arXiv subjects
Publications and source records attributed to Joseph Ray Clarence G. Damasco.
We show by construction that every rhombic lattice $Γ$ in $\mathbb{R}^{2}$ has a fundamental domain whose symmetry group contains the point group of $Γ$ as a subgroup of index $2$. This solves the last open case of a question raised in [3] on fundamental domains for planar lattices whose symmetry groups properly contain the point groups of the lattices.
It is shown that most lattices $Γ$ in $\mathbb{R}^2$ and $\mathbb{R}^3$ possess a fundamental domain $F$ for the action of $Γ$ on $\mathbb{R}^2$, respectively $\mathbb{R}^3$, having more symmetries than the point group $P(Γ)$, i.e., the group $P (Γ) \subset O(d)$ fixing $Γ$. In particular, $P (Γ)$ 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.