Periodic triangulations of \(\mathbb{Z}^{n}\)
type: Item
instance of: scholarly article
title: Periodic triangulations of \(\mathbb{Z}^{n}\)
zbMATH Open document ID: 1451.52010
DOI: 10.37236/8298
published in: The Electronic Journal of Combinatorics
publication date: 2020-06-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1810.10911
review text: This article considers certain decompositions of \(\mathbb{R}^n\) into (infinitely-many) \(n\)-dimensional polytopes, two of whose are required to meet along a (possibly empty) common face. When all these polytopes are simplices, such a decomposition of \(\mathbb{R}^n\) is called a triangulation. The authors are interested in the triangulations made up of simplices whose vertices belong to the lattice \(\mathbb{Z}^n\), and call them triangulations of \(\mathbb{Z}^n\). The article aims at enumerating the periodic triangulations of \(\mathbb{Z}^n\). In other words, these triangulations are invariant under any translation by a vector with integer coordinates. In order to do that, the authors consider a facet \(F\) of some \(n\)-dimensional simplex \(S\) in such a triangulation \(\mathcal{T}\). They show that, when \(n\) is either \(3\) or \(4\), there are only finitely-many \(n\)-dimensional simplices that can be glued to \(S\) along \(F\) within \(\mathcal{T}\). In dimension 3, this allows to show that there is a unique periodic triangulation of \(\mathbb{Z}^n\) up to the action of \(\mathrm{GL}_3(\mathbb{Z})\). In dimension \(4\), the authors use the same strategy, but this time they combine it with a computer search. This allows to establish that there are exactly four periodic triangulations of \(\mathbb{Z}^4\) up to the action of \(\mathrm{GL}_4(\mathbb{Z})\). In dimension \(5\) and above, the authors show that the same argument no longer allows to prove that there are only finitely-many simplices that can be glued to \(S\) along \(F\) within \(\mathcal{T}\). Instead, they use an algorithm based on bistellar flips, local operations that transform a triangulation into another (see for instance [\textit{J. Rambau}, in: Mathematical software. Proceedings of the 1st international congress, Beijing, China, August 17--19, 2002. Singapore: World Scientific. 330--340 (2002; Zbl 1057.68150)]), in order to show that there are at least \(950\) periodic triangulations of \(\m…
reviewed by: Lionel Pournin
Mathematics Subject Classification ID: 52C07
Mathematics Subject Classification ID: 52C22
zbMATH DE Number: 7208434
zbMATH Keywords: periodic triangulations
zbMATH Keywords: tilings
zbMATH Keywords: lattices
zbMATH Keywords: bistellar flips
author: Mathieu Dutour Sikirić
author: Alexey I. Garber
MaRDI profile type: Publication
OpenAlex ID: W3032809983
arXiv ID: 1810.10911
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description: scientific article