Shape as Memory Storage

Let us see how this contrasts with the standard foundations for geometry that have existed for almost three thousand years. In the standard foundations, a geometric object consists of those properties of a figure that do not change under a set of actions. These properties are called the invariants of the actions. Geometry began with the study of invariance, in the form of Euclid’s concern with congruence, which is really a concern with invariance (properties that do not change). And modern physics is based on invariance. For example, Einstein’s principle of relativity states that physics is the study of those properties that are invariant (unchanged) under transformations between observers. Quantum mechanics studies the invariants of measurement operators. My argument is that the problem with invariants is that they are memoryless. That is, if a property is invariant (unchanged) under an action, then one cannot infer from the property that the action has taken place. Thus I argue: Invariants cannot act as memory stores. In consequence, I conclude that geometry, from Euclid to Einstein has been concerned with memorylessness. In fact, since standard geometry tries to maximize the discovery of invariants, it is essentially trying to maximize memorylessness. My argument is that these foundations to geometry are inappropriate to the computational age; e.g., people want computers that have greater memory storage, not less. As a consequence, I embarked on a 30-year project to build up an entirely new system for geometry – a system that was recently completed. Rather than basing geometry on the maximization of memorylessness (the aim from Euclid to Einstein), I base geometry on the maximization of memory storage. The result is a system that is profoundly different, both on a conceptual level and on a detailed mathematical level. The conceptual structure