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

TL;DR

This paper studies periodic triangulations of integer lattices, exploring their properties, enumeration in low dimensions, and new phenomena in dimension five, including non-Delaunay and non-regular triangulations.

Contribution

It provides a full enumeration of periodic triangulations up to dimension 4 and uncovers new phenomena in dimension 5, including the existence of non-Delaunay and non-regular triangulations.

Findings

Full enumeration of triangulations in dimensions up to 4.

Existence of non-Delaunay, centrally-symmetric triangulations in dimension 5.

Found 950 periodic triangulations in dimension 5.

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.