence, -one and the same- clause, -one and the same- word, etc.” 719 In other words, in language there are many specimens of each sentence, each clause, each word, etc. These specimens he named variants, and the entities of which they are specimens, invariants. This specification is valid for functions too. What becomes important is how the invariants are recognised from variants, and a method that establishes the criteria for the reduction.

716 717 718 719

“Procedure is a class of operations with mutual determination.” Hjelmslev, 31. Hjelmslev, 61. Hjelmslev, 61. Hjelmslev, 62.

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Fig. 114

SPECIMEN INSTANCE

Variants and invariants.

SPECIMEN INSTANCE

“CLASS” OF SPECIMENS

SPECIMEN INSTANCE SPECIMEN INSTANCE

INVARIANT

VARIANTS

Variants and invariants To understand whether two elements are invariants or variants of the same invariant, Hjelmslev compared how the changes in the content plane correspond to the changes in the expression plane. He gave an example of two chains, pet and pat, which is illustrated in the following figure. If the difference between pet and pat, established by the correlation 720 between e and a, entails differences within the content plane (or vice versa), then these two chains are invariants. 721 Fig. 115

Process of determining whether two entities are variants or invariants.

CONTENT

EXPRESSION

RELATION

pet

p e t

e CORRELATION

pat

CONTENT PLANE

p a t

a

EXPRESSION PLANE

720 “either-or” function, alteration, equivalence. 721 “There is a difference between invariants in the expression plane when there is a correlation to which there is a corresponding correlation in the content plane so that we can register a relation between the expression-correlation and the content-correlation.” Hjelmslev, 65.

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Correspondingly, the same question can be asked from the perspective of the content: if there are differences in content, what does this entail in the expression plane? 722 Hjelmslev added that this reasoning was not valid only for word chains, but for all the invariants of the language: sentences, clauses, etc. Figure 116 illustrates the general case of such an exchange: Fig. 116 CONTENT

Generalised schema describing the process of determining whether two entities are variants or invariants.

EXPRESSION

RELATION

x1 x2 x3

x1 x2 x3

x2 CORRELATION

y1 y2 y3 y4

y1 y2 y3 y4

CONTENT PLANE

y2 y3

EXPRESSION PLANE

To make a parallel with the natural communication model, we can say that if two functives are variants, they should belong to the same equivalence class. Thus, they would be discriminated under the same bridges. Invariants, on the other side, would correspond to entities belonging to different equivalence classes, thus discriminated under different kind of bridges. The aim of inventorying Mechanical inventorying is applied in the content plane in the same way as in the expression plane. Hjelmslev gave an example of the entities of content, which are listed in Figure 117. He explained how in the particular stage of the procedure, entities marked in light gray—ram, ewe, man, woman, boy, girl, station and mare—must be eliminated from the inventory of elements, because they can be described univocally as relational units that include elements he, she, sheep, human being, child and horse.

722 “Thus, in practice there are two different invariants of content if an exchange of one for the other can entail a corresponding exchange in the expression plane.” Hjelmslev, 66.

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Fig. 117

ram

Example of the process of mechanical inventorying.

ewe

st a l l i o n

man mare

child

sheep

horse

ram

ewe

man

he-sheep

she-sheep

he

st a l l i o n he-horse

child

woman

mare

she-horse

he

she

woman she

sheep

horse

horse

girl

human being

boy

girl

he-child

she-child

human being he

sheep child

boy

she

human being he

she

He demonstrated that ram and he-sheep are variants, by applying the same procedure he did with expression-chains. In this example, differences in the expression plane do not correspond to the differences in the content plane, as ram and he-sheep are different expressions of the same content. Fig. 118

CONTENT

Variants and invariants in the content plane.

EXPRESSION

RELATION

x ram y

x ram y

ram CORRELATION

x he-sheep y

CONTENT PLANE

x he-sheep y

he-sheep

EXPRESSION PLANE

The aim of an inventorying process is to generalise as much as possible within each stage of the analysis, by reducing the actual specimens to variants, and variants to invariants. Reduction is performed until the

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restricted inventories are reached. These inventories can be then seen as the combinatorial basis for creating larger elements. 723 The reduction yields another interesting consequence. If an entity can be reduced to an invariant or exchanged with the variant corresponding to the same content, both the invariant and the variant can be said to define this entity. 724 This implies that the definition of any object is possible due to its relational property. 725 Interplay between the planes: Mutation Given that variants and invariants both depend on the relation between expressioncorrelation and content-correlation, Hjelmslev needed to provide a formal definition. The correlation in one plane, which has a relation to a correlation in the other plane of language, he called commutation. A relation and a shift within a chain, which have a relation to the corresponding relation and to the corresponding shift within a chain in the other plane of a language, he referred to as permutation. Hjelmslev chose the term mutation to generalise the notions of commutation and permutation. Commutation is defined as a mutation within a system; permutation as mutation within a process. 726 PROCESS

THEORY

SYSTEM

PERMUTATION

MUTATION

COMMUTATION

Fig. 119

To formally define mutation, Hjelmslev first needed to define the context in which the mutation applies. To establish such a context, he introduced the notion of a rank of derivates. If derivates of a class are of the same degree and belong to the same process or system, they constitute a rank. We can speak of mutation only if the derivates belong to the same rank, meaning that they are on the same level of abstraction. Mutation is then a “function existing between first-degree derivates of one and the same class, a function that has relation to a function between other first-degree derivates of one and the same class and belonging to the same rank.” 727 723 “Thus, in practice the procedure consists in trying to analyze the entities that enter the unrestricted inventories purely into entities that enter the restricted inventories.” and “The task will then consist in carrying the analysis further until all inventories have been restricted, and restricted as much as possible.” Hjelmslev, 71. 724 “But that which is established as equivalent to a given entity, when that entity is so reduced, is actually the definition of that entity, formulated in the same language and in the same plane as that to which the entity itself belongs.” Hjelmslev, 72. 725 “Analysis we can then define formally as description of an object by the uniform dependences of other objects on it and on each other.” Hjelmslev, 38. 726 Hjelmslev, 73. 727 Hjelmslev, 74.

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Mutation from the system/process perspectives.

THEORY

Fig. 120

Mutation.

content plane

MUTATION relation

CLASS

COMPONENT

relation

COMPONENT

expression plane

MUTATION

He defined commutation as a mutation between the members of a paradigm, and permutation as a mutation between the parts of a chain. Fig. 121

SYSTEM

PROCESS

Commutation and permutation.

content plane

PERMUTATION

COMMUTATION

relation

correlation

CHAIN

PARADIGM

PART

PART

MEMBER

relation relation

PERMUTATION

MEMBER

correlation expression plane

relation

COMMUTATION

In the absence of mutation between the members of a paradigm, Hjelmslev speaks of substitution.

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SYSTEM

Fig. 122

Substitution. content plane

COMMUTATION correlation

PARADIGM

MEMBER

MEMBER

correlation relation

SUBSTITUTION

✘

expression plane

From the standpoint of mutation, Hjelmslev redefined the notions of invariants and variants. Invariants can be defined as correlates with mutual commutation, and variants as correlates with mutual substitution. 728 In order to address the character of a structure of an individual language that makes it different from another language, Hjelmslev introduced the commutation test. This test establishes the number of invariants that enter into each category of the language. 729 He returned to the example of paradigms of morphemes and addressed it from the perspective of variants and invariants: The content elements ‘tree’ and ‘wood (material)’ are variants in Danish but invariants in German and French; the contentelements ‘wood (material)’ and ‘wood (forest)’ are invariants in Danish but variants in French. 730 Fig. 123

træ skov

Baum

arbre

Holz

bois

Wald

Variants and invariants need not be matching in different languages.

forêt

728 Hjelmslev, 74. 729 “The number of invariants within each category is established by the commutation test.” Hjelmslev, 74. 730 Hjelmslev, 74.

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Hjelmslev argued that before him, linguistics did not understand the role of commutation, which is why expression was taken to be the starting point of an analysis. He argued that language operates on both planes, of content and of expression, and that their interplay was of utmost importance. 731 linguistic and non-linguistic phenomena Similarity and difference between languages Hjelmslev suggested that the similarity between languages is entirely independent of factors external to language. He defined similarity to be their structural principles and differences to be the concrete rendering of those principles. 732 Therefore, both similarities and differences depend on the principle of formation and not on the substance that is formed. What is said to be common to all languages and described as purport—the unknown, uninformed substance—is not a positive entity, but something susceptible to any possible formation whatsoever. Purport can thereby not be addressed directly, but only through its formations. 733 Our senses and reason allow us to access only the specific formations of the purport. An example of such a sense is sight within the context of the visible spectrum of light. Without organs of sight we would not be able to speak about light in terms of a sensory experience, thus the conception of light would be completely different. Hjelmslev saw this as the reason why linguistics should not take substance as the basis of the analysis and the reason for its failure to describe language. What separates linguistics from non-linguistics If language is a form, then it is the task of linguistics to analyse it. This suggests that the form has a function to something that is not part of language 734 and that is a substance. Hjelmslev suggested that the substance should be the object of investigation of all non-linguistic sciences, for example physics, or anthropology. When one claims to have knowledge of certain phenomena, it means that one has extracted their forms by

731 “The important thing is that, whether at the moment we are interested especially in the expression or especially in the content, we understand nothing of the structure of a language if we do not constantly take into first consideration the interplay between the planes.” Hjelmslev, 75. 732 “The purport is formed in a specific fashion in each language, and therefore no universal formation is found, but only a universal principle of formation.” Hjelmslev, 76. 733 “The purport is therefore in itself inaccessible to knowledge, since the prerequisite for knowledge is an analysis of some kind; the purport can be known only through some formation, and thus has no scientific existence apart from it.” Hjelmslev, 76. “Differences between languages do not rest on different realizations of a type of substance, but on different realisations of a principle of formation, or, in other words, on a different form in the face of an identical but amorphous purport.” Hjelmslev, 77. 734 “language is a form and that outside that form, with function to it, is present a nonlinguistic stuff, Saussure’s ‘substance’— the purport.” Hjelmslev, 77.

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“mining” the purport, but also their linguistic forms, which is part of the language. Therefore, to have scientific knowledge also means that it is possible to communicate it; otherwise it cannot be considered to be knowledge. For that reason, Hjelmslev suggested that linguistic and nonlinguistic descriptions must be undertaken independently of each other. If, on one side, we have linguistic entities and, on the other, nonlinguistic entities, the substance of both planes can be considered in two ways: as the physical entity and as the conception held by the users of the language. 735 This is why Hjelmslev would not have said that the analysis of the linguistic content purport simply belonged to the domain of physics. For him, all sciences deal with the purport by means of substances, thus they all must deal with a linguistic content. 736 He concludes, very ambitiously: With the relative justification provided by a particular point of view, we are thus led to regard all science as centred around linguistics. 737 In this regard, it is very important to realise that the linguistics Hjelmslev is referring to is not the established linguistics of his time, but rather his own attempt to establish a new conception of science based on an algebraic, linguistic basis. Glossematics: an algebra of language The fact that linguistics deals exclusively with form has another important consequence: The kind of substance it orders is irrelevant. What is necessary for linguistics is to be able to symbolise substance as a variable. 738 This also means that the science of expression of such linguistic should symbolise its phonetic and phenomenological premises in the same way. 739 Such a linguistics would be one whose science of expression is not phonetics, and whose science of content is not semantics. It would be an immanent algebra of language. 740 To distinguish his attempt of establishing a new theory of language from previous attempts, Hjelmslev gave it a new name: glossematics. It comes from the term glossemes which stand for the irreducible invariants of his theory of language. 741

735 This double existence was already previously described with the notion of figuræ. 736 “An exhaustive description of the linguistic content-purport actually requires a collaboration of all the non-linguistic sciences; from our point of view, they all, without exception, deal with a linguistic content.” Hjelmslev, 78. 737 Hjelmslev, 78. 738 Similarly as in mathematics, the symbol x is often taken as an unknown variable. 739 “just as the various special, non-linguistic sciences can and must undertake an analysis of the linguistic purport without considering the linguistic form, so linguistics can and must undertake an analysis of the linguistic form without considering the purport that can be ordered to it in both planes.” Hjelmslev, 78. 740 “Such a science would be an algebra of language, operating with unnamed entities, i.e., arbitrarily named entities without natural designation, which would receive a motivated designation only on being confronted with the substance.” Hjelmslev, 79. 741 “We use glossemes to mean the minimal forms which the theory leads us to establish as bases of explanation, the irreducible invariants.” Hjelmslev, 80.

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Interplay between linguistic and non-linguistic Thus far, we have been speaking about linguistic form and linguistic substance or purport. However, the notions of form and substance also bear a more general meaning. A non-linguistic analysis of the purport should also lead to the recognition of the form. Such a form would be quite similar to the linguistic form, but of different nature. For this reason, science should also be interested in the form of any particular purport formation. Form is thereby rendered as a relational concept and substance is metaphysical, suggesting that it is possible to recognise different hierarchies when mining a purport. Some will be linguistic hierarchies, and some non-linguistic. For Hjelmslev, it is obvious that there needs to be functions between such hierarchies 742, because all knowledge produced so far exists on account of having entered our written and spoken language. The linguistic hierarchy Hjelmslev terms a linguistic schema and the resultants of the non-linguistic hierarchy, when they are ordered to a linguistic schema, a linguistic usage. Therefore, if a form, in the general sense, happens to be a language, he refers to it as a linguistic schema. A function between these two hierarchies is considered a manifestation. Linguistic usage can thereby be said to manifest the linguistic schema. Fig. 124

LINGUSITIC SCHEMA

Linguistic usage manifests the linguistic schema.

LINGUISTIC USAGE

architectonics of linguistic phenomena After establishing the general definitions, Hjelmslev entered a stage in his work where these definitions needed refining in order to account for as many processes specific to natural language as possible. He expanded the notion of variants into varieties and variations and spoke about the ascent of their articulation. He consequently recognized a localised (irreducible) variety that marks the stage where further analysis is no longer fruitful. After that, with the notion of a sum, he expanded on the relational identity and possibilities of its categorisation. To address the problem of multiple contextual roles those entities have, he spoke of syncretism. Finally, to address the questions of ambiguity that requires interpolation of the inaccessible functives, he spoke of catalysis.

742 “The non-linguistic analysis of purport must, then, through a deduction (in our sense of the word), lead to the recognition of a non-linguistic hierarchy, which has function to the linguistic hierarchy discovered through the linguistic deduction.” Hjelmslev, 81.

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Variants It is a fascinating characteristic of language that a thing can be expressed in an infinity of different ways. 743 To address this aspect, Hjelmslev further elaborated on the topic of variants. He began by saying that the universal property of any functive is its ability to be articulated into an arbitrary number of variants. According to Hjelmslev, this articulation is possible because every variant has two distinct roles in the chain: either as a variety or as a variation. The basic idea of a variety (a bound variant) is that a variant in the chain can have a specific dependence to another variant in the chain, by which they must be regarded as a pair. 744 To establish a formal definition, he defined them as solitary variants 745, according to the function they contract. VARIETY variant VARIETY variant

A A

Fig. 125

A and B are varieties and can only exist as a pair of constants.

B B

The idea of a variation (a free variant) is that a variant in the chain can always be replaced by a specimen of the same kind. 746 For example, different variations can stand for the same content, or can be different pronunciations of the same expression. To give a formal definition, he defined them as combined variants 747, according to the function they contract. VARIATION VARIATION variant variant

A A

Fig. 126

A and B are variations and can exist independently of each other. (as it is the case with variables)

B B

743 “Any functive in the linguistic schema can, within the schema and without reference to the manifestation, be subjected to an articulation into variants.” Hjelmslev, 81. 744 varieties— “bound/conditioned variants that appear only in certain environments in the chain.” Hjelmslev, 81–82. “Any entity of expression has as many bound variants (varieties) as it has possible relations in the chain.” Hjelmslev, 82. 745 “Varieties are defined as solidary variants, since a given variety always presupposes and is presupposed by a given variety of another invariant (or of another invariantspecimen) in the chain.” Hjelmslev, 82. 746 variations— “free variants that appear independently of environments.” Hjelmslev, 81–82. “Any entity of expression has as many free variants (variations) as it has possible specimens.” Hjelmslev, 82. 747 “Variations are defined as combined variants, since they are not presupposed by, and do not presuppose, any definite entities as coexisting in the chain; variations contract combination.” Hjelmslev, 82.

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Articulation into variants is important and necessary, in both content and expression, as the isomorphic structure between structures of the content plane and the expression plane must be respected. In both planes of language, the first articulation is always the articulation from invariants into varieties, which is repeated until the very last variety is reached. He referred to this variety as a localised variety. 748 The second articulation will be the articulation of the varieties (starting from localised variety) into variations, which is repeated until the last variation is reached. 749 He named this variety an individual. 750 presupposes

Fig. 127

Invariants are first articulated into varieties, then varieties into variations.

presupposes

ARTICULATION

...

ARTICULATION

specifies

specifies

INVARIANTS

VARIETIES

VARIATIONS

Hjelmslev concluded that when the last individual is reached, we have reached the end of a possible description. 751 748 “If the articulation of an invariant into varieties is carried out to each individual “position,” an irreducible variety is reached, and the articulation into varieties is exhausted. A variety that thus cannot be further articulated into varieties we shall call a localized variety.” Hjelmslev, 83. 749 “… for both planes of a language, in deference to the requirement of the simplest possible description, it is important to insist that the articulation into variations presupposes the articulation into varieties, since an invariant must first be articulated into varieties and after that the varieties into variations: the variations specify the varieties.” Hjelmslev, 82. 750 “If the articulation of a localised variety into variations is carried out down to the individual specimen, an irreducible variation is reached, and the articulation into variations is exhausted. A variation that thus cannot be further articulated into variations we shall call an individual.” Hjelmslev, 83. 751 “If the transitive specification cannot be continued, and the hierarchy ends as exhausted in an articulation of varieties into variations that cannot again be articulated into varieties, it will be possible to say in a certain epistemological sense that the object under consideration is no longer susceptible of further scientific description.” Hjelmslev, 83.

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Relational identities and their categories: Sums After expanding the notion of a variant on the basis of the functions it contracts, Hjelmslev did the same with the notion of a class. He showed that a class also cannot be seen in isolation and must be analysed within its context. He defined a minimal context for a class, in which every class is seen as a pair contracted by a function. This is quite similar to the notions of information and identity Zafiris introduces with the partition paradigm, where he requires the recognising of two entities to obtain one information unit. A class that has a function to one or more other classes within the same rank, Hjelmslev defined as a sum. THEORY

Fig. 128

Sums can only be established by functions within the same rank.

functions establishment

CLASS

CLASS

CLASS

COMPONENT COMPONENT

COMPONENT COMPONENT

COMPONENT COMPONENT

SUM

SUM

SUM

The sum within a process he named a unit and, within a system, a category. He defined a unit as the “chain that has relation to one or more other chains within the same rank.” 752 PROCESS

Fig. 129

Sums within the process.

functions establishment

CHAIN PART

CHAIN PART

UNIT

PART

CHAIN PART

UNIT

PART

PART UNIT

On the same basis, he defined a category as the “paradigm that has correlation to one or more other paradigms within the same rank.” 752 Hjelmslev, 85.

