Spherical Tilings with Transitivity Properties

Abstract

H. S. M. Coxeter’s work on regular and uniform polytopes is, perhaps, his best-known contribution to geometry. By central projection one can relate each of these polytopes to a tiling on a sphere, and the symmetry properties of the polytopes then lead naturally to various transitivity properties of the corresponding tilings. The main purpose of this paper is to classify all tilings on the 2-sphere with these transitivity properties (and not just those obtained from three-dimensional polytopes). Our results are exhibited in Tables 3 and 4. Here we enumerate all “types” of tilings whose symmetry groups are transitive on the tiles (isohedral tilings), on the edges (isotoxal tilings), or on the vertices (isogonal tilings). The word “type” is used here in the sense of “homeomeric type” for details of which we refer the reader to recent literature on the subjects of patterns and plane tilings, especially [18] and [20].