Original Paper

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry

, Volume 52, Issue 1, pp 133-161

Regular polyhedra of index two, I

  • Anthony M. CutlerAffiliated withNortheastern University
  • , Egon SchulteAffiliated withNortheastern University Email author 

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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.

Keywords

Regular polyhedra Kepler–Poinsot polyhedra Archimedean polyhedra Face-transitivity Regular maps on surfaces Abstract polytopes

Mathematics Subject Classification (2000)

Primary 51M20 Secondary 52B15