Lattice triangulations of \(\mathbb{E}^3 \) and of the 3-torus
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.
scientific article; zbMATH DE number 6105464
scientific article