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Chapter 3 Configurations in the N-dimensional Orthogonal Pseudo-polytopes 3.1. Introduction 3.2. Configurations for the Nd-opp's and the Equivalence Relation R F

The problem of enumerating the colorings of an nD hypercube with two colors has an interesting history. Mathematicians in the 1870’s were considering classifying the logical propositions that can exist. In Boolean Algebra, in the functions of n logical variables, each variable can assume a value of either True or False, leading to 2 n combinations in the domain of the function. If we associate a spatial dimension in n with each variable (True ≡ 1 and False ≡ 0), then each element of the domain can be represented as a point with n coordinates of 0’s and 1’s. Attempts to enumerate equivalence classes of functions were carried out in the 1800’s, apparently without realizing that 2-colorings of the nD hypercube were also being counted. In 1871, W. S. Jevons presented an article that described the cases that arise in 2-colorings of the 2D cube and the 3D cube, but from the perspective of Boolean functions. Jevons only considered 192 of the 256 propositions corresponding to 2-colorings of the 3-cube; he found 16 of the 22 equivalence classes [Banks04]. In 1877, Clifford claimed a solution of counting cases for 2-colorings of the 4D hypercube. He reported that the 2 16 = 65,536 distinct functions were grouped into 396 equivalence classes [Clifford68]. Clifford’s results were showed incorrect until 1940 when Pólya demonstrated how his enumeration methods could be applied to this problem in an automated manner. He enumerated the 402 classes of 2-colorings of the 4D hypercube, and showed where Clifford’s manual tabulations had overcounted in some cases and undercounted in others [Pólya87]. In 1987 this topic was revisited, in the 2D and 3D cases, by Lorensen and Cline through their famous Marching Cubes Algorithm [Lorensen87]. Finally, the 4D case has been studied by Roberts & Hill [Roberts99], Bhaniramka et al [Bhaniramka00] and Banks & Stockmeyer [Banks06].