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Fig. 130

SYSTEM

Sums within the system.

functions establishment

PARADIGM MEMBER

PARADIGM

MEMBER

MEMBER

CATEGORY

PARADIGM

MEMBER

MEMBER

CATEGORY

MEMBER

CATEGORY

To show how invariants and variants relate to the concept of a category, he took the notion of correlation into account. If the correlation is a commutation, the category is an invariant. If the correlation is a substitution (suspended commutation), the category is a variant. Since sums require a function contracting two classes, he introduced an establishment as the relation that exists between a sum and a function entering into it. Therefore, the sum is established by the function, and the function establishes the sum. Fig. 131

SYSTEM

Commutation and substitution between the members of a paradigm.

COMMUTATION

content plane

COMMUTATION

correlation

correlation

PARADIGM

PARADIGM

MEMBER

MEMBER

MEMBER

correlation relation

COMMUTATION

MEMBER

correlation expression plane

relation

✘

SUBSTITUTION

By concluding that every entity is a sum—meaning that each object’s identity is necessarily a relational one—Hjelmslev established the same pattern throughout the Prolegomena. 753 Such a point of view, where each entity can be seen as articulating itself in a two-fold way, became very influential in the philosophy of post-structuralism. Gilles Deleuze adopted it from Hjelmslev and named it a double articulation. 754

753 Further examples are: process and system, arbitrariness and appropriateness, expression and content, substance and form, etc. 754 Deleuze’s expression: “God is a Lobster, or a double pincer, a double bind.” Deleuze and Guattari, “10,000 B.C.: The Geology of Morals,” 45. … corresponds to Hjelmslev’s formal definition: “Every entity is a sum.” Hjelmslev, Prolegomena to a Theory of Language, 85.

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This further underlined his assertion that only functions have scientific existence. 755 Once he defined sums, Hjelmslev considered the possibility of their categorisation. One possibility would rely on the basis of the analysis (the types of bridges in the sense of Zafiris), and would yield a functional category. 756 Different bases of analysis (bridges) would yield different functional categories. The second possibility for categorisation would be one operating within a functional category. Every functional category can be clustered according to the relations contracted by the functives within it. For example, he recognises functives that appear as: selected, selecting, both selected and selecting, or neither selected nor selecting. Such a category he named a functival category. 757 If we make a parallel with Zafiris’s partition paradigm, a functional category would correspond to a clustering of a functional category according to the dependence (arrows) between the entities. Fig. 132

PARTITION AS THE BASIS OF THE ANALYSIS

Possible functival categories (indicated with arrows between functives) within a functional category.

CHAIN

PARTITION

ion inat

erm det

FUNCTIVE FUNCTIVE 1

at

FUNCTIVE 3

ion

FUNCTIVE 4

FUNCTIVE 4

FUNCTIVE 3

det

4 co m bin

erm

FUNCTIVE 2

ina

tion

determination FUNCTIONAL CATEGORY

755 Hjelmslev, Prolegomena to a Theory of Language, 85. 756 “category of the functives that are registered in a single analysis with a given function taken as the basis of analysis.” Hjelmslev, 86. 757 “By functival categories we mean the categories that are registered by articulation of a functional category according to functival possibilities.” Hjelmslev, 86.

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Syncretism In language, certain functives of a different function often share the same expression-form. In conventional grammar, such a phenomenon is known as syncretism. Hjelmslev defined it as a suspended commutation. 758 For example, in Latin, there is a syncretism between the nominative and the accusative for the nouns in the second declension. Two functives, in the nominative and the accusative would fail the commutation test, but this would not be the case with the feminine nouns in the same declension. In this case, there is a suspended mutation or an overlapping between the nominative and the accusative in the neuter, where context must be provided to understand the function of the noun. The category of functives that exhibit overlapping, Hjelmslev defined as a syncretism. 759 Fig. 133

Syncretism and overlapping.

SECOND DECLENSION—LATIN FEMININE nominative genitive dative accusative ablative vocative

NEUTER nominative genitive dative accusative ablative vocative

domus domī domō domum domō domus

NO OVERLAPPING (commutation)

nominative commutation

accusative

templum templī templō templum templō templum

OVERLAPPING (suspended mutation)

nominative INVARIANTS

VARIANTS

overlapping

accusative

SYNCRETISM

INVARIANT (category)

The important thing to note in this example is that the necessary condition for the overlapping between nominative and accusative is the neuter, which is a variant. Here, the neuter is solitary with the nominativeaccusative overlapping, and such solidarity he defined as a dominance.

758 Hjelmslev, 87. “suspension/absence—condition when the functive is suspended or absent.” Hjelmslev, 88. 759 “A suspended mutation between two functives we call an overlapping, and the category that is established by an overlapping we call (in both planes of a language) a syncretism.” Hjelmslev, 88.

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Fig. 134

Dominance stands for solidarity (interdependence) between a dominant and a syncretism.

nominative NEUTER VARIETY variant

DOMINANT

SOLIDARITY

overlapping

accusative

DOMINANCE

SYNCRETISM

Since there are cases in language where there is a choice whether to respect a certain overlapping or not, Hjelmslev made a distinction between an obligatory and an optional dominance. In order to avoid the “real” definition of these terms, he defined them by means of the function between the dominant and the syncretism. If the dominant is a variety, the overlapping is obligatory and the function between the dominant and the syncretism is solidarity. If the dominant is a variation, the overlapping is optional, and the function between the dominant and the syncretism is a combination. 760 DOMINANT—DOM1 inv—I1 inv—I2 ... inv—Im ... inv—In

Fig. 135

DOMINANT—DOM2 inv—I1 inv—I2 ... inv—Im ... inv—In

aaa bca ... aaa ... ffev

OBLIGATORY OVERLAPPING (suspended mutation is obligatory)

OPTIONAL OVERLAPPING (suspended mutation is optional)

inv—I1 obligatory overlapping

DOM1 VARIETY bounded variant

DOMINANT

SOLIDARITY

OBLIGATORY DOMINANCE

inv—Im SYNCRETISM

Difference between an obligatory and an optional dominance lies in the type of function contracted by the dominant and the syncretism.

aaa bca ... bbb OR aaa ... ffev

inv—I1 optional overlapping

DOM2 VARIATION free variant

DOMINANT

COMBINATION

OPTIONAL DOMINANCE

inv—Im SYNCRETISM

As syncretism always involves overlapping, the question is, can the function containing the overlapping functives be resolved through a local context. Hjelmslev leaves both options open, which makes syncretism either resoluble or irresoluble. Resolution occurs by introducing the syncretism-variety that does not contract the overlapping. 761

760 “We can now simply define an obligatory dominance as a dominance in which the dominant in respect to the syncretism is a variety, and an optional dominance as a dominance in which the dominant in respect of the syncretism is a variation.” Hjelmslev, 89. 761 “If, despite the syncretism, we can explain templum in one context as nominative and in another context as accusative, that is because the Latin syncretism of nominative and accusative in these instances is resoluble.” Hjelmslev, 91.

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As an example of an irresoluble syncretism he gives the Danish word top, which can be pronounced both as [t p] and [t b]. Resoluble syncretism can be illustrated on the previously mentioned examples of Latin nouns domus and templum. Unlike domus, which is a feminine noun, templum, as a neuter noun, contracts an overlapping. If they are combined, domus helps to resolve the syncretism for the whole group. Now we have obtained domus templum in nominative, and domum templum in accusative. When considered as a group, both functives have obtained a unique expression for their function. The chain with unresolved resoluble syncretisms, Hjelmslev called actualised, and the one with resolved resoluble syncretism an ideal chain. Both actualised and ideal chains can be resolved. The difference in the point of view depends on the actualisation; the chain can be regarded as resolved, or to be resolved. Fig. 136

Resolution of a syncretism.

FEMININE nominative genitive dative accusative ablative vocative

domus domī domō domum domō domus

NO OVERLAPPING (commutation)

implies

NEUTER nominative genitive dative accusative ablative vocative

templum templī templō templum templō templum

nominative genitive dative accusative ablative vocative

domus templum domī templī domō templō domum templum domō templō domus templum

NO OVERLAPPING (commutation)

OVERLAPPING (suspended mutation)

ACTUALIZED CHAIN

RESOLUTION

IDEAL CHAIN

Hjelmslev showed how the resolution of a syncretism closely relates with the traditional conception of a logical conclusion. He noted that when we resolve a syncretism, the resolved entity can be seen as the implication of the syncretism. Therefore, in a logical sense, we can see a premised proposition as the resoluble syncretism of its consequences. The logical conclusion, then, consists in resolving a given syncretism through articulation. A premised proposition is taken to be something that in itself includes the potentiality of being resolved. Additionally, Hjelmslev engaged with the question of relationship between the class and the segment. He tried to distance himself from Russell’s notion of the “class as many,” where a paradigm is considered to be the mere addition of its members. His notion of the “class as one” is something different from its members, namely a syncretism

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of its members. By the resolution of the syncretism, a class as one is transformed into a class as many. Such a syncretism between things he calls a concept. 762 Interpolation of the unknown: Catalysis Just like the syncretism depends on the syncretism-variety to resolve the ambiguity created by the overlapping, catalysis 763 interpolates the functives inaccessible to knowledge by utilising the linguistic evidence provided by the data. There can be many reasons why inaccessible functives exist in language. In Latin, for example, the preposition sine always selects the ablative 764 case. However, it can happen that in an incomplete, interrupted or damaged text in Latin a sine is found, but not an ablative. Hjelmslev suggested to eliminate such incalculable accidents. 765 However, there are many other cases where texts are complete and undamaged, but in which some of the functives are still inaccessible. Interruptions in language are sometimes deliberate. Such is the case with aposiopeses. 766 For Hjelmslev, sentences like “How nice!”, “If I only had!”, “Because!”, etc. should be a completely legal part of the system of language, even though they do not provide enough information in isolation. In order to make sense of them, we must register their relations to the context. 767 In the example of sine and ablative, there is enough linguistic evidence to assume that if a sentence breaks with sine, the ablative can be interpolated. 768 This is where Hjelmslev’s theory incorporates uncertainty and prediction. A missing entity needs to be predicted, and this requires data on the basis of which we can make assumptions. But we must also be careful not to presuppose too much if we do not have enough evidence. Hjelmslev further refined the definition by saying that an inaccessible functive in the case of catalysis could be actually seen as an irresoluble syncretism, a category of overlapping entities, whose function

762 “concept—a syncretism between things (namely, the things that the concept subsumes)” Hjelmslev, 93. 763 “catalysis—interpolation of certain functives which would in no other way be accessible to knowledge.” Hjelmslev, 94. 764 Motion away from something. 765 Hjelmslev, 94. 766 “Aposiopesis (Classical Greek: ἀποσιώπησις, “becoming silent”) is a figure of speech wherein a sentence is deliberately broken off and left unfinished, the ending to be supplied by the imagination, giving an impression of unwillingness or inability to continue.” Lanham, A Handlist of Rhetorical Terms, 20. 767 “The requirement of an exhaustive description, however, obliges us, while we register these aposiopeses and the like, also to recognize them as such, since the analysis must likewise register the outward relations which the actually observed entities have, the cohesions that point beyond the given entity and to something outside it.” Hjelmslev, Prolegomena to a Theory of Language, 94. 768 “…we can still register a cohesion (selection) with an ablative.” Hjelmslev, 94–95.

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matches this particular context. 769 Irresoluble here means that we cannot depend on any other functive to resolve the ambiguity, thus we depend on the evidence (data) to provide the grounds for performing the interpolation. He formally defined catalysis as a “registration of cohesions through the replacement of one entity by another to which it has substitution.” 770 In our example, the sine is found in the text without an ablative, which needs to be interpolated. In this case, sine will be the replaced entity. The ablative case, together with any cohesive syncretism that needs to be interpolated, is defined as the encatalysed entity. Finally, a replacing entity will consist of a replaced entity (sine) together with the encatalysed entity. Fig. 137

SELECTION

sine

Example of catalysis.

ablative

CATALYSIS (replacement)

sine

CATALYZED ENTITY (replaced entity)

sine

CATALYZED ENTITY (replaced entity)

+ ablative ENCATALYZED ENTITY

cohesive syncretisms REPLACING ENTITY

Algebraic nature of a language To make the theory fully self-referential, Hjelmslev took advantage of the formal definition of catalysis and used it in a more general sense. With purport, he had already

769 “What is introduced by catalysis is, then, in most instances not some particular entity but an irresoluble syncretism between all the entities that might be considered possible in the given “place” in the chain.” Hjelmslev, 95. 770 Hjelmslev, 95. Substitution entails that if there is no commutation between functives, they are variants of the same invariant, and therefore can be freely exchanged one for another. Hjelmslev, 74.

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illustrated the relation between form and substance. Now he was able to reformulate it, by saying that form is encatalysed to the substance. He also used the term catalysis to express that a language was encatalysed to a text in a formal procedure. 771 By introducing catalysis, Hjelmslev had defined enough so that he could summarise the process of a textual analysis: Linguistic theory prescribes a textual analysis, which leads us to recognize a linguistic form behind the “substance” immediately accessible to observation by the senses, and behind the text a language (system) consisting of categories from whose definitions can be deduced the possible units of the language. 772 The system consists of multiple strata of categories (linguistic hierarchy), contracting functions with uniform relations. 773 It is obtained in a formal deductive procedure, starting from the text (process) in its undivided and absolute integrity. Categories are obtained by classifying functives of each level of the procedure according to their functional and functival possibilities. 774 In respect to the hierarchy of categories and according to the contracts they enact, the hierarchy of inventories is established by registering respective units of the language. 775 Inventories start as unrestricted and, with the registration of the figuræ, are reduced to restricted. This allows us to regard the units of the language, as a system of figuræ with the rules of transformation operating on them in a purely formal way. 776 Hjelmslev notes that “these rules are set up without consideration of the substance in which the figuræ and units are manifested.” 777 This is what makes it possible for the theory to consider different languages as different manifestations of the same linguistic form. 778 In other words, specific substance-hierarchy, characteristic for different languages, can be seen as the specific manifestation of the form-hierarchy, according to the universal principle of formation, which is common to all languages. To affirm the necessary role of self-reflection in his theory, Hjelmslev asserted that “linguistic theory must come to contain within itself its own definition.” 779 771 “The kernel of this procedure is a catalysis through which the form is encatalyzed to the substance, and the language encatalyzed to the text.” Hjelmslev, 96. 772 Hjelmslev, 96. 773 … bridges in terms of Zafiris. 774 … by means of functors, in terms of Zafiris. 775 “A syntagmatic sum we shall call a unit, a paradigmatic sum a category.” Hjelmslev, 84–85. 776 Hjelmslev, 96. 777 “the linguistic hierarchy and, consequently, the linguistic deduction as well are independent of the physical and physiological and, in general, of the non-linguistic hierarchies and deductions that might lead to a description of the “substance”. Therefore, one must not expect from this deductive procedure any semantics or any phonetics, but both for the expression of a language and for the content of a language only a “linguistic algebra,” which provides the formal basis for an ordering of deductions of non-linguistic ‘substance’.” Hjelmslev, 96. 778 “On the basis of the arbitrary relation between form and substance, one and the same entity of linguistic form may be manifested by quite different substance-forms, as one passes from one language to another; the projection of the form-hierarchy on the substancehierarchy can differ essentially from language to language.” Hjelmslev, 97. 779 Hjelmslev, 98.

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The end point of such a deductive analysis consists in registering the irreducible elements. He introduced this stage with an example, by starting with a selection, taken as the basis of the analysis. In that case, there will be a stage within a procedure where a selection will be used for the last time. At that stage, we will be registering and inventorying virtual elements which he named taxemes. With a fair amount of certainty, he assumed that they would be usually manifested by phonemes (minimal sounds). Taxemes, even though they are the smallest virtual units, can be further partitioned on the basis of something he called universal division. This can be seen as the simplest possible mechanical division. To describe this process, Hjelmslev started with nine taxemes, known as phonemes (see Figure 138). According to the “two special rules for qualitative division,” 780 he placed them into a three by three matrix. The members that established such matrix, he called “dimensions.” Furthermore, nine taxemes could be described as the product of six members within the dimensions (3 × 3). From that follows that the members of dimensions are the parts of taxemes, thus smaller parts than the taxeme itself. He concluded that they were invariants, the members of the two solidary categories. He saw these final elements as the end point of the analysis and defined them as glossemes. 781 at, et, st, ap, ep, sp, ak, ek, sk TAXEMES

Fig. 138

Taxemes represented within a matrix.

a

e

s

t

at

et

st

p

ap

ep

sp

k

ak

ek

sk

SO

LI

DA R

IT

Y

DIMENSIONS (categories)

t, p, k, a, e, s GLOSSEMES (invariants)

780 Hjelmslev, 99. 781 “When it may be carried out, it will be the members of the dimensions and not the taxemes that are the end-points of the analysis; these end-points we call glossemes.” Hjelmslev, 100.

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The process of registering taxemes and glossemes concludes the textual analysis (syntagmatic deduction). After that, he undertakes the paradigmatic 782 deduction, which articulates language into categories and distributes the newly obtained taxemes within them. 783 Finally, he reminds us of the crucial element of his theory—that the two planes of a language must have a completely analogous structure. a generic infrastructure for communication Extending the notion of natural language Thus far, by focusing on natural language, the possibility of extending the theory was not exhausted. Hjelmslev suggested the possibility of applying the linguistic theory to structures of analogous form, which was a step in the direction of its universal applicability. For him, this was possible exactly because the theory was minimally specific and the linguistic form was viewed without regard for the substance. According to Hjelmslev, the effect of the historical supremacy of phonetics was the idea that spoken language consisted mostly of sounds. Hjelmslev referred to a counter argument: … in reality, as the Zwirners say, not only the so-called organs of speech (throat, mouth, and nose), but very nearly all the striate musculature, cooperate in the exercise of “natural” language. 784 What this means is that the linguistic form provides the ability of manifesting itself in different substances, for example alternating from words to text, hand gestures or different signs. Also, in some phonetic orthographies (in, for example, Serbo-Croatian), there is a one-to-one correspondence between the written and the spoken word. His claim went even further by showing how different systems of expression can correspond to one and the same system of content. 785 For Hjelmslev this is not surprising at all, as he considers the entities of linguistic form as being algebraic in nature. As such, the task of the linguistic theoretician is to calculate the possible expression systems in general for a given content and vice versa. Hjelmslev’s main task in this regard was to determine and define the structural principles of language from which it is possible to deduce a calculus whose categories are individual languages. To describe the relation (selection) between these different hierarchies and their derivates, Hjelmslev used the term manifestation.

782 On the side of the system. 783 “Here the language is articulated into categories, into which the highest-degree taxeme categories of the textual analysis are distributed, and from which, through a synthesis, can be deduced the possible units of the language.” Hjelmslev, 101. 784 Hjelmslev, 103. 785 “different systems of expression can correspond to one and the same system of content.” Hjelmslev, 105.

