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arXiv · 1305.1798

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

Joseph Ray Clarence G. Damasco·Dirk Frettlöh·Manuel Joseph C. Loquias
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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.

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Joseph Ray Clarence G. Damasco, Dirk Frettlöh, Manuel Joseph C. Loquias. 2013-05-08. Highly symmetric fundamental domains for lattices in R^2 and R^3. https://arxiv.org/abs/1305.1798

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