mechanical CAD - or it might be a complex real-world scene that confronts the visual system. The remarkable fact about the human cognitive system is that, when presented with a highly complex structure, such as a real-world scene, it is able to convert the complexity into an entirely understandable form. This exemplifies the general problem that we will investigate:

0. The conversion of complexity into understandability. The basic purpose is to give a generative theory of complex shape such that the complexity is entirely accounted for, and yet the structure is completely understandable.
0. Understandability and intelligence. Deep consideration reveals that understandability of a structure is achieved by maximizing transfer and recoverability.
0. The mathematics of understandability. A significant portion of the book is the development of a mathematical theory of how understandability is created in a structure.

When putting together the statement in (2) and the theory of aesthetics in section 2, one sees that, according to our theory of geometry, aesthetics is basic to the conversion of complexity to understandability. Thus, for example, the computer-vision problem, for complex scenes, is solved by aesthetics, rather than the techniques currently developed in the research literature. Furthermore, a consequence of the statement in (3) is that we will be giving a mathematic theory of aesthetics.

4 Object-Oriented Theory of Geometry

Geometry of the last 3,000 years is not object-oriented. A principal reason is that object-oriented programming allows the identification and tracking of objects through histories of complex modification (i.e., allows for recoverability or memory storage of action) and the congruence/invariance program defeats this; e.g., the adult body and egg could not be the same geometric object in Klein’s sense of object. One of the purposes of our book is the development of an object-oriented formalization of geometry. The result is a theory of geometry that has fundamentally opposite characteristics from previous geometry. The object-orientedness is formulated in a rather novel use of group theory: Groups are seen as descriptions of asymmetries rather than symmetries. One important consequence is that the theory provides an entirely new formulation of the meaning of symmetry-breaking, in which the group expands, on symmetry-breaking, rather than reduces, as it does in modern physics. This increases the descriptive power of the theory, because an expanding group provides greater number of algebraic operators.

5 Transfer

A generative theory of shape characterizes a shape by a sequence of actions needed to generate it. According to our theory, aesthetics is increased by transferring