Convex-Faced Combinatorially Regular Polyhedra of Small Genus

Symmetry. 2011;4(1):1-14 DOI 10.3390/sym4010001


Journal Homepage

Journal Title: Symmetry

ISSN: 2073-8994 (Print)

Publisher: MDPI AG

LCC Subject Category: Science: Mathematics

Country of publisher: Switzerland

Language of fulltext: English

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Jörg M. Wills
Egon Schulte


Blind peer review

Editorial Board

Instructions for authors

Time From Submission to Publication: 11 weeks


Abstract | Full Text

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.