Self-dual regular 4-polytopes and their Petrie-Coxeter-polyhedra
type: Item
instance of: scholarly article
title: Self-dual regular 4-polytopes and their Petrie-Coxeter-polyhedra
zbMATH Open document ID: 0649.52005
DOI: 10.1007/BF03322401
published in: Results in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: \textit{H. S. M. Coxeter} [Proc. Lond. Math. Soc., II. Ser. 43, 33-62 (1937; Zbl 0016.27101)] discovered that to every self-dual convex regular 4- polytope P, there may be associated a regular skew-polyhedron M. The automorphism group of M is that of P twisted by \(Z_ 2\). The authors define an incidence-polytope as a combinatorial analog of a polytope, and generalize Coxeter's construction to self-dual regular 4-incidence polytopes of type \(\{\) p,q,p\(\}\). Here ``type'' lists the orders of certain generators of the automorphism group of the polytope. The method uses twisting operations on certain groups. The associated regular map M is called the Petrie-Coxeter polyhedron of P. (Quotients of) Euclidean and hyperbolic tesselations may arise in this fashion.
Mathematics Subject Classification ID: 52Bxx
zbMATH DE Number: 4060221
zbMATH Keywords: Petrie-Coxeter polyhedra
zbMATH Keywords: self-dual regular 4-incidence polytopes
zbMATH Keywords: tesselations
author: Peter McMullen
author: Egon Schulte
reviewed by: Mikhail G. Katz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4060221