Chiral 4-polytopes in ordinary space
type: Item
instance of: scholarly article
title: Chiral 4-polytopes in ordinary space
zbMATH Open document ID: 1385.52010
author: Daniel Pellicer
published in: Beiträge zur Algebra und Geometrie
publication date: 2018-02-28T00:00:00Z
review text: Let \(\mathcal{P}\) be a polytope (finite or infinite) in \(3\)-space, let \(G(\mathcal{P})\) be its symmetry group (of euclidean isometries) and let \(G^+(\mathcal{P})\) be the even subgroup of \(G(\mathcal{P})\) that permutes flags that differ by an even number of adjacent steps. A polytope \(\mathcal{P}\) is ranked, so that a \(4\)-polytope has vertices, edges, \(2\)-faces and facets. Then \(\mathcal{P}\) is (geometrically) regular if \(G(\mathcal{P})\) is transitive on all the flags of \(\mathcal{P}\), and chiral if \(G(\mathcal{P})\) induces two orbits on the flags so that adjacent flags belong to different orbits; in the latter case, \(G(\mathcal{P}) = G^+(\mathcal{P})\). Observe that \(G(\mathcal{P})\) can differ from the automorphism group \(\Gamma(\mathcal{P})\), consisting of the combinatorial symmetries; thus \(\mathcal{P}\) can be combinatorially regular, but only chiral as a geometric polytope. The author begins by introducing several Euclidean nets (connected graphs with discrete vertex sets admitting a full lattice of translational symmetries), and giving a survey of the basic theory of polytopes and their symmetries. He then describes three chiral (infinite) \(4\)-polytopes, giving details of their symmetry groups (in particular their generators) and geometric structure; the euclidean nets play a part in the latter. (One of these was the subject of the author's earlier paper [Ars Math. Contemp. 12, No. 2, 315--327 (2017; Zbl 1382.52011)].) He then proves that these polytopes are combinatorially chiral; in fact, they have chiral facets. Since finite polyhedra in \(3\)-space cannot be chiral, nor can polygons or tessellations in the plane, these polytopes provide the strongest possible refutation of a claim by the reviewer in [Discrete Comput. Geom. 32, No. 1, 1--35 (2004; Zbl 1059.52019)] that polytopes of full rank could not be chiral; a finite chiral \(4\)-polytope of full rank was found by \textit{J. Bracho} et al. [Discrete Comput. Geom. 52, No. 4…
reviewed by: Peter McMullen
Mathematics Subject Classification ID: 52B15
zbMATH DE Number: 6845410
zbMATH Keywords: chiral polyhedron
zbMATH Keywords: chiral 4-polytope
zbMATH Keywords: abstract polytope
zbMATH Keywords: Euclidean net
MaRDI profile type: Publication
full work available at URL: s13366-017-0342-x
OpenAlex ID: W2610447370
cites work: A finite chiral 4-polytope in \(\mathbb R^4\)
cites work: Regular Skew Polyhedra in Three and Four Dimension, and their Topological Analogues
cites work: A combinatorial theory of Gruenbaum's new regular polyhedra. I: Gruenbaum's new regular polyhedra and their automorphism group
cites work: A combinatorial theory of Grünbaum's new regular polyhedra. II: Complete enumeration
cites work: Q3701635
cites work: Regular polyhedra - old and new
cites work: Q3998721
cites work: Q4865660
cites work: Regular polytopes in ordinary space
cites work: Regular polytopes of full rank
cites work: Abstract Regular Polytopes
cites work: Three-periodic nets and tilings: regular and related infinite polyhedra
cites work: Developments and open problems on chiral polytopes
cites work: Chiral polyhedra in ordinary space. I
cites work: A chiral 4-polytope in R^3
cites work: Combinatorial structure of Schulte's chiral polyhedra
cites work: Polygonal Complexes and Graphs for Crystallographic Groups
cites work: Q4314293
cites work: Chiral polyhedra in ordinary space. II
cites work: Polyhedra, complexes, nets and symmetry
cites work: Q3974989
DOI: 10.1007/S13366-017-0342-X
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description: scientific article; zbMATH DE number 6845410