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

TL;DR

This paper classifies semi-equivelar maps on the sphere, showing they correspond to well-known polyhedral boundaries and establishing their geometric realizability, with a unique exception among semi-equivelar maps.

Contribution

It provides a complete classification of semi-equivelar maps on the sphere, identifying all such maps with known polyhedral boundaries and proving their geometric realizability.

Findings

Exactly one semi-equivelar map on 2^2 is not vertex-transitive.

All semi-equivelar maps on 2^2 are geometrizable and correspond to semi-regular tilings.

Semi-equivelar maps on 2^2 are characterized by an inequality involving their face types.

Abstract

A vertex-transitive map $X$ is a map on a surface on which the automorphism group of $X$ acts transitively on the set of vertices of $X$. 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 surfaces of Euler characteristics $-1$ $\&$ $-2$ which are not vertex-transitive. It is known that the boundaries of Platonic solids, Archimedean solids, regular prisms and antiprisms are vertex-transitive maps on $\mathbb{S}^2$. Here we show that there is exactly one semi-equivelar map on $\mathbb{S}^2$ which is not vertex-transitive. More precisely, we show that a semi-equivelar map on $\mathbb{S}^2$ is the boundary of a Platonic solid, an Archimedean solid, a regular prism, an antiprism or the pseudorhombicuboctahedron. As a consequence, we show that all the semi-equivelar maps on $\mathbb{RP}^2$ are vertex-transitive. Moreover, every semi-equivelar map on $\mathbb{S}^2$ can be geometrized, i.e., every semi-equivelar map on $\mathbb{S}^2$ is isomorphic to a semi-regular tiling of $\mathbb{S}^2$. In the course of the proof of our main result, we present a combinatorial characterization in terms of an inequality of all the types of semi-equivelar maps on $\mathbb{S}^2$. Here, we present self-contained combinatorial proofs of all our results.