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The constant in a manifestation can, with reference to Saussure, be called the form; if the form is a language, we call it the linguistic schema. The variable in a manifestation can, in agreement with Saussure, be called the substance; a substance that manifests a linguistic schema we call a linguistic usage. 786 From linguistics to semiotics As a generalisation of linguistics, Hjelmslev used the term semiotic and defined it formally: Semiotic is a hierarchy, any of whose components admits of a further analysis into classes defined by mutual relation, so that any of these classes admits of an analysis into derivates defined by mutual mutation. 787 What this implies is that without mutation there cannot be a semiotics. 788 This renders language a special case of semiotics, whose characteristics concern linguistic usage. He credited Saussure for introducing semiotics as an abstract transformation structure. What remained to be defined were the boundaries between semiotics and non-semiotics, and which of the two should be assigned to language. In order to make this distinction, he provided formal definitions of language and text: A language may be defined as a paradigmatic whose paradigms are manifested by all purports… 789 …text, correspondingly, as a syntagmatic whose chains, if expanded indefinitely, are manifested by all purports. 790 This means that language is in fact a semiotic into which all other semiotics may be translated 791, thus, a structure of a higher order than a semiotic. This is because languages can be manifested by any purport whatsoever, by means of which all of knowledge can be reproduced in language. 792 Semiotic and non-semiotic systems Hjelmslev suggested that the main criterion for deciding whether a structure is a semiotic or not, is whether the exhaustive description necessitates operating with two planes, or whether the operation with one plane is sufficient. Similarly, the natural communication model also requires operating on two levels.

786 Hjelmslev, 106. 787 Hjelmslev, 106. 788 And mutation presupposes operating with two planes—expression and content. 789 “By a purport we understand a class of variables which manifest more than one chain under more than one syntagmatic, and/or more than one paradigm under more than one paradigmatic.” Hjelmslev, 109. 790 Hjelmslev, 109. 791 “both all other languages, and all other conceivable semiotic structures.” Hjelmslev, 109. 792 “in a language, and only in a language, we can work over the inexpressible until it is expressed.” Hjelmslev, 109.

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We can conclude that both Hjelmslev and Zafiris consider operating on two levels as the necessary requirement for communication. To decide whether a structure requires one or two planes, Hjelmslev devised a procedure of tentatively setting up two planes and comparing their structure. If the two planes had exactly the same structure, with oneto-one relations between the functives of one plane in respect to the other, the planes were conformal 793, which indicated that operating on a single plane was sufficient. Thus, if a structure requires operating on two planes, the planes must not be conformal. He defined the procedure known as a derivate test, which is of similar significance as the commutation test. After applying the derivate test, Hjelmslev concluded that many structures, which modern theory considered to be semiotics, were in fact not. According to him, a game of chess was not a semiotic structure: If we looked at the expression and content planes, the functional net would be entirely the same in both. Symbolic systems This raises a question about the character of structures like chess: What are they if not semiotics? In this regard, he defined structures that are interpretable 794, but not biplanar 795, as symbolic systems. With symbolic systems, the focus is on substance rather than on form. The implication is that a semiotic, since it considers the substance as a variable in respect to the form, is in fact an algebraic structure, while a symbolic system is not. In the context of symbolic systems, Hjelmslev defined the term symbol as an entity that is isomorphic with its interpretation. To illustrate this, he gave an example of the Thorvaldsen’s Christ as a symbol for compassion; a hammer and sickle as a symbol for communism; scales as a symbol for justice; onomatopoeia in the sphere of language, etc. In general, a symbol has a much broader meaning, but for him it was advantageous to be able to apply the word precisely to interpretable nonsemiotic entities. Another very important difference between a symbol and a sign, is that a symbol cannot be further analysed into figuræ. Since it is an entity simpler than a sign, a symbol can only be interpretable. towards the universal applicability of semiotic systems At the end of his Prolegomena, Hjelmslev proposes an algebraic method of further extending the notion of a semiotic structure. 796 By initially

793 “Two functives are said to be conformal if any particular derivate of the one functive without exception enters the same functions as a particular derivate of the other functive, and vice versa.” Hjelmslev, 112. 794 Content purport may be ordered to them 795 The simplicity principle does not permit to encatalyze a content form into them. 796 A generalisation of linguistics.

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defining his theory as a necessarily minimal kernel, Hjelmslev made the possibility of its extension simple and elegant. Semiotics as planes of a language First, Hjelmslev claimed to have deliberately reduced the theory presented so far, referring to it as a denotative semiotic. It is a semiotic, the planes of which neither is semiotic. Thus, the two planes of the denotative semiotic are the content and the expression plane. LINGUSITIC SCHEMA

Fig. 139

Determination between the content and the expression plane.

LINGUISTIC USAGE

Then he introduced more complex semiotics where at least one of the planes is a semiotic. A semiotic whose expression plane is a semiotic he defined as a connotative semiotic, and semiotics whose content plane is a semiotic as a metasemiotic. This gesture bears similarity with the notion of complex keys, introduced in Zafiris’s natural communication model. Both connotative semiotics and metasemiotics can be seen as motivic keys incorporating another key. Hjelmslev explored the relation between a semiotic and non-semiotic and between a semiotic and a language. He suggests that the important thing to consider is the relationship between two or more semiotics. In practice, he suggested it was rare to find a text composed in one definitive semiotic, but rather in a mixture of several. Various parts (or parts of parts) of a text can be composed in different stylistic forms, styles, value-styles, media, tones, idioms, vernaculars, national or regional languages, physiognomies, and so forth. All of them are in fact solidary categories. What this implies is that every functive in the denotative language (language that we have explored so far) must be defined in respect to all of these categories. Thus, by the combination of members of different categories, hybrids can arise. By defining every text in respect to all the registered categories and hybrids, Hjelmslev found it easy to account for the variety and richness of different texts. Functives defined in respect to all such categories he named connotators. 797 Furthermore, any textual derivate could be translated to any form established by these categories. 798 Yet, if one semiotic is a language and the other one is not, translation not possible. This is why translatability 799 is one of the most important characteristics of a language. 797 “The individual members of each of these classes and the units resulting from their combination we shall call connotators.” Hjelmslev, 116. 798 “Any textual derivate (e.g., chapter) can be translated from one stylistic form, style, valuestyle, medium, tone, vernacular, national or regional language, physiognomy to another.” Hjelmslev, 117. 799 “Given unrestrictedness (productivity) of the text, there will always be “translatability,” which here means expression-substitution, between two signs each belonging to a sign-class of its own, this sign-class in its turn being solidary with its respective connotator.” Hjelmslev, 117.

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Hjelmslev showed that the function between sign classes and their connotators is the sign function. Sign classes act as the expression of the content that are connotators. Thus, the semiotic schema and usage which we refer to as Danish language will be the expression for the connotator “Danish.” CONTENT PLANE

EXPRESSION PLANE

CONNOTATORS

SIGN CLASSES

“DANISH”

DANISH LANGUAGE

Connotative semiotics and metasemiotics were defined earlier on the bases of content and expression. Since neither are formally definable concepts—as their existence can only be considered in opposition to each other—Hjelmslev needed to define semiotics and metasemiotics formally. He articulated the class of semiotics into classes of scientific semiotics and non-scientific semiotics: Scientific semiotic: a semiotic that is an operation… 800 non-scientific semiotic we understand a semiotic that is not an operation. 801 This allowed him to define a range of possible semiotics by algebraically altering the structure of the involved concepts and relations. Connotative semiotic is a non-scientific semiotic one or more of whose planes is (are) (a) semiotic(s). 802

800 Hjelmslev, 120. Instead of saying that content plane is a semiotic. “Operation— description that is in agreement with the empirical principle” (process that leads to the comprehension of an object and is free of contradiction (self-consistent), exhaustive, and as simple as possible) Hjelmslev, 31. 801 Hjelmslev, 120. Instead of saying that expression plane is a semiotic. 802 Hjelmslev, 120.

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Fig. 140

An example of a connotator.

EXPRESSION PLANE

An example of the connotative semiotic is a national language, which stands as a symbol for the nation, local dialect as a symbol for the canton, etc. Fig. 141

Connotative semiotic.

CONTENT PLANE

CONTENT PLANE

Metasemiotic is a scientific semiotic one or more of whose planes is (are) (a) semiotic(s). 803 A metasemiotic takes semiotic as its object and allows describing a language by means of the language. 804 In Hilbert’s terms, metasemiotics would correspond to a meta-language. The main purpose of Hjelmslev’s theory is to make his theory self-referential, thus the theory its own object. A semiotic that treats a semiotic is actually linguistics.

803 Hjelmslev, 120. 804 “Usually a metasemiotic will be (or can be) wholly or partly identical with its object semiotic. Thus the linguist who describes a language will himself be able to use that language in the description.” Hjelmslev, 121.

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EXPRESSION PLANE

Fig. 142

Metasemiotic.

CONTENT PLANE

EXPRESSION PLANE

Fig. 143

Meta-(scientific) semiotic.

EXPRESSION PLANE

EXPRESSION PLANE

Semiotics of a higher order Meta-(scientific semiotic) as a metasemiotic with a scientific semiotic as an object semiotic. 805 Meta-(scientific semiotic) would be a semiotic whose object is a metasemiotic. Meta-(scientific semiotic) could, for example, consider linguistics as the typological variety of structures that can define a language.

CONTENT PLANE

CONTENT PLANE

EXPRESSION PLANE

805 Hjelmslev, 120. A semiotic that enters as a plane into a semiotic is said to be an object semiotic of that semiotic.

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EXPRESSION PLANE CONTENT PLANE

CONTENT PLANE

CONTENT PLANE

Fig. 144

Semiology.

EXPRESSION PLANE

Semiology as a metasemiotic with a non-scientific semiotic as an object semiotic. 806

Semiology can be considered a semiotic whose object is a connotative semiotic. Semiology would treat those semiotics that are defined on the basis of the relation between connotators and sign classes. For example, semiology would deal with semiotics that treat languages as special cases of semiotics. And finally, we can use the designation metasemiology of a meta(scientific semiotic) whose object semiotics are semiologies. 807 Metasemiology would be a semiotic dealing with structures that treat linguistics. Therefore, of all structures that treat linguistic as a special case of a structure that treat language, metasemiology would be just one specimen of that typology. Hjelmslev concludes that a metasemiology is a complete description of the semiotic of semiology and there should be no need to consider structures of a higher level of abstraction. 808

806 Hjelmslev, 120. 807 Hjelmslev, 120. 808 “In deference to the simplicity principle, metasemiologies of higher orders, on the other hand, must not be set up, since, if they are tentatively carried out, they will not bring any other results than those already achieved in the metasemiology of the first order or before.” Hjelmslev, 125.

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Fig. 145

CONTENT PLANE

Metasemiology.

CONTENT PLANE

CONTENT PLANE

EXPRESSION PLANE

EXPRESSION PLANE

CONTENT PLANE

EXPRESSION PLANE

According to Hjelmslev, metasemiology must not directly deal with elements that are defined on the lower level of abstraction. It must be able to see such structures abstracted and treat them algebraically. Thus, what is irreducible at the lower level of abstraction becomes a variation at a higher level of abstraction. Essentially, metasemiology provides formal means for continuing analysis, when the analysis of more specific structures is exhausted. At the extreme, the variants that it treats are not necessarily of linguistic nature, but rather physical. Thus, metasemiology is, in practice, identical with the description of substance.  809 At the end of the analysis, metasemiology, in the same way like as linguistics, recognises irreducible invariants. Yet, in metasemiology, the only possible description of the functives would be a purely statistical description. Outlook of Hjelmslev’s theory The last chapter of the prolegomena, conveyed Hjelmslev’s final perspective on the theory of language. He warned about the dangers of disciplinarisation of the sciences and humanities and offered glossematics as an avenue for potential unification. In order to understand the system of language, he pointed out the necessity of broadening our perspective into a level of abstraction that would accommodate seeing the language as a structural specimen of a higher order. Furthermore, if the notion of language was to be

809 “…ultimate variants of a language are subjected to a further, particular analysis on a completely physical basis. In other words, metasemiology is in practice identical with the so-called description of substance.” Hjelmslev, 124.

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extended, it would be possible to see science from the same position, rendering linguistics as a totality-concept and allowing it to account for everything that can be articulated. Hjelmslev noted that even though such a linguistics could explain things in a self-consistent manner, it was not a closed system. 810 His notion of a system implied a constant gluing. Thus, a linguistic theory could only be complete if there was nothing left to glue, which would contradict his idea of a language as a living thing. Finally, he concluded his work by giving account of the language as the expression of a universal human characteristic: Linguistic theory here takes up in an undreamed-of way and in undreamed-of measure the duties that it imposed on itself (pp. 8, 19–20). In its point of departure linguistic theory was established as immanent, with constancy, system and internal function as its sole aims, to the apparent cost of fluctuation and nuance, life and concrete physical and phenomenological reality. A temporary restriction of the field of vision was the price that had to be paid to elicit from language itself its secret. But precisely through that immanent point of view and by virtue of it, language itself returns the price that it demanded. In a higher sense than in linguistics till now, language has again become a key-position in knowledge. Instead of hindering transcendence, immanence has given it a new and better basis; immanence and transcendence are joined in a higher unity on the basis of immanence. Linguistic theory is led by an inner necessity to recognize not merely the linguistic system, in its schema and in its usage, in its totality and in its individuality, but also man and human society behind language, and all man’s sphere of knowledge through language. At that point, linguistic theory has reached its prescribed goal: Humanitas et universitas 811

810 “The smallest system is a self-sufficient totality, but no totality is isolated.” Hjelmslev, 126. 811 Hjelmslev, 127.

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AN INSTRUMENT FOR COMMUNICATION SELF-ORGANIZING MODEL

III

i

SElf-orgaNiziNg MaP

In a 1976 paper, neurobiologist David Willshaw and physicist Christoph von der Malsburg made a hypothesis that the relation between two laminar structures 812 in the brain is characterised by topographical, systemto-system, rather than cell-to-cell mapping. 813 Their model was based on a microscopic mechanism that utilised concepts of neural excitation, inhibition and self-organisation. Inspired by ideas from neurobiology and neural computation, Finnish physicist and engineer, Teuvo Kohonen generalised the idea of Willshaw and Malsburg in his 1982 paper, “Self-organized formation of topologically correct feature maps.” He stratified his model in two domains and showed that the logic of self-organisation is “readily generalisable to mechanisms other than neural.” 814 To complement the theory, he introduced the self-organizing map algorithm (SOM) aiming to address the question of “how symbolic representations for concepts could be formed automatically.” 815 His initial discovery was that structure-preserving mapping can be automatically established between input signals in the event space and a topological network responding to those signals. 816 Steinbuch’s cybernetic notion of a learning matrix (Lernmatrix) inspired Kohonen to define the response layer of his model as a memory matrix. 817 Input event Ak

fig. 146

Kohonen’s illustration of a system, which implements an ordered mapping. (Kohonen, 1982)

Relaying network

S1

Si

Interactive network

Sn

Processing units

h1

hi

hn

Output responses

812 Willshaw and Malsburg, “How Patterned Neural Connections Can Be Set Up by SelfOrganization,” 433. Sheets of brain cells, namely pre- and postsynaptic. 813 Willshaw and Malsburg, 442. 814 Kohonen, “Self-Organized Formation of Topologically Correct Feature Maps,” 59. 815 Kohonen, 59. 816 Kohonen, 59. 817 “Another early construct which was introduced for the explanation of brain functions and implementation of artificial intelligence is the Learning Matrix due to Steinbuch. It is a system of crossing signal lines, with an adaptive connection at each crossing.” Kohonen, Self-Organization and Associative Memory, 74.

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som as representation Kohonen showed that different network topologies could be implemented as a response layer of his model. 818 The topology of a two-dimensional lattice revealed some advantages in this respect: It yielded good results, allowed for easy visualisation 819, and complied with the idea of brain maps. 820 The output domain of Kohonen’s algorithm was named a map, and the algorithm the self-organizing map. 821 The naming also implied that the communication between different domains was in fact a mapping. The two-dimensional representation of the self-organizing map algorithm influenced its interpretation in the artificial intelligence circles. 822 In the mid 90s, the availability, memory and speed of computers increased enough to accommodate for the high computational cost of the algorithm. Researchers attained the means to conduct more experiments and demonstrated novel capabilities of the algorithm, many of which were completely unrelated to visualisation and analysis. 823 Thirty years later, Kohonen acknowledges a surprisingly wide variety of applications and modifications of the algorithm and tries to create a consensus by describing the algorithm as simply an “automatic data-analysis method.” 824 Visualisation and clustering Unlike in abstract algebra where the term map captures functionality 825 in terms of functors, in the context of Kohonen’s two-dimensional network topology, the term has a topographic connotation. It implies a graphic representation, which is to be visually perceived and then interpreted. Figure 147 can serve as an example: it shows the cartographical representation of Wikipedia’s word frequency mapped onto a low-dimensional landscape.

818 Kohonen, “Essentials of the Self-Organizing Map,” 55–56. 819 “… the 2-D grid allows rough visual presentation and interpretation of the clusters.” Vesanto and Alhoniemi, “Clustering of the Self-Organizing Map,” 599. 820 “It is believed that many other kinds of maps exist in the hippocampus or other parts of the brain system.” Kohonen, Self-Organization and Associative Memory, 121; … for example tonotopic maps in auditory cortex of the brain, Kohonen, 120. 821 “The earliest SOM models, tending to replicate the detailed neural-network structures, were intended for the description and explanation of the creation of brain maps.” Kohonen, “Essentials of the Self-Organizing Map,” 55. 822 Elaborated in the next sub-chapter. 823 Such as: Barreto and Araújo, “Identification and Control of Dynamical Systems Using the Self-Organizing Map” and Barreto and Souza, “Adaptive Filtering with the SelfOrganizing Map: A Performance Comparison.” 824 Kohonen, “Essentials of the Self-Organizing Map,” Abstract, 52. 825 Zalamea, Synthetic Philosophy of Contemporary Mathematics, 245.

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Fig. 147

A typical representation of a self-organizing map as a visualisation method based on Wikipedia featured article data (Denoir, 2013)

In 1993, Alfred Ultsch, one of the leading researchers on the topic, introduced his Unified distance matrix (U-matrix) as a technique utilising the SOM’s response layer as a basis for clustering. 826 In Ultsch’s paper, however, Kohonen’s initial fascination with automatic concept formation is absent. The algorithm is explained simply as a tool for mapping between spaces of different dimensionality with the promise of visualising the low-dimensional target space. 827 In a 1996 paper, Kohonen seemed to agree with this interpretation and affirmed that the strongest features of his algorithm were visualisation and data analysis.  828 In a similar manner, Vesanto concurred that the algorithm has shown itself valuable in many applications, but most prominently in visualising highdimensional spaces. 829 In one of his early papers he made an overview of different visualisation methods. 830 A year later, Vesanto employed a kmeans algorithm on top of the SOM response layer and concluded that in many cases SOM clustering yields better results than other clustering

826 Ultsch, “U*-Matrix: A Tool to Visualize Clusters in High Dimensional Data.” and Ultsch, “Clustering With SOM: U*C.” Ultsch has created two more algorithms which improved the clustering on top of the SOM, namely U*-matrix and U*C-matrix. 827 “Through a learning process, this neural network creates a mapping from a N-dimensional space to a two-dimensional plane of units (neurons).” Ultsch, “SelfOrganizing Neural Networks for Visualisation and Classification,” 307. 828 “Accordingly, the most important applications of the SOM are in the visualization of high-dimensional systems and processes and discovery of categories and abstractions from raw data. The latter operation is called the exploratory data analysis or data mining.” Kohonen et al., “Engineering Applications of the Self-Organizing Map,” 1358. 829 “Visualization of complex multidimensional data is indeed one of the main application areas of the SOM.” Vesanto, “SOM-Based Data Visualization Methods,” 112. 830 Vesanto, 115–24.

