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Symmetry 2012, 4(1), 208-218; doi:10.3390/sym4010208

Self-Dual, Self-Petrie Covers of Regular Polyhedra

Northeastern University, 360 Huntington Ave, Boston, MA 02115, USA
Received: 17 January 2012 / Revised: 21 February 2012 / Accepted: 23 February 2012 / Published: 27 February 2012
(This article belongs to the Special Issue Polyhedra)
Download PDF [234 KB, uploaded 28 February 2012]


The well-known duality and Petrie duality operations on maps have natural analogs for abstract polyhedra. Regular polyhedra that are invariant under both operations have a high degree of both “external” and “internal” symmetry. The mixing operation provides a natural way to build the minimal common cover of two polyhedra, and by mixing a regular polyhedron with its five other images under the duality operations, we are able to construct the minimal self-dual, self-Petrie cover of a regular polyhedron. Determining the full structure of these covers is challenging and generally requires that we use some of the standard algorithms in combinatorial group theory. However, we are able to develop criteria that sometimes yield the full structure without explicit calculations. Using these criteria and other interesting methods, we then calculate the size of the self-dual, self-Petrie covers of several polyhedra, including the regular convex polyhedra.
Keywords: abstract polyhedron; convex polyhedron; duality; map operations; mixing; Petrie polygon; Petrie dual abstract polyhedron; convex polyhedron; duality; map operations; mixing; Petrie polygon; Petrie dual
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Cunningham, G. Self-Dual, Self-Petrie Covers of Regular Polyhedra. Symmetry 2012, 4, 208-218.

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