Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry

, Volume 52, Issue 1, pp 133–161

Regular polyhedra of index two, I

Authors

  • Anthony M. Cutler
    • Northeastern University
    • Northeastern University
Original Paper

DOI: 10.1007/s13366-011-0015-0

Cite this article as:
Cutler, A.M. & Schulte, E. Beitr Algebra Geom (2011) 52: 133. doi:10.1007/s13366-011-0015-0

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.

Keywords

Regular polyhedraKepler–Poinsot polyhedraArchimedean polyhedraFace-transitivityRegular maps on surfacesAbstract polytopes

Mathematics Subject Classification (2000)

Primary 51M20Secondary 52B15

Copyright information

© The Managing Editors 2011