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techniques. 831 However, he found this to be a fortunate side-effect of the SOM’s powerful visualisation abilities. fig. 148

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SoM aPPlicaTioNS BEyoND rEPrESENTaTioN In order to reach a better understanding of the capabilities of self-organizing maps, work by other researchers focused on the applications of the algorithm reaching beyond mere analysis and representation 832, including: nonlinear time series prediction, function approximation,

831 “In [16], partitive methods (i.e., a small SOM) greatly outperformed hierarchical methods in clustering imperfect data.” Vesanto and Alhoniemi, “Clustering of the Self-Organizing Map,” 588. “Another benefit is noise reduction. The prototypes are local averages of the data and, therefore, less sensitive to random variations than the original data.” Vesanto and Alhoniemi, 588. “The experiments indicated that clustering the SOM instead of directly clustering the data is computationally effective approach.” Vesanto and Alhoniemi, 599. 832 “…However, most of this methods do not use the representational power of the SOM to build the prediction models.” Barreto and Araújo, “Identification and Control of Dynamical Systems Using the Self-Organizing Map,” 1257.

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control of dynamical systems 833, nonlinear adaptive filtering 834, data exploration (in industry, finance, natural sciences, and linguistics), digital signal processing/transmission 835, and nonparametric encoding for high-dimensional spatiotemporal dynamics. 836 In his doctoral thesis on pre-specific modelling, Moosavi provided a rich list of indexes around the SOM algorithm. 837 Fig. 149

Moosavi’s extensive list of indexes around the SOM. (Moosavi, 2015)

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833 Barreto and Araújo show that “the SOM can be successfully used to approximate dynamical input-output mappings, with minor modifications in the original algorithm.” Barreto and Araújo, 1244. 834 For example: identification and equalization of communication channels, Barreto and Souza, “Adaptive Filtering with the Self-Organizing Map: a Performance Comparison,” 787. 835 Kohonen, “Essentials of the Self-Organizing Map,” 52, Abstract. 836 Moosavi, “Computing with Contextual Numbers.” 837 Moosavi, “Pre-Specific Modeling,” 71, table 2.

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Types of SOM Fixed topology

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II self-organizing model If the dominant theoretical interpretations of the algorithm are correlated to the diversity of its applications indexed by Moosavi, there appears to be an unsettling discrepancy between the platitude of realized applications and the actual demonstrated capacity. How can a tool for visualisation or an automatic data-analysis method be so successfully used for function-approximation, control of dynamical systems, or prediction of time-series? The important hypothesis of this work is that the insistence on the terms map and mapping to describe Kohonen’s algorithm reduces it to a visualisation and analytical tool, hence obscuring the actual capacity demonstrated in applications. Such interpretations systematically neglect higher-level architectonics on top of which the algorithm is conceived, and thus, in our opinion, invoke erroneous theoretical consequences. The ambition of this work, motivated by the work of Hovestadt, Bühlmann, Moosavi, Wassermann, Zafiris, and

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Hjelmslev 838, is to propose an alternative theoretical interpretation of the self-organizing map from a more abstract mathematical perspective. The hypothesis is that the very architectonics inherent to the algorithm, not only its specific implementations, makes it unique among the computational frameworks. Such an interpretation should provide an adequate account for its seemingly extraordinary capacities. reason and coexistence The Google Ngram viewer, a web application that counts the occurrences of words in Google’s library of books, shows an increased rise of usage of terms system, computer and cybernetics in the period from the late 1950s until the end of 1990s. 839 Fig. 150

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The idea of a system is not new, but its usual interpretation within the computer science circles is usually credited to Norbert Wiener. This is not very surprising as computer science “directly applies the concepts of cybernetics to the control of devices and the analysis of information,”

838 see Birkhäuser’s Applied Virtuality Book Series edited by Hovestadt and Bühlmann; Zafiris, Natural Communication; Moosavi, “Pre-specific Modeling”; The Putnam Program: Language & Brains, Machines and Minds (blog). 839 Google Books, “Ngram Viewer,” accessed May 1, 2017, https://books.google.com/ ngrams. For the detailed explanation of the application see: Michel et al., “Quantitative Analysis of Culture Using Millions of Digitized Books.”

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such as cellular automation, decision support, robotics and simulation. 840 Cybernetics had a crucial influence on artificial intelligence, computer modelling and artificial neural networks, under which the SOM is usually classified. 841 Hovestadt argues that the cybernetic notion of a system is conceived upon the notions of reason and causality (in a Newtonian sense), which makes it incompatible with today’s body of thinking conceived upon the invention of electricity and the development of quantum physics. 842 Notions of reason and a cause are also constituents of the scientific method. In his prolegomena, Hjelmslev described the role of these notions in defining the limits of scientific knowledge: Science always seeks to comprehend objects as consequences of a reason or as effects of a cause. But if the object can be resolved only into objects that may all indifferently be said to be consequences or effects of all or none, a continued scientific analysis becomes fruitless. 843 A book, Coding As Literacy 844, appearing in Birkhäuser’s Applied Virtuality Book Series, discussed the potential of self-organizing maps as a model 845 from the perspective of computational literacy. In the chapter on elements of digital architecture, Hovestadt takes a quantum perspective to invert the customary interpretation of the SOM as a map. He gave an example of the self-organised map of Zurich where “the constellation of elements changes according to the analytical/chronological position of the observer.” 846 In his doctoral thesis, Moosavi further investigated the notion of coexistence in the context of urban data streams and demonstrated the scope of applicability of data-driven models. 847 Inspired by the ideas of Wassermann 848, Hovestadt and Bühlmann, this work reinterprets self-organizing maps by identifying the role of the algorithm within a more general, self-organizing model. This model abandons the idea of systematicity based on reasoning and facilitates the notion of coexistence in the world of data.

840 Wikipedia, s.v. “Cybernetics,” last modified August 30, 2017, 17:55, https://en.wikipedia. org/wiki/Cybernetics. 841 Wikipedia, s.v. “Cybernetics.” 842 As an overview of Hovestadt’s position on the topic see: Hovestadt, “Elements of Digital Architecture,” and “Cultivating the Generic: A Mathematically Inspired Pathway for Architects.” 843 Hjelmslev, Prolegomena to a Theory of Language, 84. 844 Bühlmann, Hovestadt, and Moosavi, Coding as Literacy–Metalithikum IV. 845 “The self-organizing map articulates the logical form of chronological elements in probability values… Therefore we suggest that we should not talk about a self-organizing map but a self-organizing model.” Hovestadt, “Elements of Digital Architecture,” 108. 846 Hovestadt, 109. 847 If we assume “… each of the prototypical forms in the trained SOM metaphorically as a word or a letter in a language, a trained SOM in coexistence with data can be interpreted as a pre-specific dictionary of the target phenomena.” Moosavi, “Pre-Specific Modeling,” 73. 848 See: The Putnam Program: Language & Brains, Machines and Minds (blog).

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kohonen’s algorithm as a key If we put the emphasis on the implementation of Kohonen’s algorithm, we can acknowledge the existence of two data-structures resembling different structural domains. 849 Kohonen’s input layer is implemented as an n-dimensional list of m-dimensional vectors, where n and m are positive integers. The response layer (or map) 850 is implemented as a two-dimensional array (the common encoding of a graph/matrix) of m-dimensional vectors, where the dimensionality m of the input layer vectors matches the dimensionality of the response-layer vectors. 851 Kohonen’s crucial discovery demonstrated by the algorithm is that the input signal alone is sufficient to enforce self-organisation in the response layer, but only if the response layer has a specific architecture and implements a local feedback mechanism. 852 The response layer (map) should then be interpreted by acknowledging that the mechanism imposed on the set of input vectors forces unlabelled classes to emerge in the response layer, with the effect that the topological distances within the response layer resemble similarities between the classes. The interesting discovery is that the schema of Kohonen’s algorithm can be accurately represented by means of a key of the same anatomy as Zafiris’s motivic key. Here, the input and output layers could be interpreted as different domains, and mapping and interpretation as adjoint communication bridges. input layer

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849 “This principle is a generalization of the formation of direct topographic projections between two laminar structures known as retinotectal mapping.” Kohonen, “SelfOrganized Formation of Topologically Correct Feature Maps,” 59. 850 … also known as lattice in the case of a two-dimensional response-layer. 851 Kohonen, 61. 852 Kohonen, 59.

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On the level of technical implementation, the elements of this schema appear to be too specific if compared with Zafiris’s motivic key. However, this is an important first step towards building a more abstract perspective. As a next step, I will demonstrate that successful SOM applications are in fact taking advantage of inverting or reinventing one or more elements of key defined as such. Modifications of the input layer In their paper “Neural methods for non-standard data,” Hammer and Jain explored the possibilities of different structures serving as an input layer of the algorithm. Instead of vectors, they explored the possibility of utilising strings, sequences, trees, graphs or functions. 853 In his paper “Computing with contextual numbers” Moosavi created a two-level input layer structure by replacing the original layer with a one-dimensional self-organizing map, acting as a non-parametric encoder. 854 These approaches are very powerful as they allow much greater freedom in the constitution of the encoding. Modifications of the response layer The work of Alfred Ultsch involves many experiments on the output layer of the selforganizing map. To minimise the learning error, he explored different network topologies in the response layer, for example hexagonal and rectangular grids. 855 U-matrix, U*-matrix and U*C-matrix algorithms directly exploited the response layer of the SOM. They handled the response layer, its distances and densities, as an input for clustering. Algorithms like neural gas and growing neural gas, replaced the twodimensional lattice with a flexible and adaptive response layer. Such a response layer could subsume any topology as it dynamically adapted itself to the variability of the input signals. 856 Modifications of the mapping/learning The generative topographic mapping algorithm (GTM) was a counterpart to the SOM, which tried to overcome its limitations by rethinking the mapping between the domains. For that purpose, it uses a “constrained mixture of Gaussians whose parameters can be optimised using the EM (expectation-maximisation) algorithm.” 857

853 Hammer and Jain, “Neural Methods for Non-Standard Data,” 289. 854 Moosavi, “Computing with Contextual Numbers,” 13. 855 Ultsch and Herrmann, “The Architecture of Emergent Self-Organizing Maps to Reduce Projection Errors,” 2. 856 Fritzke, “A Growing Neural Gas Network Learns Topologies,” 6. 857 “…lack of a theoretical basis for choosing learning rate parameter schedules and neighbourhood parameters to ensure topographic ordering, the absence of any general proofs of convergence, and the fact that the model does not define a probability density.” Bishop, Svensén, and Williams, “GTM: The Generative Topographic Mapping,” 216.

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Modifications of the interpretation Without attempting any interpretation of the results of SOM training, Barreto and Souza utilised the nodes of the response layer as the reference inventory for prediction by using the k-nearest neighbours algorithm. 858 The vector, whose missing value (dimension) was to be predicted was first introduced into the response layer and compared to those of its vectors without the missing value. The influence of every node was calculated according to their relative distance and the missing value interpolated. For the control of dynamical systems, Barreto and Araújo established a feedback loop between the input and response layer, thus enforcing the dynamic control of a system. 859 coexistence key The intention behind presenting these applications was to demonstrate that the modification or replacement of the particular elements of SOM architecture need not change the form of the key used to describe its architecture. In this regard, what differentiates SOM from other machine learning algorithms lies in its structural properties. Hjelmslev’s differentiation between variants and invariants can help us to understand why this might be the case. His conception and terminology supports the argument that all the presented modifications of the original algorithm are in fact variants by the virtue of placing a specific algorithmic procedure to a specific position within the key. Consequently, the invariant property of the self-organizing model is reflected in the position of an element within the key, and is defined relationally in respect to other elements. This in an indicator that there is more to the SOM than just an algorithm. It is a computational framework resting upon a mathematical contract established by the motivic key and its invariant anatomy. We can define such a key as a coexistence key and such a framework as the self-organizing model. The coexistence key maintains a contract established upon mathematical symmetries, while the self-organizing model furnishes its elements with computational procedures. In the manner of Zafiris’s motivic key, the coexistence key stratifies the self-organizing model in two levels, as shown on Figure 152. The domain A is defined as an unordered set of observables, and domain B as a self-referential system 860 of objects and arrows. According to Kohonen, the input signal alone is sufficient to enforce self-organisation in the response layer, if the response layer has a “specific architecture” and

858 Barreto, “Nonstationary Time Series Prediction Using Local Models Based on Competitive Neural Networks,” 1146, Abstract. 859 Barreto and Araújo, “Identification and Control of Dynamical Systems Using the SelfOrganizing Map,” 1245. 860 The notion of the system is taken from Hjelmslev, meaning premised system. Hjelmslev, Prolegomena to a Theory of Language, 16.

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implements a local feedback mechanism. 861 The proposed stratification gives new insights to Kohonen’s requirement. If we are interested in what would be the invariant property of Kohonen’s “specific architecture”— concerning we have acknowledged that many different structures could successfully serve as the response layer—an answer comes from the category theory. A domain which allows to be self-organised must be selfreferential. As we have seen with growing neural gas algorithm, any topological structure allows for self-organisation as long as every node has a reference to itself and to all the other nodes. 862 At a computational colloquy held at the CAAD chair, Hovestadt presented a series of experiments on the coexistence between data and the wide range of self-referential topological structures, ranging from linear chains to multiple graphs. To represent the model, I suggest using the category theory instead of graph theory, as it better suited for representing its structural properties. entropic

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Metaphorically speaking, domains A and B can be regarded as different “natures” of things. 863 From the perspective of Hjelmslev’s glossematics, an obstacle or a problem observed in the domain A would be a specific purport formation (substance), apparent to us by the virtue of its content-form encatalysed within the same domain. By introducing another domain, we are seeking to find a new purport formation within the domain B describing the same obstacle or a problem by the virtue of a different content-form. In order to establish coexistence between the domains A and B, the model uses encoding and decoding bridges. For the encoding bridge, 861 Kohonen, “Self-Organized Formation of Topologically Correct Feature Maps,” 59. 862 Kohonen, 60, figure 2. 863 This is in the sense that physical laws that apply in the quantum scale, seem to be completely different from those that apply in the macroscopic scale, and according to that they could be considered as different “natures” of things.

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the self-organizing model requires a generalisation procedure that can produce a partition spectrum from an indiscrete, unordered set of observables. The self-organizing map algorithm provides that functionality. In the process of self-organisation, the pockets of negentropy emerge from the entropic data set resulting with a partition spectrum. 864 Once the spectrum is obtained the obstacle or a problem is represented by means of a new cipher (a category) within the spectral domain B. Let us assume that in the domain B there exists a procedure allowing to bypass the obstacle, or solve the problem. In order to interpret what was resolved in the spectral domain B, the self-organizing model uses a decoding bridge. The decoding bridge requires a projection procedure that utilises virtual distinctions from the partition spectrum to distribute the original set of observables (encoded as the unordered set) into a category of sets corresponding to actual distinctions. The actual distinctions give the interpretation to the solution/obstacle within the original domain. form and substance: encoding the black box According to Zafiris, the notion of data assumes a geometrical encoding. Hjelmslev sees data as a content-substance: formed purport encatalysed by means of the content-form. 865 This suggests that the encoding principle itself is a form. In a chapter from A Thousand Plateaus, Gilles Deleuze has multiplied this elementary gesture until he stratified “the whole earth” and beyond, by means of coding and territorialisation. 866 This gesture introduces a very important modelling question: What is the role of a form in the context of a stratified model? Zafiris gives an example of applying his model to quantum physics, where the motivic key enables communication between two physical levels: atomic and microscopic. On an atomic level, no technical means are sufficient to localise an electron and its interactions. Thus, the whole domain can be thought of as a black box. On a microscopic level, physicists observe interference patterns and think of them as a reflection of unknown events happening on the atomic level. For Zafiris, modelling consists of establishing a pair of functors from the observational universe to the microscopic universe and back. 867 In this way, we can regard these two levels as functives, one of which is giving form to another. In this way, a form is given to something that cannot be observed.

864 Resolution of this spectrum is directly proportional to the number of nodes (objects) that were selected to constitute the network, and its number can be larger or smaller than the number of elements in the initial set of observables. 865 Hjelmslev, Prolegomena to a Theory of Language, 54. 866 Deleuze and Guattari, “10,000 B.C.: The Geology of Morals,” 45–48. 867 Zafiris, Natural Communication.

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This perspective must be investigated from the standpoint of the coexistence key. By applying the same gesture, we consider the domain A of the coexistence key as a “black box” characterised by a set of observables. We can assume that the observables describe some meaningful process, but the semantics is not explicitly present in the observables. Starting from the domain A, the application of the generalisation bridge will establish a partition spectrum. The resulting spectrum is constructed in another cipher, and at the same time juxtaposed with our original set of observables (a black box). The new cipher of domain B, known as a category, comes with a greater operative and symbolic potential than the cipher of the original domain (set). However, we are still missing the process of formalisation. In order to give form to the domain B, the system of objects and arrows must be composed in such a way that it resembles relations between the observables in the domain A. The role of the generalisation bridge is to give form to the domain A within the domain B. In the same way, we can see the projection bridge as responsible for obtaining semantics in the domain A according to the form of the domain B. By doing so, the projection bridge closes the path and establishes coexistence between levels A and B. procedural catalysis of syncretisms In order to formally define the process of formalisation, Hjelmslev introduced the term catalysis. The term implies that “form is encatalysed to substance and language encatalysed to the text.” 868 If we apply the concept of catalysis to the self-organizing model, instead of saying that the domain B gives form to the domain A, we can say that the self-organisation procedure encatalyses the form of the category to the set of observables. If the first criterion that makes self-organisation possible is self-reference, the second criterion would be the possibility of information exchange between the nodes of the system. Such property, Kohonen calls a local feedback. 869 In the self-organisation procedure, objects within the system are pushed/pulled towards each other as they exchange information. The mechanism which enforces this behaviour in the SOM algorithm is known as the distance function. 870 This function provides a metrical criterion of similarity. By applying the concept of catalysis, we can say that the distance function locally encatalyses the form to the set of affected observables. Influenced by the distance function, nodes receiving the local feedback begin to share the same-expression form. This is especially visible between two directly connected nodes of the system.

868 Hjelmslev, Prolegomena to a Theory of Language, 96. 869 Kohonen, “Self-Organized Formation of Topologically Correct Feature Maps,” 59. 870 Kohonen, “Essentials of the Self-Organizing Map,” 53.

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Such behaviour corresponds very well to the Hjelmslev’s definition of syncretism. 871 It is a category established upon the suspended mutation which he called overlapping. 872 Therefore, we can regard the distance function as the form which is encatalysed locally to the set of observables and establishes overlapping between the nodes within the system. In other words, the nodes are organised in such a way, that they will be “pushed” to share the same expression-form according to the applied distance function. Hjelmslev uses the notion of syncretism to define the term concept: concept—a syncretism between things (namely, the things that the concept subsumes). 873 According to the definition, and by the virtue of the distance function, we can define the population of nodes that exhibit overlapping, a concept. This can be illustrated with an example of an experiment that the author conduced in 2012. Here, the self-organizing map algorithm was used to explore design possibilities of a generic parametric model. The model consisted of six parameters that defined the shape and colour of a single geometric figure. This figure was replicated in one thousand instances with random parameter values and the SOM algorithm was used to organise them into a two-dimensional lattice. As Figure 153 shows, the outlined collection gathers locally similar shapes, but as we “zoom out” and start to consider the population of shapes, we can notice the difference between shapes belonging to different collections. From the standpoint of expression-form, each two neighbours within a single collection are overlapping. However, if we select two shapes belonging to different collections, they are most likely not overlapping. Those that overlap, can be said to establish a category defined by the overlapping. Thus, their collection, as the one marked on the image, can be regarded as the concept. Fig. 153

Interpretation of Hjelmslev’s notion of a concept within a partition spectrum.

