Shape Representation:

• A Generative Theory of Shape, 2001.
• Group Theory and Architecture 1: Nested Symmetries, 1999.
• Group Theory and Architecture 2: Why Symmetry/Asymmetry ? 1999.
• Symmetry, Causality, Mind, 1992.
• Shape and Causal-History, 1992.
• Processes at Discontinuites, 1989.
• Inferring Causal History from Shape, 1989.
• A Process Grammar for Shape, 1988.
• 3-D Symmetry-Curvature Duality Theorems, 1987 & 1990.
• Symmetry-Curvature Duality, 1987.
• A Limitation Theorem for the Differential Prototypification of Shape, 1987.

Perception :

• The Foundations of Aesthetics, 2006.
  
• Nested Structures of Control: An Intuitive View, 1987.
• A Theory of Information Structure II: A Theory of Perceptual Organization, 1986.
• A Theory of Information Structure I: General Principles, 1986.
• Principles of Information Structure Common to Six Levels of the Human Cognitive System, 1986.
• Generative Systems of Analyzers, 1985.
• Perceptual Organization as Nested Control, 1984.

The Foundations of Aesthetics

1. maximizing transfer of structure, 
2. maximizing recoverability of the generative operations.

In this book, the author develops a generative theory of shape with two properties fundamental to intelligence: maximizing transfer of structure, and maximizing recoverability of generative operations. The theory is applied in considerable detail to CAD, perception, and robotics. A significant aspect of this book is the development of an object-oriented theory of geometry. This includes a group-theoretic formulation of object-oriented inheritance. In particular, a class of groups is developed called "unfolding groups", which define any complex shape as unfolded from a maximally collapsed version of itself called an "alignment kernel". The group is decomposed into levels corresponding to the inheritance hierarchy within the complex object. This achieves one of the main goals of the theory - the conversion of complexity into understandability. The advantages of the theory are demonstrated with lengthy studies of robot manipulators, perceptual organization, constructive solid geometry, assembly planning, architectural CAD, and mechanical CAD/CAM

Keywords: Geometric Shape, Shape Theory, Geometric Objects, Computer Vision, Robot Navigation, Visual Perception, Shape Generati- on, Geometric Structure, Group Theory, Complex Shape, Wreath Products, Erlanger Program, Geometric Invariants.

From "James Johnson" <James-Johnson@nyc.rr.com>:

• The purpose of the book is to develop a generative theory of shape that has two properties regarded as fundamental to intelligence - maximizing transfer of structure and maximizing recoverability of the generative operations. These two properties are particularly important in the representation of complex shape - which is the main concern of the book. The primary goal of the theory is the conversion of complexity into understandability. For this purpose, a mathematical theory is presented of how understandability is created in a structure. This is achieved by developing a group-theoretic approach to formalizing transfer and recoverability. To handle complex shape, a new class of groups is developed, called unfolding groups. These unfold structure from a maximally collapsed version of that structure. A principal aspect of the theory is that it develops a group-theoretic formalization of major object-oriented concepts such as inheritance. The result is an object-oriented theory of geometry.
  The algebraic theory is applied in detail to CAD, perception, and robotics. In CAD, lengthy chapters are presented on mechanical and architectural design. For example, using the theory of unfolding groups, the book works in detail through the main stages of mechanical CAD/CAM: part-design, assembly and machining. And within part-design, an extensive algebraic analysis is given of sketching, alignment, dimensioning, resolution, editing, sweeping, feature-addition, and intent-management. The equivalent analysis is also done for architectural design. In perception, extensive theories are given for grouping and the main Gestalt motion phenomena (induced motion, separation of systems, the Johannson relative/absolute motion effects); as well as orientation and form. In robotics, several levels of analysis are developed for manipulator structure, using the author's algebraic theory of object-oriented structure.

