Professor Michael Leyton



Books by Michael Leyton:


Leyton's book "A Generative Theory of Shape" (Springer, 554 pages).



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M. Leyton:

A Generative Theory of Shape

554 pages

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 itself. 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 a mathematical language that brings interoperability into the very foundations of geometry.

The book gives extensive applications of the theory to CAD/CAM, human and machine vision, robotics, software engineering, and physics. 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. In robotics, several levels of analysis are developed for manipulator structure and kinematics. In software, a new theory is given of the principal factors such as text and class structure, object creation and modification, as well as inheritance and hierarchy prediction. In physics, a new theory is given of the conservation laws, and motion decomposition theorems in classical and quantum mechanics.

The full book can be viewed electronically at the Springer website:

Springer-Verlag: Leyton's book


Leyton's book on paintings, published by Springer (Sept 2006).






M. Leyton:

The Structure of Paintings


Michael Leyton has developed new foundations for geometry in which shape is equivalent to memory storage. A principal argument of these foundations is that artworks are maximal memory stores. At the basis of this geometry are Leyton's fundamental laws of memory storage, and these laws are shown to determine the structure of artworks. That is, the central argument is that artworks are structured so that they allow the maximal extraction of stored memory. Furthermore, the book demonstrates that the emotion expressed by an artwork is actually the memory extracted by the laws. Therefore, the laws of memory storage allow the systematic and rigorous mapping not only of the compositional structure of a painting, but also of its emotional expression. This fundamentally opposes the view that the emotional expression of an artwork is undefinable. Leyton's methodology makes the structure and emotional content of an artwork fully definable, rich, systematic and complete. The argument is supported with detailed analyses of paintings by Picasso, Raphael, Cezanne, Gauguin, Modigliani, Ingres, De Kooning, Memling, Balthus and Holbein.


Leyton's book on morphology (Springer, 544 pages)..




M. Leyton:

Process Grammar:
The Basis of Morphology

544 pages


Leyton's Process Grammar has been applied by scientists and engineers in many disciplines including medical diagnosis, geology, computer-aided design, meteorology, biological anatomy, neuroscience, chemical engineering, etc. This book demonstrates the following:
The Process Grammar invents several entirely new concepts in biological morphology and manufacturing design, and shows that these concepts are fundamentally important to biological morphology and manufacturing design. Conventional morphological theories have entirely failed to recognize these important concepts.
The Process Grammar has process-inference rules that give, to morphological transitions, powerful causal explanations that have never been given before in other morphological theories.
Remarkably, the book gives a profound unification of biological morphology and vehicle design.
The book invents over 30 new CAD operations that realize fundamentally important functions of a product.
A crucial fact is that the Process Grammar is an example of the laws in Leyton's Generative Theory of Shape which give the ability to recover the design intents for which the shape features of a CAD model were created. The book demonstrates that the Process Grammar recovers important design intents in biological morphology and manufacturing design. In large-scale manufacturing systems, the recovery of design intents is important for solving the interoperability problem and product lifecycle management.
This book is one of a series of books in Springer that elaborates Leyton's Generative Theory of Shape.


Leyton's book "Symmetry, Causality, Mind" (MIT Press, 630 pages).



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M. Leyton:

Symmetry, Causality, Mind

630 pages


From reviews of Leyton's book "Symmetry, Causality, Mind" (MIT Press, 630pages, paperback)

"This is a remarkable book. Its claim is that perception is none other than the recovery of causal history. One cannot but be struck by the depth, novelty and brilliance of Leyton's accounts, page after page, of even the most minute and ordinary of perceptual phenomena - claims which contradict virtually every previous treatment of these phenonena." Professor Eleanor Rosch, University of California, Berkeley.

"Leyton's work is a most engaging and utterly original treatment of some classical problems in perception and cognition." Professor Barbara Landau, Johns Hopkins University.

"This is a superb book. The writing is extraordinarily clear and the theory is very original. It challenges the reader to think about basic questions and presents new concepts and bold new principles that give a new perspective on visual perception in particular and cognitive representation more generally." Wayne Wickelgren, Professor, Columbia University.

For ordering information on the book, at Amazon books, click: Leyton's book at


Leyton's book on architecture, published by Birkhauser (July 2006).


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M. Leyton:

Shape as Memory:
A Geometric Theory of Architecture


Description of Book:

In his mathematical books and papers, Michael Leyton has developed new foundations to geometry that are directly opposed to the conventional foundations that have existed for almost 3,000 years, from Euclid to modern physics, including Einstein. Whereas the conventional foundations minimize the memory contained in the geometric object, these new foundations maximize it. A fundamental conclusion is that shape is equivalent to memory storage.

In this book, Leyton shows that his new foundations for geometry result in new foundations for architecture. Whereas architecture, for thousands of years, has been based on the minimization of memory storage – the aim of conventional geometry – the new architecture he proposes is based on the maximization of memory storage – the aim of the new geometry. He elaborates the structural principles by which buildings can be designed as maximal memory stores. This reformulates the relation of architecture to computation: the computational process becomes one in which the mind undergoes self-creation by reading and writing itself as history. The architectural principles proposed by Leyton are the means by which buildings can be read and written as the self-creation of mind. Leyton illustrates these new architectural principles with his own administration buildings.


