Vertex-Faithful Regular Polyhedra

TL;DR

This paper classifies and constructs abstract regular polyhedra with automorphism groups acting faithfully on vertices, focusing on cases where the number of vertices is prime, twice a prime, or a prime squared.

Contribution

It introduces a classification of regular polyhedra based on vertex set size and constructs the smallest examples for prime squared vertices.

Findings

Classified all regular polyhedra with prime or twice prime vertices.

Constructed the smallest regular polyhedra with prime squared vertices.

Showed that each non-flat polyhedron covers a vertex-faithful one.

Abstract

We study the abstract regular polyhedra with automorphism groups that act faithfully on their vertices, and show that each non-flat abstract regular polyhedron covers a "vertex-faithful" polyhedron with the same number of vertices. We then use this result and earlier work on flat polyhedra to study abstract regular polyhedra based on the size of their vertex set. In particular, we classify all regular polyhedra where the number of vertices is prime or twice a prime. We also construct the smallest regular polyhedra with a prime squared number of vertices.