1. Transfer
2. Recoverability, p. 35
3. Mathematical Theory of Transfer, I, p. 77
4. Mathematical Theory of Transfer, II, p. 115
5. Theory of Grouping, p. 135
6. Robot Manipulators, p. 161
7. Algebraic Theory of Inheritance, p. 175
8. Reference Frames, p. 185
9. Relative Motion, p. 213
10. Surface Primitives, p. 229
11. Unfolding Groups, I, p. 239
12. Unfolding Groups, II, p. 257
13. Unfolding Groups, III, p. 271
14. Mechanical Design and Manufacturing, p. 299
15. A Mathematical Theory of Architecture, p. 365
16. Solid Structure, p. 397
17. Wreath Formulation of Splines, p. 423
18. Wreath Formulation of Sweep Representations, p. 443
19. Process Grammar, p. 455
20. Conservation Laws of Physics, p. 467
21. Music, p. 477
22. Against the Erlanger Program, p. 495
23. Appendix A. Semi-direct Products, p. 531
24. Appendix B. Symbols, p. 539
25. References, p. 541
26. Index, p. 549

web link

In a forthcoming book, I give a comprehensive theory of design based on group theory. Whereas the book itself requires an advanced knowledge of group theory, the present series of articles will give the material in an intuitive form, and build up any needed group theory, in tutorial passages. The articles will begin by assuming that the reader has no knowledge of group theory, and we will progressively add more and more group theory in an easy form, until we finally are able to get to quite difficult topics in tensor algebras, and give a group-theoretic analysis of complex buildings such as those of Peter Eisenman, Zaha Hadid, Frank Gehry, Coop Himmelblau, Rem Koolhaas, Daniel Libeskind, Greg Lynn, and Bernard Tschumi. This first article is on a subject of considerable psychological relevance: nested symmetries.

web link

From Piero Scaruffi's Annotated Bibliography of Mind-related Topics:

• "Leyton's idea is that shape is used by the mind to recover the past. Shape is time. Shape equals the history that created it. By studying the psychological relationship between shape and time, Leyton offers a working model of how perception is converted into memory. There is a relationship between perceived asymmetry, inferred history and environmental energy. The energy of a system corresponds to memory of the causal interactions that transferred to the system. Shape, or asymmetry, is a memory of the energy transferred to an object in causal interactions. All vision is the recovery of the past: vision simply "unlocks" time from the image. In general, perceptual representations are representations of stimuli in terms of causal histories. This is also true of cognitive representations. Any cognitive representation is the description of a stimulus as a state in a history that causally explains the stimulus to the organism. A cognitive system is a system that creates and manipulates causal explanations."

From the MIT Press:

• Michael Leyton's arguments about the nature of perception and cognition are fascinating, exciting, and sure to be controversial. In this investigation of the psychological relationship between shape and time, Leyton argues compellingly that shape is used the the mind to recover the past and as such it forms a basis for memory. He elaborates a system of rules by which the conversion to memory takes place and presents a number of detailed case studies--in perception, linguistics, art, and even political subjugation--that support these rules.
  Leyton observes that the mind assigns to any shape a causal history explaining how the shape was formed. We cannot help but perceive a deformed can as a dented can. Moreover, by reducing the study of shape to the study of symmetry, he shows that symmetry is crucial to our everyday cognitive processing. Symmetry is the means by which shape is converted into memory.
  Perception is usually regarded as the recovery of the spatial layout of the environment. Leyton, however, shows that perception is fundamentally the extraction of time from shape. In doing so, he is able to reduce the several areas of computational vision purely to symmetry principles. Examining grammar in linguistics, he argues that a sentence is psychologically represented as a piece of causal history, and archaeological relic disinterred by the listener so that the sentence reveals the past. Again through a detailed analysis of art he shows that what the viewer takes to be the experience of a painting is in fact the extraction of time from the shapes of the painting. Finally, he highlights crucial aspects of the mind's attempt to recover time in examples of political subjugation.

Web link.

Establishes the "Symmetry-Curvature Duality Theorem":

• For a regular simply connected planar curve, to each extrema of curvature there exist a symmetry axis which ends at that extrema.

Uses the SLS (smoothed local symmetries of Brady & Asada) as a construction process (example) of symmetries.

Studies the Codon representation of Hoffman & Richards.

SLS-Codon Theorem:

• The SLS of a codon is unique and terminates at the point of maximal curvature.

Details on evolutes: they bound (on each side) the loci of maximally inscribed circles.

Note on information theory:

• The "creation of greater information by curvature is compensated by the creation of greater redundancy" (and thus, paradoxically! lesser information content, according to Shannon's theory) "of the associated symmetry."
• See the discussion by Arnheim on Art and Entropy.

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