From the perspective of glossematics, the relational property of the model that enforces that each individual shape locally reflects the arrangement 871 “…the category that is established by an overlapping we call (in both planes of a language) a syncretism.” Hjelmslev, Prolegomena to a Theory of Language, 88. 872 Hjelmslev, 88. 873 Hjelmslev, 93.

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of all the others, we can call a connotator. Therefore, in the previous example, each individual shape is considered to be the connotator of the category it belongs to. After investigating computational models in architecture within the broader context of mathematical and computational modelling, a body of indexes pointing towards the contemporary models of communication have been established. Two such models were covered, coming from mathematics and language. They helped to crystallise ideas of a quantum-characterisation and stratification, and recognise the mathematics of category theory as an adequate framework for describing communication models. This helps in understanding the potential of the self-organizing map to serve as a communication bridge. Finally, on the basis of the model of natural communication and glossematics, the self-organizing model has been defined as a computational framework that establishes coexistence between different levels of the model, and allows for communication. Modern world

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AN EXPERIMENT COMMUNICATION AND NATURES OF ARCHITECTURAL REPRESENTATION

IV

The aim of this experiment is to address the question of how similarity between spaces in architecture could be accounted for within a paradigm of communication. The hypothesis is that the question of similarity cannot be adequately posed within the established computational models in architecture, due to their reliance on a pre-given notion of space in which the architectural representations are submerged. Being able to speak about the similarity of spaces requires addressing the problem outside of the given space of representation, finding or constructing another level of representation of the same phenomena, where the question can be posed and answered once the coexistence between two levels is established. Speaking in terms of Hjelmslev, we can say that we are interested in finding a new content and expression-form for the same content purport. By applying the self-organizing model, I attempt to show that this is indeed possible. I nature(s) of architectural representation To describe the problem of similarity between architectural spaces, it is first necessary to address the question of architectural representation in the context of spatiality. In this respect, a quick overview of the established models of representation will be given, which should be helpful in determining the appropriate strategy of encoding the problem computationally. technical drawings The first written mention of architectural representation was recorded in Vitruvius’ Ten Books on Architecture. Vitruvius wrote about design in a form of recipes, but also expressed the idea of the “second basis of architecture,” dispositio or arrangement, which consisted of ichnography (plan), orthography (elevation), and scenography (perspective). 874 This arrangement was for the first time challenged in the 15th century. In his seminal work, De Re Aedificatoria, published in 1485, Alberti wrote on the inappropriateness of painterly effects for architectural design. Instead of perspective he insisted on using orthographic projection. 875 The orthographic projection was considered non-intuitive at the time, since what it represented did not correspond to human perception. Still, it was a great advantage for those who understood it, as it preserved the actual measures and geometric relations between the shapes representing objects. 876 This was a clear 874 Vitruvius, On Architecture (Loeb Classical Library 251), 25. 875 Luce, “Raphael and the Pantheon’s Interior: A Pivotal Moment in Architectural Representation,” 48. “Orthographic projection refers to a system of interrelated twodimensional views of a building. This system includes the views from above or a horizontal cross section of a building (the plan), the views from the side of a building (the elevation) and the views of vertical ‘cuts’ or cross sections of the building (the section).” Farrelly, Representational Techniques, 68. 876 Evans, “Architectural Projection,” 21.

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indicator of the changing tendency in representation, which started shifting from illustration towards a formal description of architecture by means of technical drawing. By the virtue of this shift and Alberti’s vision that an architect should not be producing buildings but drawings of buildings 877, Carpo credits Alberti as the inventor of the architectural profession. 878 The trend of formalising architectural representation continued in the 16th century. Sebastiano Serlio introduced a model that defined eight types of projections, each of them variations on a planar or elevational view. He used a central perspective only for the sake of illustration. 879 A similar attitude continued with Palladio, but was finally challenged by Daniele Barbaro, whose book, La pratica della perspettiva (1568), pushed architectural research in the direction towards formal representations of scenografia. The result of this effort was that it was finally possible to construct perspective in an equally rigorous way as orthogonal projections. 880 physical models A physical model is an another prominent type of architectural encoding. Smith describes it as the mechanism that helps architects develop an understandable scale with which to measure an unknown thing, such as a future building. 881 The first recorded use of physical models dates back to the 5th century BC, when Herodotus, in the Book V-Terpsichore, made a reference to a model of a temple. 882 However, such artefacts were still used only for illustration and communication purposes. Evidence suggests that the full scale three-dimensional prototypes were used in antiquity, when the translation between different scales was not mathematically well understood, and occasionally wooden models in Middle Ages. 883 Only in the 14th century did this form of representation become relevant as an architectural encoding. Filippo Brunelleschi primarily designed in three dimensions and used models extensively. His model of the Florence cathedral at the end of 14th century serves as an example of the changing role of physical models, from illustrative to constructive. 884 Smith describes models like Brunelleschi’s as the renaissance “small-scale model machines.” 885 The 20th century saw a major resurgence in the use of the physical models in architecture. Dunn refers to the work of Walter Gropius, “who in founding the Bauhaus in 1919,

877 Evans, 21. 878 Carpo, preface to The Alphabet and the Algorithm, X. 879 Tavernor, “Brevity Without Obscurity,” 10. 880 Tavernor, 9. 881 Smith, Introduction to Architectural Model as Machine, XXI. 882 Dunn, Architectural Modelmaking, 14. 883 Dunn, 14. 884 Dunn, 15. 885 Smith, Architectural Model as Machine, 25.

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was keen to resist the prevailing preoccupation with paper designs in favour of physical models to explore and test ideas quickly.” 886 diagrammatic representation A diagram is defined by the Oxford English Dictionary as “a simplified drawing, showing the appearance, structure, or workings of something, a schematic representation.” 887 It is an abstract representation that puts emphasis on the essential parts of interest, or points to specific features. For Ben van Berkel, the essence of the diagrammatic technique in architecture was to introduce into a work “qualities that are unspoken, disconnected from an ideal or an ideology, random, intuitive, subjective, not bound to a linear logic—qualities that can be physical, structural, spatial or technical.” 888 Since they are not focused on formally representing the “real thing,” 889 diagrams play a role of assemblages of “solidified situations, techniques, tactics, and functionings.” For van Berkel, a diagram is an architectural manifold. He illustrated this claim with the example of Jeremey Bentham’s panoptic prison plan from the 18th century, which expressed “a number of cultural and political circumstances cumulating in a distinctive manifestation of surveillance.” 890 Vidler provides evidence of this claim by giving an account of Corbusier who referred to diagrams as tools that provided “insider vision” to architecture, unlike drawings which he considered “architecture’s trap.” 891 At the turn of the 21st century, the diagrammatic representation began to be considered “the matter of architecture” itself, rather than its formal representation. 892 The position adopted here is not far from this: I believe that an architectural diagram is in fact not the instrument for encoding spatiality of architectural objects. Rather it as an instrument for encoding architectural concepts by means of a graphic art form. representation criteria This chapter gave a brief overview of the standard architectural representations. Together with digital representations of computational models presented before, we now have at our disposal a wide range of encodings that can aid in closely examining the problem of similarity between architectural spaces. In order to concisely describe the essential

886 Dunn, Architectural Modelmaking, 30. 887 The New Oxford Dictionary of English, 11th ed. (2001), s.v. “diagram.” 888 Berkel and Bos, “Techniques: Network Spin, and Diagrams,” 369. 889 “The diagram is not a blueprint. It is not the working drawing of an actual construction, recognisable in all its details and with a proper scale.” van Berkel and Bos, 369. 890 van Berkel and Bos, 369. 891 Vidler, “Diagrams of Diagrams,” 12. 892 Somol, “Dummy Text, or the Diagrammatic Basis of Contemporary Architecture,” 7.

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idea of the experiment, certain requirements need to be put upon the choice of the architectural representation. These are: • The representation must give an adequate and comprehensive account of an architectural space, its proportions and spatial measurements. • The representation should be standardised and its codes and ciphers, familiar. • Actual encodings of a large number of buildings, in terms of data, should be generally available and easy to obtain. In respect to these requirements, physical models could serve as an adequate representation if they were readily available in a digitised form. Nor are architectural diagrams adequate, as their codes and ciphers follow no conventions and vary greatly from project to project. Their role is not to accurately convey the spatial characteristics of the building. Consequently, technical drawings and computer generated (solid) 3D models appear to be best suited for the purposes of the experiment. Technical drawings are advantageous in multiple aspects: They are much easier to acquire; they can all be represented in a standardised way as a digital image; they convey the significant part of the architectural legacy, which makes them familiar and well understood. At the end of the experiment it will be clear that, by means of certain technical modifications, the experiment could be easily extended to address the problem of similarity of spaces by means of a three-dimensional representation. floor plan representation An architectural plan is usually described as an orthographic projection of a three-dimensional object acquired by cutting the object with a horizontal plane. 893 This projection is the geometrical basis of the technical encoding that encapsulates spatial and functional characteristics of a building for the purpose of communication. Encoding a (future) building into a plan asks the designer to follow a number of conventions that institute the encoding as a language of design. The conventions include scale, orientation and position of the horizontal section 894, a system of measurements, usage of lines and their thicknesses, symbols, texts, etc. 895 According to the function and purpose of the representation, the same plan can be reproduced by using different conventions. In this respect, presentation drawings will usually look very different from construction drawings. In order to standardise the encoding, these conventions should be acknowledged and their usage consistent. 896

893 “In other words, a plan is a section viewed from above.” Farrelly, Representational Techniques, 69. 894 “A plan is an imagined horizontal cross section of a room or building 1.2 metres above ground or finished floor level.” Farrelly, 100. 895 Farrelly, Representational Techniques, 85. 896 Farrelly, 71.

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Scale One of the most useful conventions that makes the distinction between different plans possible 897 is the notion of scale. 898 In the case of a printed (or drawn) plan, a scale indicates the ratio between the measure of an object indicated on the plan to its corresponding measure in the physical space. In the case of a computer drawing simulation (CAD) where scale does not play a role in design (since the drawing can be freely zoomed in or out), the scale is indicated symbolically and using geometric abstraction. The plan only shows the elements that are relevant for its purpose, and omits those that are not. In the context of this project, the question of the scale at which floor plans should be rendered must be addressed. Conventions of representation Conventions within different contexts also regulate how geometric abstraction is applied within plans. For example, conventions used to produce official construction drawings are regulated by the law and must be obeyed, in order to be validated. Presentation drawings, on the other side, allow much more freedom in terms of conventions, as their primary goal is to communicate spatial ideas and relations. 899 To illustrate this point more clearly, a few examples of different floor-plan articulations are presented in Figure 154. Fig. 154

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Different conventions of a floor plan representation.

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897 For example, difference between a building plan and urban plan. 898 Farrelly, 34. 899 Farrelly, 87.

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The floor plans one and two, known as blueprints, provide a necessary information for the actual realisation of the building on a building site. They are used to communicate spatial information to contractors, engineers, and construction workers. Image three depicts the two apartments from the famous American television series, Friends, made with no particular function in mind, except to communicate the idea. Image four is the standard representation of apartment floor plans in real estate advertisements, where names of the spaces and their dimensions are considered to be important. Finally, image five is a representation of a floor-plan taken from an architecture book. Their completely different intentions make it of very little value and of too much difficulty to compare such different representations in this work. Therefore, my requirement at this stage that the dataset should comprise floor plans whose representation follows the same conventions. II

computational precedents

typical computational floor plan encodings The task of establishing the notion of similarity between floor plans comes with the difficulty of computationally encoding a plan. This problem is far from new and, in this respect, the majority of CAAD literature points to March and Steadman’s encoding in The Geometry of Environment (1970). According to Coates, this work is considered to be the “bible of the mathematical approach for architecture.” 900 To encode the architectural plan, March and Steadman used graph-representation 901, where the network of nodes and edges represented architectural elements and their relationships. 902 They illustrated the idea on which the model rests, with an example of three different floor plans designed by Frank Lloyd Wright. All three plans, seemingly different, were shown to be specimens of one and the same organisational schema which was modelled by the graph. 903 Coates argues that the ideas presented in this book were a necessary groundwork for many of the topdown design methods research of the 1960s and 1970s. At the same time, Christopher Alexander’s article “A City is Not a Tree” (1965) presented a critique of the so called “top-down” network approach and attempted to establish a bottom-up encoding on the basis of set theory. 904

900 Coates, programming.architecture, 161. 901 Graph, theory, which formed the basis of this approach was invented by the Swiss mathematician Leonhard Euler in the 18th century. Kalay, Architecture’s New Media, 257. 902 March and Steadman, The Geometry of Environment, 28, figure 1.14. 903 March and Steadman, 27–28. 904 Alexander, “A City Is Not a Tree,” 401–27.

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Fig. 155

Three Frank Lloyd Wright’s house plans (right) represented as variations of a single organisational schema. (Mitchell, 2016)

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Both March and Steadman’s and Alexander’s approaches rely on the same set-theoretical assumptions about the architectural object, only applied in different modes of reasoning. Alexander’s example makes this explicit by describing a semilattice as just a special case of the set structure. 905 In the first example, the elements of the graph must be scrutinised. The notion of a “living room,” in the case of March and Steadman, is a clearly and objectively defined concept, which takes part in the constitution of a floor plan object. This suggests that the idea of a living space is something already pre-given, its boundary within a plan can be clearly recognised, and its specimens simply counted. Relation-wise, this example also renders the connections between spaces trivial and explicit: a “door” object manifests a connection between rooms. To be fair, a such simplified representation often matches the cogency of many architectural articulations. However, the interest here is not to provide an account for the simplest possible description, but rather the very opposite.

905 Alexander, 401.

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STaTE-of-THE-arT I consider Benjamin Dillenburger’s doctoral dissertation “Raumindex” (2016), as the state of the art in computational representation and analysis of floor-plans. Dillenburger provided a “system for computer aided retrieval of floor plans” as the support structure for research and planning. 906 His idea of encoding went one step beyond the previous solutions and involved two approaches of characterising a plan. In the first case, his model comprises a complex hierarchical network of relations between geometric and logical elements. Properties of the elements such as: “isPartOf”, “hasPart”, “hasBoundary”, “isPublic”, “isExterior”, “isRainproof”, “isOpenable,” involve binary choices. 907 His database, consisting of more than 2000 models, together with a database-like query language, should allow the user to ask for a specific configuration of a floor plan and retrieve the potential solutions from the database. fig. 156

Dillenburger’s illustrations of his floor plan model. (Dillenburger, 2016)

906 Dillenburger, “Raumindex,” Abstract, II. 907 Dillenburger, 78.

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In the second part of his dissertation, Dillenburger characterises floor plans on a more abstract level. He computes their properties on the basis of their geometry and assigns these properties to floor plan objects. His analyses include openness, visibility, distance of every point in space from the entrance, centrality, traffic volume. 908 By normalising the distribution of the measured values for a large number of apartments, he obtains a number of histograms characterising each plan. As this distribution can be described numerically, he uses it as the basis for calculating the similarity between floor plans by means of the self-organizing map algorithm. fig. 157

Dillenburger’s floor plan characterisation based on the properties computed upon the floor plan geometry. (Dillenburger, 2016)

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I previously presented three characteristic levels of abstraction employed in the computational encoding of floor plans. The least abstract 908 Dillenburger, 199–234.

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way encodes a plan as a set of objects and a set of relations interconnected as part of a graph. The elements of the set of objects are the representations of “concrete” entities, like living room, bedroom, opening, wall, etc. On the second level, floor plans are represented in an object-oriented paradigm (OOP), by utilising the hierarchy of classes of geometrical elements and the binary representation of their properties. Finally, the most abstract way consisted of encoding a floor plan as a list of numerical values regarding their externally defined objective properties, such as openness, visibility, centrality, traffic volume. III spectral characterisation of an abstract object On the basis of the self-organizing model, as an implementation of the natural communication 909 model and glossematics 910, this chapter will attempt to reach a higher level of abstraction in characterising floor plans. By applying the self-organizing model to a collection of digital representations of floor plans, a finite collection of elementary geometric expressions is extracted and symbolised. Any floor plan created by the same conventions, can thereby be defined on this symbolic basis. data acquisition and data processing In the previous subchapter we concluded that the effectiveness using floor plans, as the objects of our interest in this experiment, depends on certain requirements: • Floor plans should be in the same scale. • Floor plans should follow the same conventions of representation. • Floor plans should put emphasis on the geometric description of a floor plan and utilise only elementary symbols. • Floor plans should be in the same graphical format.

909 Implementation that uses the self-organising map algorithm as the encoding bridge, stratifies model in two domains, employs mathematics of category theory and characterises object from the partition paradigm. 910 … as the language theory that provides a framework for characterising textural processes algebraically.

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Floor plan data The 2009 book typologie+ 911 contained 189 floor plans that satisfied the requirements and that were digitally scanned, cropped in high resolution at the same scale and stored in raster format (Figure 158). Fig. 158

Floor plan data set obtained from scanning the typologie+ book. (Ebner et al., 2009)

911 Ebner, Peter, Eva Herrmann, Roman Höllbacher, Markus Kuntscher, and Ulrike Wietzorrek. typologie+: Innovativer Wohnungsbau. Basel: Birkhäuser, 2009.

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The ambiguities of representation The digitized floor plans employ the convention of displaying objects by lines. They use line thickness to symbolise different aspects of the drawing: Thicker lines represent walls, while thinner lines represent the geometry of the floor, for symbols for fixtures and for furnishings. As we can see in the upper two images of Figure 159, such a representation can lead to ambiguities when computationally extracting parts of the plan. In order to reduce this ambiguity, the areas bounded with thicker lines were filled, allowing the later analysis to account for the thickness of solid elements in the drawings.

Fig. 159

Floor plan representation based only on lines can lead to ambiguities if the floor plan is discretised into parts.

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The next stage of the data processing was to make all of the representatives employ this new convention. This was done semi-automatically, by employing a flood fill algorithm 912, and resulted in a new data set: Fig. 160

An intermediate data set obtained by displaying the walls as solid.

