Symmetries of equivelar 4-toroids
type: Item
instance of: scholarly article
title: Symmetries of equivelar 4-toroids
zbMATH Open document ID: 1263.51016
author: Isabel Hubard
author: Alen Orbanić
author: Daniel Pellicer
author: Asia Ivić Weiss
published in: Discrete \& Computational Geometry
publication date: 2013-01-11T00:00:00Z
review text: The quotient of a regular tiling by a subgroup of full rank of its translation group is an equivelar toroid. In this paper, the authors consider such toroids obtained from the tessellations \(\{4,4\}\) and \(\{3,6\}\) of the plane, and the cubic tiling \(\{4,3,4\}\) of ordinary space, particularly the latter. The symmetry group of a toroid depends on how symmetric the translation subgroup is. For example, for \(\{4,3,4\}\) the subgroup is generated by triples \(v_1,v_2,v_3\) of lattice (integer) vectors, and these fall into various classes, subdivided into families, according to how large a subgroup of the octahedral group continues to act on the toroid. Thus in class 1, with three families, one has the full octahedral group (these are the regular toroids). Class 2 would contain chiral toroids, but there are none, and so this class is empty. Otherwise, there is a class (for some, split into subclasses) for each divisor of \(24\), with only the central symmetry of the octahedra group surviving in class 24; the full results are too elaborate to reproduce in the review. In a similar way, the classification of the toroids of types \(\{4,4\}\) and \(\{3,6\}\) depend on the subgroup structure of the dihedral groups of orders \(8\) and \(12\), respectively.
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 52B15
zbMATH DE Number: 6124644
zbMATH Keywords: equivelar
zbMATH Keywords: symmetries of toroids
zbMATH Keywords: polytope
zbMATH Keywords: cubical tessellation
reviewed by: Peter McMullen
MaRDI profile type: Publication
full work available at URL: s00454-012-9444-2
OpenAlex ID: W1981450700
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DOI: 10.1007/S00454-012-9444-2
description: scientific article; zbMATH DE number 6124644