Existence of Well-Centered Simplicial Face-to-Face Tiling of Five-Dimensional Space

Abstract

A well-centered simplex is characterized by the position of its circumcenter in the interior of the simplex. For triangles, i.e., two-dimensional simplices, well-centeredness complies with the property of acuteness. In higher dimensions these two properties are distinct. Both well-centered and acute tilings are of interest for their applications in numerical mathematics. While it has been already proven that there is no acute face-to-face simplicial tiling of five-dimensional Euclidean space, the question of the existence of well-centered tiling of five-dimensional Euclidean space remained open. In this paper, we deliver a positive answer. The proof of the main assertion is constructive. It is based on Sommerville’s construction of tetrahedral tilings, generalized for arbitrary dimension. Beside the main result, we also construct well-centered face-to-face tilings of four-dimensional Euclidean space formed by congruent copies of a single 4-simplex.