convex-faced combinatorially regular polyhedra of small genus

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ID: 161741
2011
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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.
Reference Key
wills2011symmetryconvex-faced Use this key to autocite in the manuscript while using SciMatic Manuscript Manager or Thesis Manager
Authors ;Jörg M. Wills;Egon Schulte
Journal journal of hospitality and tourism management
Year 2011
DOI
10.3390/sym4010001
URL
http://www.mdpi.com/2073-8994/4/1/1/
Keywords Keywords not found

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