Power Set Unions

Symbolic Models for Discrete Spaces

Karen Dawn Colins

Atomic Energy of Canada Limited

Abstract

In ancient times, geometry was the science of figures in space. The quantum of both figure and space was a primitive concept named "point". Points were assumed to be of zero dimension, an entity with no intrinsic property.  Another primitive concept, "line", was defined as an entity having just one intrinsic property, length. In modern times, figures have come to be recognized as spaces; and geometry has evolved to become the science of the relations of points determining spaces. The primitive concepts, point and line, have been subsumed in the more general concept of the simplex, allowing discrete spaces to be modelled as simplicial complexes. In such models, the geometry of a discrete space is determined by assigning to an abstract simplicial complex 𝒮, a particular set 𝒱 of constrained content-values for a set of i-faces of the k-simplices. Although the effects on geometry, that occur by the choice of 𝒱, have been examined fairly extensively, particularly in the context of quantum gravity, the effects that occur by the choice of 𝒮 have received less attention. Representing 𝒮 as a union of power sets, and taking advantage of Mathematica's strengths for symbolic computation, experimentation, and visualization, symbolic models of discrete spaces can be explored in new ways, and new insights gained.