Lattice triangulations of \(\mathbb{E}^3 \) and of the 3-torus
type: Item
instance of: scholarly article
title: Lattice triangulations of \(\mathbb{E}^3 \) and of the 3-torus
zbMATH Open document ID: 1258.52009
author: Ulrich Brehm
author: Wolfgang Kühnel
published in: Israel Journal of Mathematics
publication date: 2012-11-13T00:00:00Z
review text: This paper concerns a few questions on tilings of Euclidean spaces and the 3-dimensional torus. The tiles here are topological simplices with curvilinear edges. The authors investigate lattice triangulations of Euclidean 3-space whose vertices form a lattice of rank 3 and such that these triangulations are invariant under all lattice translations. The tiles are not assumed to be Euclidean polyhedra but only topological polyhedra. In 3-space there is a unique standard lattice triangulation with straight edges by Euclidean tetrahedra and there are infinitely many non-standard lattice triangulations where the tetrahedra have certain curvilinear edges. This implies that there are triangulations of 3-space which do not carry any flat discrete metric equivariant under the lattice. Furthermore, the authors investigate lattice triangulations of the 3-dimensional torus as quotients by a sublattice. The standard triangulation admits such quotients with any number \(n \geq 15\) of vertices. The unique one is of 15 vertices pairwise connected by an edge. It turns out that for any odd \(n \geq 17\) there is an \(n\)-vertex triangulation with pairwise connected vertices of the 3-torus as a quotient of a certain non-standard lattice triangulation. Combinatorially, one can obtain these neighborly 3-tori as slight modifications of the boundary complexes of the cyclic 4-polytopes.
Mathematics Subject Classification ID: 52B70
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 52C22
zbMATH DE Number: 6105464
zbMATH Keywords: tilings of Euclidean spaces
zbMATH Keywords: 3-dimensional torus
zbMATH Keywords: lattice triangulations
zbMATH Keywords: Euclidean 3-space
reviewed by: Oleg Karpenkov
MaRDI profile type: Publication
full work available at URL: s11856-011-0180-8
OpenAlex ID: W1972006430
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Wikidata QID: Q125676419
DOI: 10.1007/S11856-011-0180-8
description: scientific article; zbMATH DE number 6105464