Chirality and projective linear groups
type: Item
instance of: scholarly article
title: Chirality and projective linear groups
zbMATH Open document ID: 0806.52010
DOI: 10.1016/0012-365X(94)90387-5
author: Asia Ivić Weiss
author: Egon Schulte
published in: Discrete Mathematics
publication date: 1995-02-16T00:00:00Z
review text: This paper treats (abstract) polytopes of rank \(n\) which are defined by the properties of the face lattice of ordinary \(n\)-dimensional convex polytopes. A polytope is called chiral if it is in a certain sense symmetrical by rotation but not by reflexion, thus giving rise to the phenomenon of enantiomorphism. Important examples of regular or chiral polytopes of rank 3 correspond to maps on the torus. They are derived from the regular tessellations \(\{4,4\}\), \(\{3,6\}\) or \(\{6,3\}\) of the plane. Here the authors construct infinite series of chiral or regular polytopes of rank 4 whose facets or vertex figures are toroidal maps. This is done by the use of the groups \(\text{PSL}_ 2 (\mathbb{Z}_ m)\), \(\text{PSL}_ 2 (\mathbb{Z}_ m [i])\) and \(\text{PSL}_ 2 (\mathbb{Z}_ m [\omega])\), where \(i^ 2 = - 1\) and \(\omega^ 2 + \omega + 1 = 0\). The types of these polytopes correspond to regular honeycombs in hyperbolic three-space. Each of the five types \(\{4,4,3\}\), \(\{4,4,4\}\), \(\{6,3,3\}\), \(\{3,6,3\}\), \(\{6,3,6\}\) is studied in detail. A final section concerns the lengths of Petrie polygons of regular or chiral polytopes. The paper constitutes a continuation and elaboration of previous work of the authors, published in the `Victor Klee Festschrift' [Discret. Math. Theor. Comput. Sci. 4, 493-516 (1991; Zbl 0741.51019)].
reviewed by: Johann Linhart
Mathematics Subject Classification ID: 52B15
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 626340
zbMATH Keywords: abstract polytope
zbMATH Keywords: chirality
zbMATH Keywords: projective linear group
zbMATH Keywords: toroidal maps
zbMATH Keywords: honeycombs
MaRDI profile type: Publication
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full work available at URL: 0012-365x(94)90387-5
OpenAlex ID: W1999296501
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description: scientific article; zbMATH DE number 626340