Towards a New Order of the Polyhedral Honeycombs: Part I: The New Order Introduced

Abstract

The periodic polyhedral honeycombs exhibit an elegant integrity and extraordinary interrelationship, and offer potential for diverse applications, but a satisfactory order that does them full justice appears lacking. Building on my prior order of the regular and semi-regular polyhedra [1, 2], I arrange the honeycombs according to lattice size and expansion/contraction sequences and groups, into an order that comprises three classes of symmetry. Primary polyhedra in all honeycombs are either situated at points of reference cubic arrays, and sometimes generate neutral elements disposed at mid-edges or mid-faces of reference cubic arrays; or they are situated in tetrahedral arrays, with alternating Tetrahedra or Truncated Tetrahedra. Examining lattice size for unit polyhedral edge, together with the disposition of common elements, suggests expansion/contraction groups in two classes, and the regular increase in lattice size at each stage of expansion helps validate the order. Part I provides an overview.