A Generative Theory of Shape

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.

Includes supplementary material: sn.pub/extras

Auteur
Professor Michael Leyton ison the faculty of the Center for Discrete Mathematics and Theoretical Computer Science, at Rutgers. He is president of the International Society for Mathematical and Computational Aesthetics and lives in Manhattan.

Contenu
Transfer.- Recoverability.- Mathematical Theory of Transfer, I.- Mathematical Theory of Transfer, II.- Theory of Grouping.- Robot Manipulators.- Algebraic Theory of Inheritance.- Reference Frames.- Relative Motion.- Surface Primitives.- Unfolding Groups, I.- Unfolding Groups, II.- Unfolding Groups, III.- Mechanical Design and Manufacturing.- A Mathematical Theory of Architecture.- Solid Structure.- Wreath Formulation of Splines.- Wreath Formulation of Sweep Representations.- Process Grammar.- Conservation Laws of Physics.- Music.- Against the Erlanger Program.