The Morphology of the Regular & Semi-Regular Polyhedra and Tessellations According to the Separation of Facial Polytopes

Abstract

In previous work, inspired by Critchlow, and by Grünbaum & Shephard, I 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, which could be differentiated into an upper and lower layer of 4 polytopes each, and characterized by corresponding pairs, so that upper polytope always corresponds to lower. I explored the motif of paired two-step sequences of first alternating facial separation and morphological transformation, and second facial morphological transformation and separation, which in the 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 (diamonds or rhombi). Following intensive research, I here focus on a deeper pattern of morphological transformation of the primary prototypes that is characterized by the separation of one gendered set of the negative (−ve), neutral (ntrl), or positive (+ve) facial polytopes along the Y, Z, & X axes of the cubic schema. As one set of faces separates, the other two sets morph/ project if polar/neutral, through null→regular or quasi-regular→double facial levels (0→α|β→2) of the rhombic schema or its reflection. Each facial set separates just once: d=0→1. The cubic schema reveals significant three-fold symmetry by gender. The separation of faces provides an adequate schema for the morphology of the three classes of the regular and semi-regular polyhedra of {2,3,3}, {2,3,4}, and {2,3,5} symmetry, and two classes of polygonal tessellations (tilings) of {2,3,6} and {2,4,4} symmetry.