A New Order of the Polyhedral Honeycombs II: Which Polyhedron Mates with Which?

In this second part of a series of three papers, I explore the ordering of the polyhedra that comprise the periodic polyhedral honeycombs, and consider how pairs of polyhedra can regularly combine or mate, whether proximally or distally, along √1, √2 and √3 axes of reference cubic and tetrahedral lattices. I initially do this for pairs of what I term the Great Enablers (GEs), the positive and negative tetrahedra and truncated tetrahedra; then for pairs of GEs and the Primary Polytopes (PPs), and then for pairs of PPs. The three types of mating, GE:GE, GE:PP and PP:PP, correlate with the three symmetry groups {2,3,3|2,3,3}, {2,3,3|2,3,4} and {2,3,4]2,3,4}, respectively, of the arrays; and these matings typically occur in pairs, which enjoy a one-to-one correspondence with the possible periodic honeycombs. I formally differentiate the PPs into two groups of four. This lays the groundwork for a proposed new order of the honeycombs.