Abstract In this paper, we present a simple and intuitive approach for designing a new class of space-filling shapes that we call Generalized Abeille Tiles (GATs). GATs are generalizations of Abeille vaults, introduced by the French engineer and architect Joseph Abeille in late 1600s. Our approach is based on two principles. The first principle is the correspondence between structures proposed by Abeille and the symmetries exhibited by woven fabrics. We leverage this correspondence to develop a theoretical framework for GATs beginning with the theory of bi-axial 2-fold woven fabrics. The second principle is the use of Voronoi decomposition with higher dimensional Voronoi sites (curves and surfaces). By configuring these new Voronoi sites based on weave symmetries, we provide a method for constructing GATs. Subsequently, we conduct a comparative structural analysis of GATs as individual shapes as well as tiled assemblies for three different fabric patterns using plain and twill weave patterns. Our analysis reveals interesting relationship between the choice of fabric symmetries and the corresponding distribution of stresses under loads normal to the tiled assemblies.

中文翻译：

---

广义 Abeille 瓷砖：基于织物对称性生成的拓扑互锁空间填充形状

摘要在本文中，我们提出了一种简单直观的方法来设计一类新的空间填充形状，我们称之为广义 Abeille Tiles (GAT)。GAT 是 Abeille 拱顶的推广，由法国工程师和建筑师 Joseph Abeille 在 1600 年代后期引入。我们的方法基于两个原则。第一个原则是 Abeille 提出的结构与机织织物表现出的对称性之间的对应关系。我们利用这种对应关系从双轴 2 折织物理论开始为 GAT 开发理论框架。第二个原则是使用具有更高维 Voronoi 位点（曲线和曲面）的 Voronoi 分解。通过基于编织对称性配置这些新的 Voronoi 站点，我们提供了一种构建 GAT 的方法。随后，我们对使用平纹和斜纹组织图案的三种不同织物图案的 GAT 作为单个形状和平铺组件进行比较结构分析。我们的分析揭示了织物对称性的选择与在垂直于平铺组件的载荷下的相应应力分布之间的有趣关系。