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Symmetry 2012, 4(1), 1-14; doi:10.3390/sym4010001
Article

Convex-Faced Combinatorially Regular Polyhedra of Small Genus

1,†,*  and 2
1 Department of Mathematics, Northeastern University, Boston, MA 02115, USA 2 Department Mathematik, University of Siegen, Emmy-Noether-Campus, D-57068 Siegen, Germany Supported by NSF-Grant DMS–0856675.
* Author to whom correspondence should be addressed.
Received: 28 November 2011 / Revised: 15 December 2011 / Accepted: 19 December 2011 / Published: 28 December 2011
(This article belongs to the Special Issue Polyhedra)
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Abstract

Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E3 of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g ≥ 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 ≤ g ≤ 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete; in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 ≤ g ≤ 6, admitting realizations as convex-faced polyhedra in E3. For all admissible maps in this range, save Gordan’s map of genus 4, and its dual, we rule out realizability by a polyhedron in E3.
Keywords: Platonic solids; regular polyhedra; regular maps; Riemann surfaces; polyhedral embeddings; automorphism groups Platonic solids; regular polyhedra; regular maps; Riemann surfaces; polyhedral embeddings; automorphism groups
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Schulte, E.; Wills, J.M. Convex-Faced Combinatorially Regular Polyhedra of Small Genus. Symmetry 2012, 4, 1-14.

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