Next Article in Journal
Towards Symmetry-Based Explanation of (Approximate) Shapes of Alpha-Helices and Beta-Sheets (and Beta-Barrels) in Protein Structure
Next Article in Special Issue
Self-Dual, Self-Petrie Covers of Regular Polyhedra
Previous Article in Journal
Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry
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)
Download PDF [271 KB, uploaded 28 December 2011]

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 (CC BY 3.0).
SciFeed

Share & Cite This Article

Further Mendeley | CiteULike
Export to BibTeX |
EndNote |
RIS
MDPI and ACS Style

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

View more citation formats

Related Articles

Article Metrics

For more information on the journal, click here

Comments

[Return to top]
Symmetry EISSN 2073-8994 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert