An algorithmic approach for computing fundamental domains of crystallographic groups

Abstract

A crystallographic group is a discrete subgroup of the Euclidean group $\operatorname{E}(n)$ that has a compact fundamental domain. Since such a crystallographic group $Γ$ is infinite, computing fundamental domains of $Γ$ is algorithmically challenging. We address this difficulty by targeting the computation of Dirichlet cells that can form fundamental domains of $Γ$. We show that the half-spaces defining such a Dirichlet cell can be derived from elements of $Γ$ acting on $\mathbb{R}^n$ that can be expressed as words of bounded length in a suitable generating set. Based on these results, we design an algorithm for the computation of fundamental domains of crystallographic groups and exploit it to study the construction of topological interlocking assemblies.