Polytope Typology A: The separation of facial polytopes in the morphology of the regular and semi-regular polyhedra and tessellations

Inspired by Critchlow [1], and Grünbaum and Shephard [2], previous work has proposed an integral 2.5D cubic schema of the regular and semi-regular polyhedra and polygonal tessellations of the plane for each class of symmetry. This schema is differentiated into an upper and lower layer of 4 polytopes each, and characterized by corresponding pairs of upper and lower polytopes [3]. The motif of paired two-step sequences of first alternating separation and morphological transformation of faces, and second morphological transformation and separation of faces is explored, which in 2D consideration of the 2.5D schema are disposed about the vertical axis, as characterized by the correspondence between the PPs of the lower and upper squares (rhombi). Developing earlier sustained research [3−11], this paper addresses a deeper typology of morphological transformation of the primary polytopes, involving the separation of one gendered set of the negative (–ve), neutral (ntrl), or positive (+ve) facial polytopes along the Y, Z, and X axes of the cubic schema. While one set of faces separates, the other two sets morph or project through null→regular or quasi-regular→double facial levels (0→α|β→2) of the rhombic schema or its reflection. Each facial set only separates once, faces separating by d=0→1. The cubic schema exhibits significant three-fold symmetry by gender. The separation of faces schema adequately describes the morphology of the three classes of regular and semi-regular polyhedra of {2,3,3}, {2,3,4}, and {2,3,5} symmetry, and the two classes of polygonal tessellations of {2,3,6} and {2,4,4} symmetry.