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. In certain cases we impose additional restrictions on such triangulations such as regularity or invariance under central symmetry with respect to the origin; both properties hold for Delaunay tessellations of lattices.

Full enumeration of such periodic triangulations is obtained for dimension at most $4$. 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 infinitely many possible adjacent simplices. We found $950$ periodic triangulations in dimension $5$ but finiteness of the whole family is unknown.