%0 Journal Article
%T Convex-Faced Combinatorially Regular Polyhedra of Small Genus
%A Egon Schulte
%A J£¿rg M. Wills
%J Symmetry
%D 2012
%I MDPI AG
%R 10.3390/sym4010001
%X 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 ¡Ý 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 ¡Ü g ¡Ü 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 ¡Ü g ¡Ü 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.
%K Platonic solids
%K regular polyhedra
%K regular maps
%K Riemann surfaces
%K polyhedral embeddings
%K automorphism groups
%U http://www.mdpi.com/2073-8994/4/1/1