912 see: https://www.hackerearth.com/practice/algorithms/graphs/flood-fill-algorithm/ tutorial/

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Plan orientation Depending on the context, the orientation of a plan can be an important factor regarding the quality of the represented space. The orientation is often specified in the plan with a special symbol pointing towards the north. By taking into the consideration the movement of the sun, one can usually predict effects of solar radiation (such as solar heat gain) on the space. However, as previously noted, the interest here is not in computing the objective properties of spaces according to the geometry of the plan, but in describing floor plans in the terms by which they are communicated—by means of parts that constitute their graphical encoding, whatever they may be. In this respect, the orientation of a plan in respect to the sun does not play a role, but rather the orientation of the plan in respect to the representation medium. Since the floor plans are represented as raster images, any part extracted from it can be regarded in respect to two rotations: the rotation of the part itself, and the rotation of the plan containing the rotated part. In order to reduce the complexity of having to deal with both rotations at once, the rotation of the plan was accounted for first. The procedure to standardise the rotation of a plan is illustrated in Figure 161, and requires four steps: 1 Loading the original drawing. 2 Applying the Canny edge detector 913 to produce a linear representation of a plan. 3 Extracting the straight lines from the obtained image, whose length surpasses a certain threshold, by using the probabilistic Hough transform algorithm. 914 By extracting the point coordinates, which constitute a line, the angle of each line is calculated in respect to the orthogonal reference system. The number of occurrences of each angle is recorded and the angle value with most occurrences taken as the rotation vector. 4 The original image is rotated according to the previously obtained rotation vector. The image is centred according to the statistical mean of non-white pixels and cropped on all four sides. The final image is saved.

913 see: http://scikit-image.org/docs/dev/auto_examples/edges/plot_canny.html 914 see: http://scikit-image.org/docs/dev/auto_examples/edges/plot_line_hough_transform.html

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All computation in this phase was done in the Python programming language, with the help of “scikit-image” collection of algorithms for image processing. 915 Now that all plans are rotated according to the same principle, images of floor plans are ready for the next stage of the procedure. TowarDS THE ParTiTioN SPEcTruM At this stage, it is possible to think of the collection of floor plans as an abstract object characterised by the dataset of raster images. The codes and ciphers found at this level (graphical and symbolic elements and their conventions) are familiar to every architect, but they do not provide an adequate basis for the comparison of floor plans. The aim is to establish a completely new level, where the same abstract object can be represented, but in entirely different terms. The codes and ciphers on this level are expected to provide a more adequate basis for comparison. For this reason, the choice was made to construct the new level on a symbolic basis. The first step towards this goal is the creation of communication bridges. 915 http://scikit-image.org

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Partition procedure Hjelmslev wrote about the deductive progression from class to component as a requirement for achieving a self-consistent and exhaustive description of the text. He defined it as the analysis, which according to the perspective taken was seen as either a partition of a process (text) or an articulation of a system (language). A partition was to be applied to a text in its undivided and absolute integrity. 916 In the context of the Natural Communication model, Zafiris writes about the partition point of view, where the description of an object starts from the state of total indistinguishability and advances in stages by applying communication bridges, yielding a partition spectrum at each stage. 917 In order to account using Markov chains for the probabilities of any meaningful sequence, the parts that comprise them must be distinguishable from each other, and only then be taken as a means of partitioning the sequence. With this in mind, we will consider the collection of floor plans as an unknown object manifested in its full integrity by its data. In arriving at its description, we will begin by partitioning it, gradually obtaining the partition spectrum. This procedure will take into account Hjelmslev’s principles of arbitrariness and appropriateness. From the standpoint of arbitrariness, partitioning of the data into smaller entities will be performed in a purely mechanical fashion, without any semantic consideration, resulting in an arbitrary number of parts of arbitrary size. From the standpoint of appropriateness, and with regard to previous experimentation, it is known that certain ways of partitioning and certain sizes of parts will yield better results than the others, with respect to the empirical data. Image division The first stage of the partitioning was already accomplished by separating the floor plans into different images. The partition of each individual floor plan into parts is accomplished by dividing the image in such a way, that the collection of equal size parts covers the image entirely. According to their size, increasing the number of parts over a required limit will result in overlapping parts. This redundancy proved to greatly enhance the model, as it introduced a larger number of elements and greater overall variability. To further exploit this feature,

916 Hjelmslev, Prolegomena to a Theory of Language, 12. 917 Zafiris, Natural Communication.

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overlapping is made such that the parts attempt to cover the image, starting from the image edge inwards. This mode of partitioning results with equal spacing between the overlaps. Figure 162 shows a partition of a single floor plan image. The part size is chosen to be approximately one tenth of the total image size (150 × 150 pixels). The middle illustration shows the minimum required number of parts to fully cover the image, giving very little overlap. In the third illustration, the desired number of parts is increased, resulting with larger overlaps between the parts. fig. 162

Partition procedure: The division of a single floor plan image with different overlapping between parts.

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By choosing the minimal overlapping, the floor plan image is divided into 108 parts, illustrated in Figure 163: Fig. 163

Division of a single floor plan into 108 parts.

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The increase in overlapping between the parts yields 432 elements from the same image. At this stage it is evident that the majority of the parts are empty, but this redundancy will be dealt with later. Fig. 164

Division of the same floor plan (Fig. 155) into 432 parts by means of a larger overlapping.

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When the partition procedure was applied to all 188 floor plan images, with a minimal overlapping, it produced 23,393 parts as shown in Figure 165. At this stage it becomes apparent that the large number of parts extracted provides very little information about our abstract object, appearing like white noise from a distance. We can consider this apparently entropic state of our abstract object as the state of total indistinguishability, characterised by maximum symmetry. Fig. 165

23’393 floor plan parts on a single image.

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Rotation transformation group The design of building plans often employs orthogonal grids, but there are exceptions. A simple but effective solution to account for the elements of the plan that are deviating from the established axes is to apply a rotation transformation group to each floor plan image, repeat the partition procedure and add new parts to the collection. In this way, depending on the proposed rotation angle increment value, the final number of parts will increase. Figure 166 illustrates this. The chosen rotation angle increment of an element in a figure is 15 degrees (r/12). The division of the rotation value of a full circle (360 degrees or 2r) by the selected rotation value, yields 24 new parts from a single part. Fig. 166

Application of the rotation transformation group to a single part of a floor plan.

However, if we rotate the whole image, and then repeat the partition procedure, the number of new parts cannot be predicted exactly in advance. In the current case, the rotation angle increment of 90 degrees was chosen. By applying the original partition procedure with larger overlapping the abstract floor plan object was divided into 143,372 parts. By applying the rotation transformation group, this number increases to 573,488 parts.

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Translation and mirroring invariance bridge Another problem inherent to the partitioning of images has to do with the position of the cut. This is illustrated in Figure 167. Top four squares mark four different parts of a floor plan, all enclosing a geometric shape representing a column. Regardless of the involved semantics, it is obvious that by applying a simple geometric translation we can make these shapes perfectly overlap with each other and thereby conclude that they are used to represent the same thing. It is the same case with the bottom two squares representing walls. In case of the two squares in the middle, the shapes representing building corners are almost identical to each other, except for their orientation. Making the two shapes overlap with each other requires a reflection transformation. However, deciding when and how to apply which kind of transformation is a simple task for an intelligent human, but quite a challenging task to be formalised in computer code. Fig. 167

Identifying the parts that have a different expression but display the same content.

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In this work, the aforementioned problem was solved by creating an encoding bridge that abstracts the problems of position and orientation. The encoding bridge was implemented as an algorithm that operates on each part of a floor plan image, translating it into an n-dimensional vector of frequencies, where the value of n can be freely selected. The mechanics of the algorithm are illustrated in Figure 168. The two middle square images contain an extracted part of a floor plan image of the size of 150 × 150 pixels. The upper two images show the horizontal scanning procedure, which successively distributes a number of horizontal lines over the image from top to the bottom and reads each line from left to right. 918 The bottom two images show the vertical scanning procedure, which successively distributes a number of vertical lines over the image from left to the right and reads each line from top to bottom. 919 Reading each line involves measuring the length of a continuous stretch of black pixels indicating an obstacle. Looking first at the value of the first pixel on the line, if it is white, nothing is registered and the scanning proceeds to the next pixel along the line. When a black pixel is registered, the measuring process begins (indicated by the light gray points on the line). The scanning continues pixel by pixel, until the next white pixel is registered (indicated by dark gray points). The distance between the light and the dark gray point on the same line expresses the length of the uninterrupted stretch of black pixels, which provides a measure of the obstacle. The bottom two images show the analogous procedure applied in a vertical direction. Fig. 168

Application of the translation and mirroring invariance bridge.

918 i) First line from the top—left to right; ii) Second line from the top—left to right; iii) Third line from the top—left to right, etc. 919 i) First line from the left—top to bottom; ii) Second line from the left—top to bottom; iii) Third line from the left—top to bottom, etc.

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In the previous example, for the purpose of a more comprehensible illustration, the horizontal and vertical image scanning was depicted by means of 20 horizontal and vertical lines. In practice, this limitation is not necessary and each horizontal and vertical line is scanned. The images of the size of 150 × 150 pixels were used, which implies that scanning should register 150 horizontal and 150 vertical lines. The result of the horizontal scanning is numerically represented in the form of two lists in Figure 169. Each number on the upper list represents the length of an uninterrupted stretch of black pixels. The bottom list shows the same values sorted in order from low to high. The sorting introduces an algebraic spectrum of measurements, and shows that certain lengths are more frequent than the others. The benefit of this kind of representation is that neither the location nor the orientation of the parts continues to play any role.

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array([

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In Figure 170, the results of both the horizontal and the vertical scanning procedures are presented in a graphical form. The top-left graph shows the distribution of the lengths of horizontal stretches of black pixels, displayed in the order of their measurement. The middle-left graph shows the same values sorted in order from low to high. The top-right graph shows the distribution

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Fig. 169

Original and sorted values obtained by the application of the translation and mirroring invariance bridge.

of the lengths of the vertical stretches of black pixels, displayed in the order of their measurement. The middle-right image shows the same values sorted in order from low to high. The two histograms on the bottom divide up the spectrum of measurements in a number of characteristic ranges and serve as the generalisation of the recorded distribution.

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4, 2, Fig. 43,170 1, 43, 2, 43, 2, 4, 44, 80, 2, 1, 3, 2, 45, showing 4, 44, Graphs 44, 44, 44, the 44, 44, 44, of measuring 4, 44, results 2, 44, 2, 44, 1, 43, 5, 105, a105, 105, single floor105, plan104, 105, 105, 0, 33, 6, 4, 2, 1, 50, 103, part by means 1, 51, 1, 52, 1, 52, 1, 51, the translation 1, 1, of51, 1, 52, 1, 11, 4, 1, 24, and 2,mirroring 11, 24, 1, 11, 24, 5, 1, invariance 1, 11, bridge. 25, 2, 1, 11, 2, 1, 11, 25, 2, 11, 24, 1, 2, 11, 25, 2, 1, 11, 25, 2, 4, 2, 10, 24, 2, 11, 24, 2, 4, 1, 2, 11, 24, 2, 2, 1, 4, 2, 2, 11, 24, 1, 2, 11, 1, 2, 11, 24, 2, 2, 11, 24, 2, 11, 24, 1, 2, 11, 24, 1, 1, 25, 1, 2, 27, 23, 2, 2, 2, 51, 1, 2, 51, 2, 4, 1, 1, 50, 1, 50, 2, 50, 2, 50, 4, 1, 105, 1, 105, 1, 105, 1, 1, 105, 110, 110, 2, 105, 2, 105, 7, 110, 104, 2, 1, 1, 2, 2, 0, 2, 50, 2, 50, 2, 50, 2, 2, 10, 24, 2, 10, 24, 2, 10, 0, 24, 2, 10, 24, 2, 10, 24, 4, 2, 10, 24, 2, 10, 24, 2,

In the current example, the spectrum of measurements regarding the continuous stretch of black pixels is divided into ten value ranges in horizontal and ten value ranges in the vertical direction. Once normalised, the frequency values characterising this distribution are concatenated into a twenty-dimensional characterisation of each part of a floor plan, as illustrated in Figure 171.

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Fig. 171

A sample of 63 extracted floor plan parts represented as a histogram of frequencies.

virtual and actual distinctions Generalisation bridge At this stage in the procedure, the floor plan object is represented by a large number of parts reduced to a 20-dimensional vector. Translated into frequencies on the basis of the applied encoding, the parts constituting a plan and their relations have lost the ability to be interpreted as images. Furthermore, the amount of information about the object in this entropic state is reduced to zero on the basis of which it can be characterised as a “black box.” To address this situation, the generalisation bridge of the self-organizing model is applied to the collection of elements in order to obtain a partition spectrum. As we have discussed in part III, Kohonen’s self-organizing map algorithm serves as the implementation of this generalisation bridge. The first level of the self-organizing model was defined as an unordered set of observables represented by the collection of floor plan parts. The generalisation procedure connects this level to another level, represented by a self-referential system of objects and arrows. One of the most important decisions when operating with selforganizing maps is choosing the number of objects and the architectonics of its system of relations. The system of relations was represented by means of a graph connected as a torus by means of a two-dimensional orthonormal lattice, containing 100 elements.

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Fig. 172

The same SOM response layer represented as a graph (left) and in a simplified two-dimensional grid form (right).

Partition spectrum The input layer of the self-organizing map consists of 573,488 twenty-dimensional vectors, obtained in the previous steps. The training procedure involved one million epochs until the response layer evolved into a stable configuration. The output of the SOM, in computer programming terminology, is a three-dimensional array whose first two dimensions are integers, indexing the cells of the lattice and the third is a list of 20 numerical frequency values normalised between 0 and 1, corresponding to the dimensionality of the input vectors. Figure 173 illustrates the list of values characterising a single cell of an array. Fig. 173

A single cell of an array has the same dimensionality as the input vectors. However, the values within the cell do not correspond to any actual extracted floor plan part.

array([ 0.00763766, 0.00434224, 0.01640242, 0.00082263,

0.00874811, 0.00577101, 0.01260795, 0.48120903,

0.0123074 , 0.00430378, 0.00184821, 0.01804983,

0.00274055, 0.00337214, 0.00056695, 0.00080054,

0.0129704 , 0.36586151, 0.01686004, 0.00395376])

As this representation does not provide any insight into the relations between different cells, Alfred Ultsch’s unified distance matrix was applied to the results. The U-matrix calculates the normalised sum of the distances between each cell and its immediate neighbours and assigns that value to each cell of the lattice. According to this value, the SOM response layer is visualised as a landscape of similarities, as shown in Figure 174. This image can also be regarded as the partition spectrum obtained on the basis of the generalisation procedure applied to the collection of floor plan parts.

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Fig. 174

Response layer of a trained self-organizing map interpreted as a partition spectrum.

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Distinctions In order to obtain information from the partition spectrum, we must attempt to register distinctions on it. Any attempt to do this will break the initial symmetry of relations and influence the final description. In order to reach actual distinctions, the simplest possible approach was chosen, that does not involve the procedure of iteratively refining the spectrum. 920 Instead of clustering the spectrum into a minimal number of areas—which can be done by recursively applying the generalisation procedure with increasing number of SOM cells and then refining them further—the distinctions found on the SOM’s output layer were interpreted as equivalence classes, each containing a class of virtual distinctions. Projection bridge After obtaining the equivalence classes, whose twenty-dimensional vectors imply the virtual distinctions, the projection bridge of the self-organizing model was applied to them. The projection procedure consists of selecting each of the 573,488 parts obtained on the basis of the partition procedure and comparing them to the 100 classes of virtual distinctions (represented by vectors within each equivalence class). As the encoding of the elements within the unordered set of observables matches the encoding of the classes of virtual distinctions, they

920 More complex distinction procedures have already been attempted and proved feasible, but put aside for the future work.

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can be compared by means of the Pythagorean metrical distance function. According to the criteria of metrical similarity, each observable will have the minimal distance to one, and only one, class of virtual distinctions. Since only a single class of virtual distinctions will be assigned to each floor plan, we can say that each class of virtual distinctions yields a set of actual distinctions on the basis of their metrical distance. This is illustrated in Figure 175. Here the class of virtual distinctions named S17, marked with a red square and indexed as with coordinates 1,7 creates a set of actual distinctions containing the empty parts within different floor plans. The same can be observed with the class of virtual distinctions S73, which creates a set of actual distinctions collecting floor plan parts of a different kind. At this point it is important to note that these virtual classes do not precisely match any semantic description of the involved parts. It is unknown which kinds of parts are displayed within the class S73, although they are certainly different from the elements of another class. Yet, even without this semantic characterisation, these classes can still be used to describe any floor plan created with the same conventions as those from the dataset. 1, 7

Fig. 175

Virtual distinctions with indexes (1, 7), (7, 3), (6, 2) and the corresponding sets of actual distinctions.

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Virtual distinctions with indexes (5, 9), (1, 2), (0, 3) and the corresponding sets of actual distinctions.

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To better illustrate the results of the projection bridge, the graph of the U-matrix is superimposed with an image, depicting each sets of actual 0 distinctions by 100 randomly chosen representatives. 2

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U-matrix representation overlaid with the sets of actual distinctions.

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Finally, Figure 178 shows ten thousand randomly chosen parts of floor plan images, from the initial pool of 573,488, categorised on the basis of one hundred classes of virtual distinctions.

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Fig. 178

10’000 randomly chosen floor plan parts extracted from the initial data set and clustered into 100 sets of actual distinctions.

IV modelling with conjugate symbolic domains symbolising spectral distinctions: a floor plan alphabet Conditions of a language : The search for “prime words” (“divisible” only by themselves and by unity). Take a Larousse dict. and copy all the so-called “abstract” words, i.e., those which have no concrete reference. Compose a schematic sign designating each of these words. (this sign can be composed with the standard stops) These signs must be thought of as the letters of the new alphabet. A grouping of several signs will determine (utilise colors— in order to differentiate what would correspond in this [literature] to the substantive, verb, adverb declensions, conjugations, etc.) necessity for ideal continuity, i.e.: each grouping will be connected with the other groupings by a strict meaning (a sort of

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grammar, no longer requiring a pedagogical sentence construction. But apart from the differences of languages, and the “figures of speech” peculiar to each language—weights and measures some abstractions of substantives, of negatives, of relations of subject to verb, etc, by means of standard signs. (Representing these new relations: conjugations, declensions, plural and singular, adjectivation inexpressible by the concrete alphabetic forms of languages living now and to come.). This alphabet very probably is only suitable for the description of this picture. M. Duchamp, “Bride’s Veil” (1914: 32) From the perspective of communication, and very much in the spirit of Marcel Duchamp’s quote, we can indeed utilise any substance-expression without concrete reference and think of it as an alphabet. Hjelmslev would describe this alphabet as standing in an arbitrary relation to the purport. As such, we can symbolise the 100 classes of virtual distinctions as 100 new characters in an alphabet. It is entirely possible to assign the letters of the Latin alphabet (or any other alphabet, for that matter) to these classes, but their unusually large number does not allow this to be done in an elegant way. For this reason, the letters of this alphabet are represented by combinations of the letter S and an index from 00 to 99, as shown in the Figure 179.

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fig. 179 S00

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alPHaBETic cHaracTEriSaTioN of a floor PlaN To characterise a floor plan in terms of the alphabet, we return to the initial level of representation containing unpartitioned floor plan images. Any image can be translated into the new symbolic domain, as long as it is in the same scale and employs the same drawing conventions as the images of the original dataset. If these requirements are met, the following steps need to be taken towards the successful translation: 1 The image is rotated according to the most dominant rotation value of its lines. 2 The partition procedure divides the image into a number of overlapping parts. 3 The procedure starts from the top and ends with the bottom row, while the individual parts are cut from left to the right, and their order recorded. The overlapping value between the parts can be selected freely but its variation will affect the quality of the description.

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Floor plan alphabet created on top of the partition spectrum.

