Platonic solids, Archimedean solids and semi-equivelar maps on the sphere

Abstract

A map X on a surface is called vertex-transitive if the automorphism group of X acts transitively on the set of vertices of X. A map is called semi-equivelar if the cyclic arrangement of faces around each vertex is same. In general, semi-equivelar maps on a surface form a bigger class than vertex-transitive maps. There are semi-equivelar maps on the torus, the Klein bottle and other surfaces which are not vertex-transitive. It is known that the boundaries of Platonic solids, Archimedean solids, regular prisms and anti-prisms are vertex-transitive maps on S2. Here we show that there is exactly one semi-equivelar map on S2 which is not vertex-transitive. As a consequence, we show that all the semi-equivelar maps on RP2 are vertex-transitive. Moreover, every semi-equivelar map on S2 can be geometrized, i.e., every semi-equivelar map on S2 is isomorphic to a semi-regular tiling of S2. In the course of the proof of our main result, we present a combinatorial characterisation in terms of an inequality of all the types of semi-equivelar maps on S2. Here we present combinatorial proofs of all the results.