Advisor(s)

Egon Schulte

Contributor(s)

Mark Ramras, Brigitte Servatius, Andrei V. Zelevinsky

Date of Award

2009

Date Accepted

4-2009

Degree Grantor

Northeastern University

Degree Level

Ph.D.

Degree Name

Doctor of Philosophy

Department or Academic Unit

College of Arts and Sciences. Department of Mathematics.

Keywords

Index 2, Polyhedra

Subject Categories

Polyhedra

Disciplines

Mathematics

Abstract

We classify all finite regular polyhedra of index 2, as defined in Section 2 herein. The definition requires the polyhedra to be combinatorially flag transitive, but does not require them to have planar or convex faces or vertex-figures, and neither does it require the polyhedra to be orientable. We find there are 10 combinatorially regular polyhedra of index 2 with vertices on one orbit, and 22 infinite families of combinatorially regular polyhedra of index 2 with vertices on two orbits, where polyhedra in the same family differ only in the relative diameters of their vertex orbits. For each such polyhedron, or family of polyhedra, we provide the underlying map, as well as a geometric diagram showing a representative face for each face orbit, and a verification of the polyhedron's combinatorial regularity. A self-contained completeness proof is given. Exactly five of the polyhedra have planar faces, which is consistent with a previously known result. We conclude by describing a non-Petrie duality relation among regular polyhedra of index 2, and suggest how it can be extended to other combinatorially regular polyhedra.

Document Type

Dissertation

Rights Holder

Anthony Cutler


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