A Generative Theory of Shape

Michael Leyton3-540-42717-1, 9783540427179

The purpose of this book is to develop a generative theory of shape that has two properties we regard as fundamental to intelligence –(1) maximization of transfer: whenever possible, new structure should be described as the transfer of existing structure; and (2) maximization of recoverability: the generative operations in the theory must allow maximal inferentiability from data sets. We shall show that, if generativity satis?es these two basic criteria of – telligence, then it has a powerful mathematical structure and considerable applicability to the computational disciplines. The requirement of intelligence is particularly important in the gene- tion of complex shape. There are plenty of theories of shape that make the generation of complex shape unintelligible. However, our theory takes the opposite direction: we are concerned with the conversion of complexity into understandability. In this, we will develop a mathematical theory of und- standability. The issue of understandability comes down to the two basic principles of intelligence – maximization of transfer and maximization of recoverability. We shall show how to formulate these conditions group-theoretically. (1) Ma- mization of transfer will be formulated in terms of wreath products. Wreath products are groups in which there is an upper subgroup (which we will call a control group) that transfers a lower subgroup (which we will call a ?ber group) onto copies of itself. (2) maximization of recoverability is insured when the control group is symmetry-breaking with respect to the ?ber group.

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Table of contents :
Transfer….Pages 1-34
Recoverability….Pages 35-76
Mathematical Theory of Transfer, I….Pages 77-114
Mathematical Theory of Transfer, II….Pages 115-134
Theory of Grouping….Pages 135-159
Robot Manipulators….Pages 161-173
Algebraic Theory of Inheritance….Pages 175-183
Reference Frames….Pages 185-212
Relative Motion….Pages 213-227
Surface Primitives….Pages 229-238
Unfolding Groups, I….Pages 239-255
Unfolding Groups, II….Pages 257-270
Unfolding Groups, III….Pages 271-298
Mechanical Design and Manufacturing….Pages 299-363
A Mathematical Theory of Architecture….Pages 365-395
Solid Structure….Pages 397-422
Wreath Formulation of Splines….Pages 423-441
Wreath Formulation of Sweep Representations….Pages 443-454
Process Grammar….Pages 455-466
Conservation Laws of Physics….Pages 467-476
Music….Pages 477-493
Against the Erlanger Program….Pages 495-529