Regular Polyhedra of Index Two, II

TL;DR

This paper completes the classification of finite regular polyhedra of index 2 in 3-space, focusing on those with vertices on a single orbit, identifying ten such polyhedra with specific symmetry properties.

Contribution

It provides a complete enumeration and classification of regular polyhedra of index 2 with vertices on one orbit, expanding understanding of their symmetry and combinatorial structure.

Findings

Identified ten regular polyhedra of index 2 with vertices on one orbit.

Completed the classification of all finite regular polyhedra of index 2 in 3-space.

Clarified symmetry group relationships for these polyhedra.

Abstract

A polyhedron in Euclidean 3-space is called a regular polyhedron of index 2 if it is combinatorially regular and its geometric symmetry group has index 2 in its combinatorial automorphism group; thus its automorphism group is flag-transitive but its symmetry group has two flag orbits. The present paper completes the classification of finite regular polyhedra of index 2 in 3-space. In particular, this paper enumerates the regular polyhedra of index 2 with vertices on one orbit under the symmetry group. There are ten such polyhedra.