Periodic triangulations of ${\mathbb{Z}}^n$

Abstract

We consider in this work triangulations of ${\mathbb{Z}}^n$ that are periodic along ${\mathbb{Z}}^n$. They generalize the triangulations obtained from Delaunay tessellations of lattices. Other important property is the regularity and central-symmetry property of triangulations. Full enumeration for dimension at most $4$ is obtained. In dimension $5$ several new phenomena happen: there are centrally-symmetric triangulations that are not Delaunay, there are non-regular triangulations (it could happen in dimension $4$) and a given simplex has a priori an infinity of possible adjacent simplices. We found $950$ periodic triangulations in dimension $5$ but finiteness is unknown.