Lattice triangulations of $\mathbb{E}^3$ and of the 3-torus

Abstract

This paper gives answers to a few questions concerning tilings of Euclidean spaces where the tiles are topological simplices with curvilinear edges. We investigate lattice triangulations of Euclidean 3-space in the sense that the vertices form a lattice of rank 3 and such that the triangulation is invariant under all translations of that lattice. This is the dual concept of a primitive lattice tiling where the tiles are not assumed to be Euclidean polyhedra but only topological polyhedra. In 3-space there is a unique standard lattice triangulation by Euclidean tetrahedra (and with straight edges) but there are infinitely many non-standard lattice triangulations where the tetrahedra necessarily have certain curvilinear edges. From the view-point of Discrete Differential Geometry this tells us that there are such triangulations of 3-space which do not carry any flat discrete metric which is equivariant under the lattice. Furthermore, we investigate lattice triangulations of the 3-dimensional torus as quotients by a sublattice. The standard triangulation admits such quotients with any number n ≥ 15 of vertices. The unique one with 15 vertices is neighborly, i.e., any two vertices are joined by an edge. It turns out that for any odd n ≥ 17 there is an n-vertex neighborly triangulation 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. As a kind of combinatorial surgery, this is an interesting construction by itself.