Regular Polyhedra of Index Two, I

TL;DR

This paper classifies regular polyhedra of index 2 in Euclidean 3-space, focusing on those with vertices on two symmetry orbits, revealing new geometric structures that are combinatorially regular but geometrically less symmetric.

Contribution

It provides a complete classification of regular polyhedra of index 2 with vertices on two orbits, advancing understanding of near-regular polyhedral structures.

Findings

Enumerates regular polyhedra of index 2 with two vertex orbits

Establishes combinatorial regularity versus geometric symmetry gap

Lays groundwork for classifying polyhedra with one vertex orbit in subsequent work

Abstract

A polyhedron in Euclidean 3-space is called a regular polyhedron of index 2 if it is combinatorially regular but "fails geometric regularity by a factor of 2"; its combinatorial automorphism group is flag-transitive but its geometric symmetry group has two flag orbits. The present paper, and its successor by the first author, describe a complete classification of regular polyhedra of index 2 in 3-space. In particular, the present paper enumerates the regular polyhedra of index 2 with vertices on two orbits under the symmetry group. The subsequent paper will enumerate the regular polyhedra of index 2 with vertices on one orbit under the symmetry group.