Classification of tight regular polyhedra

TL;DR

This paper classifies tight regular polyhedra, including orientably and non-orientably regular types, based on their automorphism groups, expanding the understanding of their existence and structure.

Contribution

It determines the existence of tight non-orientably regular polyhedra and fully classifies all tight regular polyhedra by their automorphism groups.

Findings

Existence criteria for tight non-orientably regular polyhedra.

Complete classification of tight regular polyhedra by automorphism groups.

Extension of known results from orientable to non-orientable cases.

Abstract

A regular polyhedron of type {p, q} has at least 2pq flags, and it is called tight if it has exactly 2pq flags. The values of p and q for which there exist tight orientably regular polyhedra were previously known. We determine for which values of p and q there is a tight non-orientably regular polyhedron of type {p, q}. Furthermore, we completely classify tight regular polyhedra in terms of their automorphism groups.