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The translation and mirroring invariance bridge translates each part into a vector, whose dimensions must match the dimensions previously defined for the data set. 5 The distance is measured between each obtained part and each virtual distinction. The part adopts the symbol from the virtual distinction to which it has minimal distance. 6 The symbolic description of the parts is distributed in the same order in which the individual parts were cut. The floor plan in Figure 180 is chosen to demonstrate the outcome of the translation into the new symbolic level. Fig. 180

A single floor plan specimen from the original data set. (adopted from Ebner et al., 2009)

By applying the previously defined steps, a matching representation of the floor plan is created within the symbolic domain. Fig. 181

A symbolic representation of the plan from the Figure 180.

S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S07—S67—S57—S57—S67—S73—S73—S74—S27—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S48—S13—S11—S21—S21—S21—S12—S32—S64—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S37—S03—S12—S11—S11—S11—S02—S37—S65—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S08—S28—S79—S96—S96—S96—S28—S28—S55—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S08—S08—S79—S71—S92—S92—S09—S08—S55—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S08—S08—S79—S26—S82—S92—S09—S08—S55—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S04—S23—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S23—S32—S08—S08—S79—S23—S11—S11—S02—S02—S21—S20—S20—S20—S23—S04—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S37—S13—S11—S20—S20—S20—S20—S20—S20—S21—S21—S20—S20—S20—S23—S32—S37—S37—S78—S23—S11—S11—S02—S02—S21—S20—S20—S11—S13—S48—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S39—S12—S20—S20—S20—S20—S20—S20—S11—S11—S20—S20—S20—S23—S32—S04—S14—S04—S23—S11—S21—S20—S20—S20—S20—S20—S12—S39—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S08—S99—S17—S17—S17—S17—S17—S26—S44—S87—S86—S96—S86—S76—S76—S86—S96—S97—S87—S45—S35—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S08—S99—S17—S17—S17—S17—S17—S16—S26—S87—S87—S96—S86—S76—S76—S86—S96—S97—S87—S45—S16—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S98—S98—S96—S86—S76—S76—S86—S96—S98—S98—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S99—S99—S81—S81—S26—S25—S80—S71—S99—S99—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S06—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S99—S99—S81—S25—S06—S06—S26—S80—S99—S99—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S68—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S98—S98—S32—S71—S53—S34—S70—S32—S99—S98—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S68 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S78—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S88—S32—S32—S53—S53—S34—S60—S32—S22—S88—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S88 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S79—S19—S19—S99—S17—S17—S17—S17—S17—S17—S26—S32—S32—S32—S53—S53—S34—S60—S32—S32—S32—S25—S17—S17—S17—S17—S17—S17—S99—S19—S19—S99 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S79—S19—S19—S99—S17—S17—S17—S17—S17—S17—S25—S81—S81—S17—S17—S17—S17—S17—S17—S81—S81—S25—S17—S17—S17—S17—S17—S17—S99—S19—S19—S99 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S79—S29—S29—S99—S53—S53—S53—S34—S60—S34—S32—S32—S04—S34—S34—S44—S43—S43—S44—S04—S32—S32—S53—S53—S43—S34—S34—S53—S99—S29—S29—S99 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S79—S41—S50—S61—S62—S52—S61—S60—S60—S61—S49—S49—S48—S34—S60—S44—S53—S53—S44—S48—S49—S49—S52—S62—S61—S60—S61—S62—S51—S41—S41—S99 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S68—S49—S49—S48—S62—S52—S61—S60—S60—S61—S49—S49—S48—S34—S60—S44—S53—S53—S44—S48—S49—S49—S52—S62—S61—S60—S61—S62—S51—S49—S49—S88 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S06—S85—S85—S44—S53—S53—S34—S60—S34—S44—S85—S85—S85—S34—S34—S34—S43—S43—S44—S04—S32—S32—S53—S44—S34—S34—S43—S53—S33—S32—S32—S06

274

TOWARDS COMMUNICATION IN CAAD

It is important to remark that the symbols used to describe the image exactly match the ones we have obtained in the previous step, as shown in previous figures. This makes it possible to assess the quality of the model and test the credibility of results. In order to make this easier, is it possible to overlap two different representations of the same floor plan in one image. S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S07—S67—S57—S57—S67—S73—S73—S74—S27—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S48—S13—S11—S21—S21—S21—S12—S32—S64—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S37—S03—S12—S11—S11—S11—S02—S37—S65—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S08—S28—S79—S96—S96—S96—S28—S28—S55—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S08—S08—S79—S71—S92—S92—S09—S08—S55—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S08—S08—S79—S26—S82—S92—S09—S08—S55—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S04—S23—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S23—S32—S08—S08—S79—S23—S11—S11—S02—S02—S21—S20—S20—S20—S23—S04—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S37—S13—S11—S20—S20—S20—S20—S20—S20—S21—S21—S20—S20—S20—S23—S32—S37—S37—S78—S23—S11—S11—S02—S02—S21—S20—S20—S11—S13—S48—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S39—S12—S20—S20—S20—S20—S20—S20—S11—S11—S20—S20—S20—S23—S32—S04—S14—S04—S23—S11—S21—S20—S20—S20—S20—S20—S12—S39—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S08—S99—S17—S17—S17—S17—S17—S26—S44—S87—S86—S96—S86—S76—S76—S86—S96—S97—S87—S45—S35—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S08—S99—S17—S17—S17—S17—S17—S16—S26—S87—S87—S96—S86—S76—S76—S86—S96—S97—S87—S45—S16—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S98—S98—S96—S86—S76—S76—S86—S96—S98—S98—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S99—S99—S81—S81—S26—S25—S80—S71—S99—S99—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S06—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S99—S99—S81—S25—S06—S06—S26—S80—S99—S99—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S68—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S98—S98—S32—S71—S53—S34—S70—S32—S99—S98—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S68 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S78—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S88—S32—S32—S53—S53—S34—S60—S32—S22—S88—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S88 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S79—S19—S19—S99—S17—S17—S17—S17—S17—S17—S26—S32—S32—S32—S53—S53—S34—S60—S32—S32—S32—S25—S17—S17—S17—S17—S17—S17—S99—S19—S19—S99 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S79—S19—S19—S99—S17—S17—S17—S17—S17—S17—S25—S81—S81—S17—S17—S17—S17—S17—S17—S81—S81—S25—S17—S17—S17—S17—S17—S17—S99—S19—S19—S99 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S79—S29—S29—S99—S53—S53—S53—S34—S60—S34—S32—S32—S04—S34—S34—S44—S43—S43—S44—S04—S32—S32—S53—S53—S43—S34—S34—S53—S99—S29—S29—S99 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S79—S41—S50—S61—S62—S52—S61—S60—S60—S61—S49—S49—S48—S34—S60—S44—S53—S53—S44—S48—S49—S49—S52—S62—S61—S60—S61—S62—S51—S41—S41—S99 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S68—S49—S49—S48—S62—S52—S61—S60—S60—S61—S49—S49—S48—S34—S60—S44—S53—S53—S44—S48—S49—S49—S52—S62—S61—S60—S61—S62—S51—S49—S49—S88 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S06—S85—S85—S44—S53—S53—S34—S60—S34—S44—S85—S85—S85—S34—S34—S34—S43—S43—S44—S04—S32—S32—S53—S44—S34—S34—S43—S53—S33—S32—S32—S06

Fig. 182

Floor plan specimen overlaid with its symbolic representation.

The following six images serve as an example of this feature. S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S06-S14-S14-S14-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S05-S04-S03-S03-S03-S06-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S16-S14-S03-S03-S03-S03-S03-S04-S16-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S06-S04-S31-S22-S03-S13-S03-S03-S03-S14-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S05-S32-S31-S31-S22-S22-S79-S68-S03-S03-S03-S05-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S05-S31-S41-S31-S14-S68-S79-S78-S04-S03-S03-S04-S06-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S05-S49-S41-S49-S14-S05-S78-S79-S78-S04-S03-S03-S04-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S16-S48-S41-S41-S48-S72-S68-S78-S78-S04-S03-S03-S03-S14-S17-S17-S17-S17-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S48-S49-S49-S41-S48-S93-S68-S68-S04-S04-S03-S03-S03-S06-S17-S17-S17-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S06-S73-S73-S73-S73-S73-S73-S73-S73-S73-S34-S34-S48-S48-S49-S41-S48-S84-S85-S85-S14-S05-S04-S03-S03-S04-S06-S17-S17-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S06-S04-S23-S20-S20-S20-S20-S20-S20-S20-S20-S20-S20-S02-S02-S12-S12-S39-S84-S85-S15-S15-S06-S06-S04-S03-S31-S32-S14-S27-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S46-S46-S23-S20-S21-S11-S11-S11-S21-S21-S21-S20-S20-S21-S12-S21-S21-S13-S85-S85-S06-S06-S16-S17-S05-S03-S31-S31-S31-S32-S05-S16-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S46-S46-S67-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S67-S66-S16-S17-S17-S17-S17-S16-S05-S42-S32-S31-S31-S31-S32-S05-S16-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S46-S46-S46-S15-S15-S35-S43-S43-S43-S44-S15-S06-S16-S06-S06-S25-S35-S44-S35-S06-S16-S17-S17-S16-S15-S44-S53-S62-S42-S32-S31-S31-S31-S04-S06 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S36-S36-S36-S25-S15-S15-S15-S35-S15-S06-S16-S16-S17-S17-S06-S15-S35-S44-S35-S15-S16-S16-S16-S15-S15-S44-S43-S53-S71-S53-S32-S31-S32-S04-S06 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S06-S44-S71-S35-S35-S25-S43-S71-S71-S72-S35-S15-S17-S17-S06-S44-S43-S53-S43-S44-S16-S16-S06-S15-S25-S44-S43-S53-S43-S44-S35-S72-S72-S14-S06 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S06-S34-S71-S71-S72-S73-S73-S73-S73-S91-S72-S35-S34-S34-S34-S34-S34-S53-S53-S43-S43-S43-S44-S06-S15-S44-S43-S44-S35-S35-S72-S72-S35-S26-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S06-S04-S23-S21-S21-S21-S20-S11-S11-S11-S11-S21-S20-S20-S20-S20-S20-S20-S20-S20-S20-S20-S22-S32-S05-S44-S71-S44-S26-S72-S72-S35-S06-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S06-S04-S23-S20-S20-S20-S20-S21-S21-S11-S21-S20-S20-S20-S20-S20-S20-S20-S20-S20-S20-S20-S22-S31-S31-S32-S24-S35-S26-S81-S81-S15-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S16-S06-S67-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S57-S67-S22-S31-S31-S31-S31-S32-S36-S36-S25-S15-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S05-S32-S31-S31-S31-S31-S32-S46-S16-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S16-S05-S32-S31-S31-S22-S36-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S16-S05-S32-S04-S05-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17

AN EXPERIMENT

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Fig. 183

Floor plan specimen overlaid with its symbolic representation.

Fig. 184

Floor plan specimen overlaid with its symbolic representation.

S17—S17—S17—S05—S49—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S00—S23—S13—S07—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S14—S49—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S49—S14—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S91—S22—S11—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S11—S11—S11—S20—S23—S14—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S91—S91—S90—S35—S26—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S16—S25—S35—S90—S90—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S91—S91—S91—S34—S06—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S06—S34—S90—S90—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S92—S92—S91—S16—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S16—S90—S90—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S92—S92—S91—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S91—S91—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S16—S06—S34—S91—S91—S91—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S91—S91—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S26—S26—S34—S91—S91—S91—S06—S34—S85—S94—S94—S94—S94—S94—S94—S94—S94—S94—S94—S94—S84—S85—S44—S06—S06—S92—S92—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S26—S26—S34—S91—S91—S91—S06—S34—S85—S94—S94—S94—S94—S94—S94—S94—S94—S94—S94—S94—S84—S85—S44—S06—S06—S92—S92—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S25—S25—S25—S91—S91—S91—S06—S34—S93—S94—S94—S94—S94—S94—S94—S94—S94—S94—S94—S94—S84—S85—S44—S06—S06—S92—S92—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S25—S25—S35—S91—S80—S80—S17—S15—S71—S71—S61—S61—S61—S53—S43—S34—S70—S60—S60—S61—S71—S71—S26—S06—S34—S92—S92—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S25—S82—S82—S91—S70—S80—S17—S68—S32—S11—S11—S11—S11—S11—S11—S11—S11—S11—S11—S11—S23—S32—S36—S35—S34—S92—S92—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S25—S92—S91—S92—S70—S71—S44—S68—S32—S11—S11—S02—S02—S11—S11—S11—S11—S11—S11—S11—S23—S78—S46—S35—S34—S91—S91—S82—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S25—S91—S91—S92—S35—S25—S44—S81—S76—S96—S96—S28—S28—S79—S96—S86—S57—S57—S57—S57—S68—S68—S46—S35—S35—S91—S91—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S26—S90—S90—S81—S06—S25—S34—S81—S35—S25—S17—S08—S08—S79—S35—S71—S91—S91—S81—S25—S68—S68—S44—S35—S35—S91—S91—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S26—S90—S90—S80—S17—S25—S34—S81—S35—S25—S17—S08—S08—S79—S35—S71—S91—S91—S81—S25—S05—S05—S35—S26—S16—S91—S91—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S26—S90—S90—S80—S17—S25—S44—S72—S72—S73—S73—S08—S08—S79—S73—S73—S73—S73—S73—S73—S78—S78—S46—S06—S17—S91—S91—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S25—S90—S90—S80—S17—S15—S44—S14—S32—S11—S21—S02—S02—S21—S21—S11—S11—S11—S11—S21—S23—S78—S46—S17—S17—S91—S91—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S25—S91—S91—S82—S17—S16—S06—S14—S32—S21—S20—S12—S12—S21—S20—S20—S21—S21—S20—S20—S23—S32—S06—S17—S17—S92—S92—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S25—S91—S91—S82—S17—S17—S17—S27—S34—S34—S34—S34—S34—S34—S34—S34—S34—S34—S44—S44—S44—S06—S17—S17—S17—S92—S92—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S25—S91—S91—S82—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S91—S91—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S25—S91—S91—S82—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S91—S91—S91—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S26—S90—S90—S62—S43—S06—S06—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S25—S71—S91—S91—S91—S34—S15—S15—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S26—S90—S90—S71—S71—S71—S34—S16—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S82—S81—S91—S70—S35—S35—S35—S15—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S26—S80—S80—S70—S71—S80—S35—S26—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S92—S92—S92—S06—S16—S25—S25—S25—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S26—S26—S26—S80—S91—S91—S44—S35—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S92—S92—S92—S17—S17—S25—S25—S25—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S26—S26—S26—S80—S90—S90—S44—S06—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S92—S92—S92—S17—S17—S25—S25—S25—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S26—S26—S26—S80—S90—S90—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S92—S92—S92—S17—S17—S25—S25—S25—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S26—S26—S26—S80—S90—S90—S17—S17—S17—S17—S17—S17—S06—S76—S86—S76—S05—S43—S43—S44—S34—S34—S35—S92—S92—S92—S17—S17—S25—S25—S25—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S25—S25—S25—S80—S90—S90—S17—S17—S17—S17—S17—S17—S46—S77—S87—S87—S87—S61—S61—S61—S61—S61—S35—S92—S92—S92—S17—S17—S26—S26—S26—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S25—S35—S35—S80—S90—S90—S17—S17—S17—S17—S17—S17—S46—S67—S57—S57—S47—S70—S70—S34—S34—S71—S35—S92—S92—S92—S17—S17—S25—S25—S25—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S15—S35—S34—S90—S90—S90—S17—S17—S17—S17—S17—S17—S06—S66—S67—S67—S34—S34—S34—S34—S34—S34—S06—S92—S91—S91—S17—S17—S25—S25—S25—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S16—S06—S34—S90—S90—S67—S57—S57—S57—S57—S57—S57—S57—S57—S57—S57—S57—S57—S57—S57—S57—S57—S57—S57—S57—S57—S67—S73—S73—S73—S83—S27 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S80—S49—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S49—S48 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S04—S49—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S59—S49—S48 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S66—S32—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S00—S10—S10—S10—S10—S32—S14

276

TOWARDS COMMUNICATION IN CAAD

S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S05—S53—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S53—S44—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S88—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S62—S52—S82—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S99—S99—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S52—S62—S62—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S99—S99—S99—S52—S52—S52—S52—S62—S62—S62—S62—S52—S62—S62—S62—S52—S52—S62—S62—S62—S62—S62—S62—S62—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S99—S99—S99—S62—S62—S62—S62—S62—S62—S62—S62—S62—S62—S62—S62—S62—S62—S62—S62—S62—S62—S62—S62—S61—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S99—S99—S99—S62—S62—S62—S61—S61—S61—S61—S61—S61—S62—S62—S61—S61—S62—S61—S61—S61—S61—S61—S61—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S99—S99—S99—S52—S52—S62—S61—S04—S44—S34—S34—S34—S53—S53—S43—S44—S43—S44—S34—S44—S53—S53—S04—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S99—S99—S99—S52—S52—S52—S04—S04—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S06—S14—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S99—S99—S99—S52—S62—S62—S82—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S25—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S07—S76—S96—S97—S98—S52—S52—S62—S62—S82—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S25—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S05—S76—S96—S97—S52—S52—S52—S62—S62—S82—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S25—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S68—S67—S57—S62—S62—S62—S62—S62—S62—S82—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S25—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S78—S78—S62—S62—S62—S61—S61—S61—S92—S92—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S25—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S78—S78—S78—S99—S99—S99—S61—S61—S61—S92—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S25—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S78—S78—S78—S99—S99—S99—S62—S62—S62—S82—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S25—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S79—S79—S79—S99—S99—S99—S62—S62—S62—S04—S14—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S06—S14—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S79—S79—S79—S99—S99—S62—S61—S61—S61—S04—S04—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S06—S04—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S68—S68—S60—S60—S60—S61—S61—S61—S61—S04—S04—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S06—S04—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S06—S34—S60—S60—S60—S34—S34—S71—S71—S71—S44—S34—S34—S34—S34—S44—S60—S60—S60—S60—S71—S71—S34—S44—S44—S71—S91—S91—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S81—S82—S81—S44—S35—S35—S64—S64—S64—S60—S60—S60—S61—S92—S71—S34—S44—S44—S71—S91—S91—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S82—S82—S82—S44—S71—S71—S55—S55—S55—S60—S60—S60—S92—S92—S71—S34—S44—S44—S71—S91—S91—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S82—S82—S82—S71—S71—S71—S55—S55—S55—S60—S90—S72—S92—S82—S16—S17—S17—S17—S25—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S82—S82—S82—S34—S71—S35—S55—S55—S55—S60—S90—S67—S64—S64—S17—S17—S17—S17—S25—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S82—S82—S92—S44—S43—S44—S55—S55—S55—S65—S65—S46—S55—S55—S17—S17—S17—S17—S25—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S82—S92—S92—S52—S52—S53—S55—S55—S55—S55—S55—S46—S55—S55—S17—S17—S17—S17—S25—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S82—S92—S92—S53—S53—S43—S55—S55—S55—S55—S55—S52—S55—S55—S17—S17—S17—S17—S25—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S82—S92—S92—S53—S53—S53—S55—S55—S55—S55—S55—S46—S55—S55—S17—S17—S17—S17—S25—S92—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S82—S32—S23—S21—S21—S20—S21—S11—S12—S21—S11—S11—S11—S21—S20—S20—S20—S20—S21—S23—S92—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S15—S32—S23—S20—S20—S20—S21—S21—S21—S21—S21—S11—S21—S21—S20—S20—S20—S20—S20—S23—S14—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S06—S32—S23—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S23—S04—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17

