Highly symmetric fundamental domains for lattices in R^2 and R^3

TL;DR

This paper demonstrates that most lattices in two and three dimensions have fundamental domains with higher symmetry than their point groups, except for certain special cases like cubic lattices in 3D.

Contribution

It establishes the existence of highly symmetric fundamental domains for most lattices in R^2 and R^3, highlighting exceptions such as cubic lattices.

Findings

Most lattices have fundamental domains with symmetry group of index 2.

Cubic lattices in 3D lack such highly symmetric fundamental domains.

Possible exceptions include rhombic lattices in 2D.

Abstract

It is shown that most lattices $\Gamma$ in $\mathbb{R}^2$ and $\mathbb{R}^3$ possess a fundamental domain $F$ for the action of $\Gamma$ on $\mathbb{R}^2$, respectively $\mathbb{R}^3$, having more symmetries than the point group $P(\Gamma)$, i.e., the group $P (\Gamma) \subset O(d)$ fixing $\Gamma$. In particular, $P (\Gamma)$ is a subgroup of the symmetry group $S(F)$ of $F$ of index 2 in these cases. Exceptions are cubic lattices in the three-dimensional case, where such an $F$ does not exist. Possible exceptions are rhombic lattices in the plane case, where the constructions presented here do not seem to work.