Regular Polyhedra of Index Two, I

Details Regular Polyhedra of Index Two, I Regular Polyhedra of Index Two, I

2010-05-26

arxiv.org/abs/1005.4911v1

Mathematics

Metric Geometry

34 pages; 6 figures; to appear in "Contributions to Algebra and Geometry"

Scientific paper

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.

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