Leyton's Published Musical Compositions

The scores of Leyton's string quartets are currently being published. The first to be published was String Quartet #5, shown below.



Susan Kay on Leyton's string quartets:
"Nothing can prepare you for the overwhelming experience of hearing Michael Leyton's string quartets. They are a fundamental revolution in harmonic and melodic structure."

Leyton's Scientific and Mathematical Influence

Leyton's mathematical work on shape has been used by scientists in over 40 disciplines including: radiology, meteorology, computer vision, chemical engineering, geology, computer-aided design, robotics, anatomy, botany, urban planning, forensic science, software engineering, architecture, abductive reasoning, linguistics, mechanical engineering, computer graphics, control theory, electromagnetism, thermodynamics, quantum field theory, archaeology, etc.

As an example: Leyton's Symmetry-Curvature Duality Theorem has been applied by scientists in over 40 disciplines, including: MRI human brain scans, dental radiographs, transmission electron microscopy, blood-cell analysis, neuronal growth models, DNA molecule-tracking, grasp determination in robotics, geological formation of volcanic islands, interactive rendering, cartoon vectorization, drainage patterns, musculoskeletal development, botanical leaf analysis, human facial expression, shock scaffolds, weather prediction, molecular dynamics, shape skeletonization, etc.

Leyton's original paper proving this theorem can be viewed at Symmetry-Curvature Duality Theorem

Another example is his Process-Grammar, which he invented in 1987. This recovers the past history of an object; e.g., the history of growth, deformation, morphology, design, etc.

For an easy introduction to his Process-Grammar, and a description of a number of its scientific applications, see the website: Process Grammar

Leyton's extensive theory of CAD, solid-modeling, object-oriented programming, mechanical engineering, has aroused great interest from all these disciplines, as can be seen on the web by the many invitations he has received from the major conferences in these areas. You can view a comprehensive mathematical elaboration of his theory in his Springer book mentioned above, A Generative Theory of Shape. The conceptual elaboration of his work is given in his other books in MIT Press and Birkhauser.

Professor Leyton is president of the following two societies:

International Society for Mathematical and Computational Aesthetics
International Society for Group Theory in Cognitive Science

He is also on the governing board of the Institute for the Fundamental Reseach in Music, in Zurich; the MittelEuorpa foundation in Bolzano Italy.

Leyton receives several invitations a year to give conference addresses in a wide range of disciplines: Computer vision, Mechanical Engineering, Mathematics, Computer-Aided Design, Cognitive Science, Aritificial Intelligence, Music Theory, Semiotics, Art, etc.

Recent invited conference addresses include:

Keynote address: Society for Applied Systems Research
- Included presentation to Leyton of Medal of Achievement
Plenary address: Conference on Representations in AI, Philips Labs.
Plenary address: 1st International Conference on Visual Form.
Plenary address: Conference of the International Semiotics Society.
Plenary address: Workshop on Topological Models, San Marino.
Plenary address: Art and Mathematics conference.
Plenary address: Non-Linear Mathematics, Russian Academy of Science.
Plenary address: 3rd International Conference on Visual Form.
Plenary address: Bridges between Science and Art.
Keynote address: Mathematical Music Theory Conference, Zurich.
Plenary address: SIAM Conference on Computer-Aided Geometric Design.
Plenary address: Shape Modeling Conference, Genova.
Plenary address: Visual and Spatial Reasoning in Design Conference, MIT.
Plenary address: Model-Based Reasoning Conference, Pavia.
Plenary address: Meta-Representation Conference, Bremen.
Plenary address: Aesthetic Perception Symposium, Aarhus.
Keynote address: IEEE International Conference, IRI.

Leyton's Artistic Exhibitions

Leyton is a well-known artist. His paintings, sculptures, and architectural projects, have been featured in international design journals and invited exhibitions. The following site gives you access to a large selection of Leyton's works:

Leyton's Paintings, Architectural Projects, Sculptures, Musical Compositions.

Eric Wiener of the East Village Guide, New York, writes:
"This Rutgers University professor is a practitioner of arts, mathematics, music and philosophy. His breathtaking artwork deserves more than your average surfing minute as the complexities will float before your retina with time."

Leyton's current work

Leyton is currently writing a 4-volume work on the foundations of science, with particular emphasis on quantum mechanics. He also continues to work on the structure of software, as well as interoperability and large-scale engineering systems integration, in the mechanical/aerospace industry.

Recent published work includes:

Leyton's Theory of Software and Interoperability.

Leyton's Theory of Music.

Postal addresses:

Professor Michael Leyton,
DIMACS Center for Discrete Mathematics,
& Theoretical Computer Science,
Rutgers University, Busch Campus,
New Brunswick, NJ 08854,

E-mail address: