Periodic Euclidean Graphs on Integer Points

TL;DR

This paper refines Bieberbach's theorem to show that n-periodic Euclidean graphs can be represented with vertices at integer points, aiding in the computer generation of crystal nets.

Contribution

It provides a new theorem demonstrating that n-periodic Euclidean graphs can be transformed to have integer vertices while preserving their symmetry groups.

Findings

Existence of integer-point representations for n-periodic graphs.

Extension of Bieberbach's theorem to Euclidean graphs.

Application to computer-generated crystal nets.

Abstract

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".