An algorithmic approach for computing fundamental domains of crystallographic groups

2026-07-02

AI Summary

This work addresses the long-standing challenge of algorithmically constructing fundamental domains for crystallographic groups—discrete infinite groups of isometries. The authors propose the first general-purpose algorithm that leverages Dirichlet cells associated with group elements of bounded word length. They prove that the bounding half-spaces of these cells can be determined by words of finite length and combine insights from group theory, computational geometry, and the action of Euclidean isometry groups to efficiently compute fundamental domains. This approach not only renders the construction of fundamental domains for crystallographic groups computationally tractable but also enables practical applications in the design of topologically interlocked structures, thereby significantly expanding their potential in geometric modeling and materials science.

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.

Problem

• crystallographic groups
• fundamental domains
• Dirichlet cells
• algorithmic computation
• Euclidean group

Innovation

• crystallographic groups
• fundamental domains
• Dirichlet cells
• algorithmic computation
• bounded word length