Periodic triangulations of \(\mathbb{Z}^{n}\)
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 \(\mathbb{Z}^5\) up to the action of \(\mathrm{GL}_5(\mathbb{Z})\). The authors also study the regularity properties of the triangulations they find: a triangulation of \(\mathbb{Z}^n\) is regular if it can be obtained by lifting the points of \(\mathbb{Z}^n\) into \(\mathbb{R}^{n+1}\) using a convex function \(\mathbb{R}^n\rightarrow\mathbb{R}\) that tells the altitude of the lifted points, by taking the convex hull of these lifted points, and by projecting the faces of that convex hull back on \(\mathbb{R}^n\). If the convex function is a (positive definite) quadratic form, then the triangulation is a Delaunay triangulation and it is necessarily centrally-symmetric with respect to the origin. While all the periodic triangulations of \(\mathbb{Z}^3\) are Delaunay, there exist non-Delaunay, non-centrally-symmetric periodic triangulation of \(\mathbb{Z}^4\). Among the obtained \(950\) periodic triangulations of \(\mathbb{Z}^5\), the authors find both non-regular triangulations and centrally-symmetric triangulations that are not Delaunay.
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