earticle

영어

Earlier discovery and presentation of an inherent order of the regular and semi-regular polyhedra that displays three interrelated classes, together with consideration of the honeycombs, has led me to posit the existence of a coherent and integral metapattern that should relate the various all-space-filling periodical honeycombs, which I advance in an earlier paper that should be read in conjunction with thispaper, as they form part of a series.Here, I approach the periodic polyhedral honeycombs byexploring how pairs of polyhedra regularly combine or mate, whether proximally or distally, along the √1, √2 and √3 axes of their reference cubic and tetrahedral lattices. This is first performed for pairs of what I elsewhere term the Great Enablers (GEs), the positive and negative tetrahedra and truncated tetrahedra; secondly, for pairs of GEs and the Primary Polytopes (PPs); and thirdly, for pairs of PPs. This reveals that these three forms 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 periodical honeycombs.These matings typically occur in naturallyoccurring pairs along each axis, so in general, a PP mates with just two PPs, though in certain cases one of these is the same as the original. These pairsof matings display a one-to-one correspondence with the possible periodic honeycombs. Differentiating the PPs into two groups of four according to their formal behavior suggests a pathway towards a proposed new order of the honeycombs.