Regular polytopes in ordinary space
type: Item
instance of: scholarly article
title: Regular polytopes in ordinary space
zbMATH Open document ID: 0876.52003
DOI: 10.1007/PL00009304
published in: Discrete \& Computational Geometry
publication date: 1997-10-27T00:00:00Z
review text: The authors give a new proof of the completeness of the enumeration of the regular apeirohedra (that is, the regular infinite polyhedra) of rank 3 in \(\mathbb E^3\). Previously the proof was obtained by \textit{A. W. M. Dress} in Aequationes Math. 23, 252-265 (1981; Zbl 0506.51010) and ibid. 29, 222-243 (1985; Zbl 0588.51022). Besides, the paper contains a complete classification of the discrete regular polytopes in \(\mathbb E^3\): the authors describe seven new discrete regular apeirotopes of rank 4 in the ordinary space, to add to the familiar honeycomb \(\{ 4,3,4\} \) of cubes, and prove that there are no others. Presentations of the symmetry groups are given of those discrete regular apeirohedra in \(\mathbb E^3\) whose groups are affinely irreducible. New pairings are introduced between the finite regular polyhedra. The notion of abstract polytope is used systematically.
Mathematics Subject Classification ID: 52B10
zbMATH DE Number: 1016881
zbMATH Keywords: regular polyhedra
zbMATH Keywords: regular polytopes
zbMATH Keywords: combinatorial polytopes
zbMATH Keywords: Grünbaum's regular polyhedra
zbMATH Keywords: regular apeirohedra
zbMATH Keywords: apeirotopes of rank 3
zbMATH Keywords: apeirotopes of rank 4
zbMATH Keywords: automorphism group
zbMATH Keywords: symmetry group
zbMATH Keywords: isometry group
author: Peter McMullen
author: Egon Schulte
reviewed by: Victor Alexandrov
MaRDI profile type: Publication
Recommended article: On regular polytopes
Recommended article: Q4309486
Recommended article: Q3706092
Recommended article: Geometric Regular Polytopes
Recommended article: Q3345360
Recommended article: Realizations of regular polytopes
Recommended article: Q3692377
Recommended article: Q4441823
description: scientific article; zbMATH DE number 1016881