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

Basudeb Datta, Dipendu Maity

Published 2018-04-18Version 1

A vertex-transitive map $X$ is a map on a closed surface on which the automorphism group ${\rm Aut}(X)$ acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. In particular, there are semi-equivelar maps on the torus, on the Klein bottle and on the surface of Euler characteristic $-1$ which are not vertex-transitive. Here we show that every semi-equivelar map on $\mathbb{S}^2$ and $\mathbb{RP}^2$ are vertex-transitive. In particular, we show that a semi-equivelar map on $\mathbb{S}^2$ is the boundary of a Platonic solid, an Archimedean solid, the prism, a drum or an antiprism (Section 2).