Algorithmic method for computing fundamental domains of crystallographic groups
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[Submitted on 2 Jul 2026]
Summary
This paper presents an algorithmic approach for computing fundamental domains of crystallographic groups. Crystallographic groups are infinite discrete subgroups of the Euclidean group with compact fundamental domains, making their computation algorithmically challenging. The authors address this by targeting Dirichlet cells, showing that the half-spaces defining such cells can be derived from group elements expressed as words of bounded length. They design an algorithm based on these results and apply it to study topological interlocking assemblies.
Source
bskyAlgorithmic method for computing fundamental domains of crystallographic groupsarxiv.orgKey quotes
· 5 pulledA crystallographic group is a discrete subgroup of the Euclidean group 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 ℝⁿ 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.
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