计算晶体群基本域的算法方法
An algorithmic approach for computing fundamental domains of crystallographic groups
AI总结：
提出一种基于Dirichlet胞腔的算法，通过有界长度的群元素推导半空间，计算晶体群的基本域，并应用于拓扑互锁结构。
摘要：
晶体群是欧几里得群$\operatorname{E}(n)$的一个离散子群，具有紧致的基本域。由于这样的晶体群$\Gamma$是无限的，计算$\Gamma$的基本域在算法上具有挑战性。我们通过针对计算可形成$\Gamma$基本域的Dirichlet胞腔来解决这一困难。我们证明了定义这样一个Dirichlet胞腔的半空间可以从$\Gamma$中作用于$\mathbb{R}^n$的元素导出，这些元素可以表示为适当生成集中有界长度的单词。基于这些结果，我们设计了一种用于计算晶体群基本域的算法，并利用它来研究拓扑互锁结构的构造。
英文摘要：
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.
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