Representing the Sporadic Archimedean Polyhedra as Abstract Polytopes

TL;DR

This paper investigates how sporadic Archimedean polyhedra can be represented as quotients of regular abstract polytopes, providing methods for such representations and analyzing their Petrie schemes.

Contribution

It introduces two methods for representing Archimedean polyhedra as quotients of regular abstract polytopes and applies them to all 14 sporadic cases.

Findings

Representations of all 14 sporadic Archimedean polyhedra as quotients obtained.

Characterization of which polyhedra have acoptic Petrie schemes.

A summarized table of the representations and properties.

Abstract

We present the results of an investigation into the representations of Archimedean polyhedra (those polyhedra containing only one type of vertex figure) as quotients of regular abstract polytopes. Two methods of generating these presentations are discussed, one of which may be applied in a general setting, and another which makes use of a regular polytope with the same automorphism group as the desired quotient. Representations of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) as quotients of regular abstract polyhedra are obtained, and summarised in a table. The information is used to characterise which of these polyhedra have acoptic Petrie schemes (that is, have well-defined Petrie duals).