A New Order in Space — Platonic and Archimedian Polyhedra and Tilings

Abstract

There are generally considered to be five Platonic (Regular) and thirteen Archimedian (Semi-Regular) polyhedra (leaving aside the Kepler-Poinsot polyhedra, and disregarding the regular prisms and antiprisms).1 These are important as they represent the most regular articulation that the space of our experience is capable of. But as far as I am aware, a fully satisfactory pattern which situates these polyhedra in their proper interrelationship has not previously been advanced. Keith Critchlow proposes the tetrahedron and truncated tetrahedron as nuclear or ‘over’ solids, together with a two-fold periodic arrangement of secondary regular ‘parent’ solids and their truncations for the remainder.2 Cundy and Rollett propose a similar arrangement,3 with the first two solids of tetrahedral symmetry, and the rest of octahedral or icosahedral symmetry. Coxeter suggests the Platonic solids fall naturally into two classes, the “crystallographic” solids (tetrahedron, octahedron and cube), and the “pentagonal” polyhedra (icosahedron, dodecahedron and Kepler-Poinsot solids).4 But in my opinion these arrangements, although valuable, do not adequately integrate all the polyhedra into a rigorous and comprehensive order. This writer proposes a three-fold arrangement, which does appear rigorous and comprehensive. Thus the regular and semi-regular polyhedra are conveniently classified into three parallel sets, according to which one of only three spatial patterns of symmetries each exhibits: {2,3,3}-fold TetraTetrahedral, {2,3,4}-fold OctaHexahedral, or {2,3,5}-fold IcosiDodecahedral symmetry. These are the symmetries of the quasi-regular octahedron, cuboctahedron, and icosidodecahedron, respectively. Four polyhedra exhibit two of these fundamental patterns, and accordingly reappear as different elements in either of the sets to which they belong. Two polyhedra each appear twice as different elements in the same set, in accord with their alternative orientations. Polyhedra which reappear are given alternative names. There are therefore in total three parallel sets of eight polyhedra each, making twenty-four polyhedra in all; there being then in effect six regular and eighteen semi-regular polyhedra. Each element of a set strictly correlates with its corresponding elements in the other sets. Further, particular patterns - which are termed aspects - are replicated across each of the sets. Within each set of eight, subsets of elements are discernable according to various rules: sequential truncation, rotation and displacement, and octaving of faces. The particular aspect which characterizes a subset of one set correlates with the respective aspects of the other sets. Properties of elements and of aspects can then be generalized across classes and this predictive ability is confirmed in practice. A gestalt pattern is advanced for each set which integrates the constituent aspects, and this is further developed to include the whole three sets. The classification is also regularly extended to recognize a fourth and fifth class, which together comprise the three regular and all but one of the eight semi-regular tilings of the planar surface, (the exception 33 · 42 being considered a degenerate case). These classes are of {2,3,6}-fold and {2,4,4}-fold symmetry, of the quasi-regular TriangularHexagonal Array and SquareSquare Array, respectively. Within either two-dimensional class some elements are repeated with different colorings as different facial arrangements are associated with the respective symmetry pattern of that class, the repeated elements being given alternative names. Discounting the degenerate exception, there are then in effect a total of four regular and twelve semi-regular tilings of the plane. But they do not exhaust the thirty-two possible uniform colorings of regular and semi-regular tilings. The harmony of interrelationship which characterizes the entire order reveals the elegant structure of space. This research is from the author's forthcoming work “New Light on the Platonic and Archimedian Polyhedra — A New Order in Space”, which will be available from the author.