  A uniformly discrete Euclidean graph is a graph embedded in a Euclidean space
so that there is a minimum distance between distinct vertices. If such a graph
embedded in an $n$-dimensional space is preserved under $n$ linearly
independent translations, it is "$n$-periodic" in the sense that the quotient
group of its symmetry group divided by the translational subgroup of its
symmetry group is finite. We present a refinement of a theorem of Bieberbach:
given a $n$-periodic uniformly discrete Euclidean graph embedded in a
$n$-dimensional Euclidean space of symmetry group $\bbbS$, there is another
$n$-periodic uniformly discrete Euclidean graph embedded in the same space
whose vertices are integer points (possibly modulo an affine transformation)
and whose symmetry group has a (not necessarily proper) subgroup isomorphic to
$\bbbS$. We conclude with a discussion of an application to the computer
generation of "crystal nets".
