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

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.

arXiv commentCompletely rewritten version, two additional authors (JD, ML) of this new version. 18 pages, 15 figures