  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.
