Date: 30 Mar 2011

Regular polyhedra of index two, I

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Abstract

A polyhedron in Euclidean 3-space \({{\mathbb{E}^3}}\) 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 \({{\mathbb{E}^3}}\) . 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.

E. Schulte was supported by NSA-grant H98230-07-1-0005 and NSF-grant DMS-0856675.