S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S14—S85—S94—S94—S94—S94—S94—S94—S94—S84—S22—S20—S20—S20—S20—S23—S14—S16—S17—S17—S17—S17—S16—S04—S23—S20—S21—S32—S04—S23—S20—S21—S32—S05—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S48—S48—S84—S94—S94—S94—S94—S94—S94—S84—S22—S11—S11—S11—S01—S23—S48—S16—S17—S17—S17—S17—S14—S03—S23—S20—S21—S32—S04—S23—S20—S11—S32—S35—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S08—S28—S84—S94—S94—S94—S94—S94—S94—S99—S22—S01—S11—S11—S02—S13—S38—S16—S17—S17—S17—S06—S03—S03—S23—S21—S21—S32—S04—S23—S21—S11—S32—S35—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S08—S08—S08—S17—S17—S17—S17—S46—S88—S99—S88—S73—S73—S73—S48—S48—S14—S17—S17—S17—S17—S04—S03—S03—S04—S43—S43—S43—S35—S15—S71—S70—S71—S35—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S08—S08—S08—S17—S17—S17—S17—S06—S43—S88—S63—S73—S73—S48—S48—S48—S06—S17—S17—S17—S05—S03—S03—S03—S43—S43—S43—S06—S06—S35—S70—S70—S35—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S08—S08—S08—S17—S17—S17—S17—S17—S44—S14—S32—S21—S21—S13—S48—S14—S17—S17—S17—S06—S04—S03—S03—S04—S35—S16—S17—S17—S35—S71—S70—S71—S16—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S08—S08—S08—S17—S17—S17—S17—S17—S06—S14—S31—S21—S20—S23—S04—S06—S17—S17—S17—S04—S03—S03—S04—S35—S15—S17—S17—S16—S35—S91—S71—S35—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S08—S08—S08—S17—S17—S17—S17—S17—S17—S05—S04—S03—S96—S21—S20—S20—S20—S20—S20—S21—S03—S03—S72—S35—S17—S17—S17—S35—S71—S91—S35—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S08—S08—S08—S17—S17—S17—S17—S17—S17—S17—S14—S03—S23—S20—S20—S20—S20—S20—S20—S21—S32—S04—S35—S06—S17—S17—S06—S70—S70—S71—S15—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S08—S08—S08—S17—S17—S17—S17—S17—S17—S17—S16—S14—S23—S20—S20—S20—S20—S20—S20—S22—S32—S06—S06—S17—S17—S17—S35—S70—S70—S35—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S08—S08—S08—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S35—S71—S70—S71—S16—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S08—S08—S08—S17—S17—S17—S17—S17—S16—S16—S16—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S27—S27—S06—S16—S71—S71—S71—S35—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S08—S08—S08—S17—S17—S17—S17—S17—S04—S04—S14—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S04—S04—S04—S35—S71—S71—S35—S17—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S08—S08—S08—S17—S17—S17—S17—S17—S04—S04—S14—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S04—S04—S04—S71—S71—S71—S06—S17—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S08—S08—S08—S17—S17—S17—S17—S17—S04—S04—S14—S16—S17—S17—S17—S17—S17—S17—S16—S16—S16—S14—S14—S04—S70—S70—S35—S17—S17—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S37—S37—S37—S44—S63—S63—S63—S52—S52—S62—S52—S52—S52—S52—S52—S52—S62—S52—S52—S62—S62—S52—S52—S61—S71—S71—S17—S17—S17—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S04—S04—S04—S44—S63—S63—S63—S52—S52—S62—S52—S52—S52—S52—S52—S52—S62—S52—S52—S62—S62—S52—S52—S71—S71—S35—S17—S17—S17—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S06—S43—S53—S44—S63—S63—S63—S52—S52—S53—S53—S53—S52—S52—S52—S52—S62—S52—S52—S53—S62—S52—S52—S43—S06—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17

AN EXPERIMENT

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Fig. 185

Floor plan specimen overlaid with its symbolic representation.

Fig. 186

Floor plan specimen overlaid with its symbolic representation.

Fig. 187

Floor plan specimen overlaid with its symbolic representation.

S06-S34-S60-S60-S60-S34-S34-S34-S34-S34-S17-S17-S68-S49-S02-S00-S00-S00-S00-S00-S00-S00-S01-S01-S00-S00-S00-S00-S00-S00-S00-S00-S00-S02-S48-S06-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S34-S60-S60-S60-S60-S60-S61-S61-S61-S70-S26-S17-S79-S41-S02-S01-S11-S11-S00-S00-S00-S01-S02-S02-S00-S00-S00-S00-S00-S00-S00-S00-S00-S02-S37-S46-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S80-S70-S60-S60-S60-S60-S61-S61-S70-S80-S80-S17-S79-S28-S28-S94-S94-S94-S94-S94-S94-S94-S28-S28-S94-S94-S94-S94-S94-S94-S94-S94-S94-S37-S37-S46-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S90-S90-S43-S43-S44-S44-S34-S34-S90-S90-S90-S17-S79-S29-S28-S44-S44-S43-S43-S43-S43-S99-S08-S08-S17-S17-S17-S17-S17-S17-S17-S17-S64-S48-S48-S26-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S90-S90-S17-S17-S17-S17-S17-S17-S90-S90-S80-S34-S68-S48-S48-S44-S06-S17-S17-S17-S17-S99-S08-S08-S17-S17-S17-S17-S17-S17-S17-S17-S26-S90-S90-S26-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S90-S90-S17-S17-S17-S17-S17-S17-S80-S80-S70-S34-S70-S90-S90-S44-S06-S17-S17-S17-S06-S88-S37-S37-S17-S17-S17-S17-S17-S17-S17-S17-S26-S90-S90-S26-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S90-S90-S17-S17-S17-S17-S17-S17-S34-S34-S60-S60-S70-S90-S90-S34-S15-S17-S17-S16-S35-S34-S04-S04-S17-S17-S17-S17-S17-S17-S17-S17-S26-S90-S90-S26-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S90-S90-S53-S53-S53-S53-S53-S53-S43-S34-S34-S34-S71-S90-S90-S94-S94-S94-S94-S94-S94-S94-S85-S04-S04-S23-S21-S20-S20-S23-S14-S17-S64-S80-S80-S26-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S80-S71-S53-S53-S53-S53-S53-S53-S43-S34-S34-S34-S35-S90-S93-S94-S94-S94-S94-S94-S94-S94-S85-S04-S37-S13-S11-S21-S20-S23-S04-S06-S65-S37-S37-S16-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S06-S44-S53-S53-S53-S53-S53-S53-S43-S44-S16-S17-S25-S80-S93-S94-S94-S94-S94-S94-S94-S94-S85-S14-S08-S39-S12-S11-S21-S23-S04-S44-S65-S37-S37-S06-S16 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S25-S92-S92-S17-S17-S17-S17-S17-S17-S17-S17-S17-S08-S09-S79-S70-S70-S70-S88-S44-S64-S48-S48-S34-S26 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S25-S92-S92-S17-S17-S17-S17-S17-S17-S17-S17-S17-S08-S28-SS7-SS7-SS7-SS7-S67-S67-S64-S04-S04-S34-S35 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S15-S92-S92-S17-S17-S17-S17-S17-S17-S17-S16-S06-S38-S85-S95-S95-S95-S94-S93-S85-S64-S48-S48-S34-S26 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S78-S48-S48-S17-S17-S17-S17-S17-S17-S17-S26-S26-S47-S67-S94-S94-S94-S93-S93-S93-S55-S09-S09-S44-S06 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S78-S37-S37-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S93-S93-S94-S94-S29-S08-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S78-S48-S84-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S48-S48-S17-S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S17-S0S-S33-S84-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S94-S85-S33-S17-S17

communication between two natures of architectural representation Since any floor plan image that satisfies the criteria defined in the beginning of this chapter can be described by means of two conjugate levels regardless of its content, we can say that we have established coexistence between the two levels of our model. By means of the adjunction between the encoding and decoding bridges we have created, it is also possible to reflect any change in one of the levels, within the corresponding conjugate level. In this regard we have established communication between different levels of our model.

Fig. 188

Communication between two levels of a model.

Level A

Level B

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S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S07—S67—S57—S57—S67—S73—S73—S74—S27—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S48—S13—S11—S21—S21—S21—S12—S32—S64—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S37—S03—S12—S11—S11—S11—S02—S37—S65—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S08—S28—S79—S96—S96—S96—S28—S28—S55—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S08—S08—S79—S71—S92—S92—S09—S08—S55—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S17—S08—S08—S79—S26—S82—S92—S09—S08—S55—S17—S17—S17—S17—S17—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S04—S23—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S20—S23—S32—S08—S08—S79—S23—S11—S11—S02—S02—S21—S20—S20—S20—S23—S04—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S37—S13—S11—S20—S20—S20—S20—S20—S20—S21—S21—S20—S20—S20—S23—S32—S37—S37—S78—S23—S11—S11—S02—S02—S21—S20—S20—S11—S13—S48—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S39—S12—S20—S20—S20—S20—S20—S20—S11—S11—S20—S20—S20—S23—S32—S04—S14—S04—S23—S11—S21—S20—S20—S20—S20—S20—S12—S39—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S08—S99—S17—S17—S17—S17—S17—S26—S44—S87—S86—S96—S86—S76—S76—S86—S96—S97—S87—S45—S35—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S08—S99—S17—S17—S17—S17—S17—S16—S26—S87—S87—S96—S86—S76—S76—S86—S96—S97—S87—S45—S16—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S98—S98—S96—S86—S76—S76—S86—S96—S98—S98—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S17—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S99—S99—S81—S81—S26—S25—S80—S71—S99—S99—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S06—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S99—S99—S81—S25—S06—S06—S26—S80—S99—S99—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S17 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S68—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S98—S98—S32—S71—S53—S34—S70—S32—S99—S98—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S68 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S78—S08—S08—S99—S17—S17—S17—S17—S17—S17—S26—S88—S32—S32—S53—S53—S34—S60—S32—S22—S88—S46—S17—S17—S17—S17—S17—S17—S99—S08—S08—S88 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S79—S19—S19—S99—S17—S17—S17—S17—S17—S17—S26—S32—S32—S32—S53—S53—S34—S60—S32—S32—S32—S25—S17—S17—S17—S17—S17—S17—S99—S19—S19—S99 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S79—S19—S19—S99—S17—S17—S17—S17—S17—S17—S25—S81—S81—S17—S17—S17—S17—S17—S17—S81—S81—S25—S17—S17—S17—S17—S17—S17—S99—S19—S19—S99 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S79—S29—S29—S99—S53—S53—S53—S34—S60—S34—S32—S32—S04—S34—S34—S44—S43—S43—S44—S04—S32—S32—S53—S53—S43—S34—S34—S53—S99—S29—S29—S99 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S79—S41—S50—S61—S62—S52—S61—S60—S60—S61—S49—S49—S48—S34—S60—S44—S53—S53—S44—S48—S49—S49—S52—S62—S61—S60—S61—S62—S51—S41—S41—S99 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S68—S49—S49—S48—S62—S52—S61—S60—S60—S61—S49—S49—S48—S34—S60—S44—S53—S53—S44—S48—S49—S49—S52—S62—S61—S60—S61—S62—S51—S49—S49—S88 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S06—S85—S85—S44—S53—S53—S34—S60—S34—S44—S85—S85—S85—S34—S34—S34—S43—S43—S44—S04—S32—S32—S53—S44—S34—S34—S43—S53—S33—S32—S32—S06

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With the resolution of this step, we have unintentionally touched upon an important question dealing with the nature of representation. If an object may be represented in a completely different domain, and still represented in its own terms, we can no longer talk about the true nature of any representation. As exotic or alien as our symbolic representation of floor plans seems to be, it has the same claim to represent our object as any other. Consequently, I suggest approaching the problem of representation from the standpoint of appropriateness and not truthfulness, just like mathematicians did 170 years ago. probabilistic notion of similarity within the symbolic This chapter opened with the hypothesis that the new symbolic domain of representation would provide a better basis for the comparison of floor plans. Unlike a floor plan image, which characterises the architectural space by means of a single object, the new symbolic basis characterises the same space by means of multiple abstract objects in a particular arrangement. As demonstrated in the chapter “Markov’s mechanisation of entropic cuts,” and further reinforced with the work of Hjelmslev, the ordering of the elements can indeed serve as a basis for probabilistic description. The symbolic representation of a floor plan was interpreted as a sequence from which a Markov chain was created. Each symbol of the alphabet in the Markov chain represents a state of the model and the probabilities for that state are calculated according to the sequence of symbols that represent the floor plan. The result is stored in a 100 × 100 matrix, containing 10,000 probability values. At this point, each of our floor plans can be characterised in probabilistic terms by means of 10,000 values taking part as the objects’ dimensions. The generalisation procedure can now be applied on floor plan objects, on the basis of this new characterisation. The obtained partition spectrum reveals the similarities between the floor plans. The result of this procedure is displayed in Figure 189.

AN EXPERIMENT

279

Finally, the measure of similarity between any two floor plans A and B taken from our original data set can be expressed by the topological distance measured on Figure 189. This value is obtained by counting the minimal number of steps between two blocks, where one contains the floor plan A and another contains the floor plan B. If the two blocks are immediate neighbours, this distance will be equal to one and it will increase if the two blocks are further apart.

AN EXPERIMENT

281

Fig. 189

Topological distance as the measure of floor plan similarity.

EPILOGUE

V

Information technology seems to be relentless in its changes and challenges posed to society. In only the last five years since this text was originally conceived, much has happened. Architectural education, not being able to evade the impact of programmable architectural tools on future architects, began integrating these tools into its curricula. In this sincere effort, however, lies a formidable lack of understanding of what these computational models are about. Technological artefacts are appropriated without reflection, and pragmatically put to use. The intention of the first part of this work was to draw attention to the emergence of computation within the context of the 19th century’s foundational crisis, and in this regard, present computational models in architecture within a broader technical and intellectual context. It is important to remember that the provided categorisation of models is not in any way meant to be absolute or immutable. On the contrary, it is a deliberately bold, but well-researched, statement that begs to be challenged. Doing so, however, will require delving deeper into the modelling concepts and mathematics that constitute them, which is exactly what we encourage. The categorisation provided here proved to be a valuable instrument for understanding why computational models in architecture continue to revolve around the same conceptual ideas, rarely surpassing what sheer computational power allows them. Without reflecting on some very old, important problems that had agitated Frege, Russell, Dedekind, Hilbert, Riemann and many others, this assessment would not be possible. An ambitious plan was set out for the rest of the work: to challenge the established computational models in architecture. The development of the Internet as a communication infrastructure has extended the capacity of education. Anyone owning a computer, tablet or smart phone, can learn how to use, for example, machine learning for a range of different applications. However, the very conception of courses on the topic still remains very much under the authority of its “native” domains (computer science and engineering), while the medium has evolved. It follows that if an architect wishes to have a deeper understanding of computation to institute his or her own modelling approach, he or she must have an equivalent of an engineering degree, incorporate the engineering vocabulary and worldview. As an alternative, the second part of this work aimed to show that architecture has much more in common with coding than what we usually believe. In this respect, the models of natural communication and glossematics were presented from the perspective of architectonics. It has been demonstrated that the architectural ability to abstract in order to integrate on a higher level can be advantageous and quite precious in understanding the advanced concepts like manifold, form, adjunction, purport, functor, functive, expression, distinction, content, and differentiation. Through an experiment, the last part of the work attempted to set the aforementioned models in motion by implementing them in an

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architectural context. While consciously accepting the risk of appearing technically understated for the sake of clarifying the concepts, it was demonstrated how a contemporary concept of communication could be applied to architectural modelling. Despite the real benefits of the symbolic representation applied to floor plans, this latter may prove to be insufficient in future research. This would miss the point of the experiment. Its object might be specific, but its aim is not: to present a novel perspective on how the concept of similarity could be derived from an object and in an object’s own terms, and, furthermore, how we can computationally encode such an idea. Another goal of this work was to provide a frame of reference for a researcher interested in challenging the established paradigms of computational modelling in architecture. In this regard, the presented material aims to differentiate between bodies of thinking characteristic for the modern world, and the contemporary quantum informational world. The following table presents this knowledge in a condensed form, hoping that it can serve as a compass for exploring the uncharted territories of computational modelling. The hope remains that what was presented and demonstrated could be enough to motivate and encourage some new ideas of why, how and in what direction can the research in CAAD go past its current stage, if it dares to wander.

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285

Modern world

Quantum /  information world

Space

Spatio-temporal embedding

Unfolding space and time

Structure

Top-down ad bottom-up

Structure-ability

Reference

Global structure

Localisation principle

Relations

Causal relations and determinism

Probability of an event

Variational principle

Principle of least action

Path-integral formulation

Role of computers

Computing machinery

Infrastructure for communication

Language

Linguistics

Glossematics

Models

Simulation

Communication between multiple levels

Language

Linguistics

Linguistics

Stratification

Single level of hierarchical functions

Communication between multiple levels

Objects

Well-defined

Emerging from indistinguishability

Mathematics

Set-theory

Category-theory

Coding

Grammars

Geometry

Machine learning

Neural networks

Cœxistence in the world of data

“What is this whole digital thing about?”, one might rightfully ask after submitting themselves to reading these pages. No one knows for sure, but it is possible to reflect on it. The way we think about objects—both in general and these specific ones we create in our programs—relies very much on our intuitions. Intuition was always there to help us make sense of things, and that very sense has provided us with the necessary stability when we needed it. Today’s information technology disrupts

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things that we like keeping stable, and it does not seem to provide any new stability. Coding is everything but intuitive, we can agree to that. We are overwhelmed by the abundance of information, which seems to be pointing to infinity in every direction we dare to look at. It is scary. An infinity without anything to contain it cannot appear in any different way. To prevent it from hurting us, we get engaged; we start trusting our senses and intuitions even more. In 2017, people decided to organise a conference to gather around the idea that the earth is flat. It is common sense, they say. This work does not attempt to offer a solution, but rather a different instrument for looking at that which we are afraid to look at. It is free to use, but nevertheless comes with a price, which might appear to be too high. It requires us to give up on our very keeper of stability: sense. In what way? By affirming all possible senses: by accepting that the infinity is in every direction we look at and every object contains it. If you accept it, you will understand that today’s world of information is the world that gives, and what it gives cannot be exhausted. Simply take everything it offers and say: “thank you.” This is what is at stake when you take a bunch of different floor plans and discretise them into an enormous number of arbitrary and meaningless cuts. From the perspective of sense, you are indeed left with nothing at all. But once you have them all cut up, meaningless and plenty, something great can happen. Now you can at least play with them without the fear of doing something disgraceful. You take these pieces of nothing at all, measure them, put them in constellations, combine them, make them dance for you if you like. They will start talking to each other and soon enough, talking to you. That is how you create meaning from that which does not make sense. Today we call it digital. And you are no longer afraid.

EPILOGUE

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IMAGE AND ILLUSTRATION CREDITS Fig. 1: Hyacinth. “Dedekind cut- square root of two,” 2015. Image source: https://upload.wikimedia.

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