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Convex-Faced Combinatorially Regular Polyhedra of Small Genus
Department of Mathematics, Northeastern University, Boston, MA 02115, USA
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; in revised form: 15 December 2011 / Accepted: 19 December 2011 / Published: 28 December 2011
(This article belongs to the Special Issue Polyhedra
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
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Schulte, E.; Wills, J.M. Convex-Faced Combinatorially Regular Polyhedra of Small Genus. Symmetry 2012, 4, 1-14.
Schulte E, Wills JM. Convex-Faced Combinatorially Regular Polyhedra of Small Genus. Symmetry. 2012; 4(1):1-14.
Schulte, Egon; Wills, Jörg M. 2012. "Convex-Faced Combinatorially Regular Polyhedra of Small Genus." Symmetry 4, no. 1: 1-14.