Naskręcki Bartosz, Malinowski Jakub, Dauter Zbigniew, Jaskolski Mariusz
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland.
Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wrocław, Poland.
Acta Crystallogr A Found Adv. 2025 Jan 1;81(Pt 1):64-81. doi: 10.1107/S2053273324010763.
This work analyzes the rules governing the growth of the numbers of vertices, edges and faces in all possible periodic tessellations of the 2D Euclidean space, and encodes those rules in several types of polynomial growth functions. These encodings map the geometric, combinatorial and topological properties of the tessellations into sets of integer coefficients. Several general statements about these encodings are given with rigorous mathematical proof. The variation of the growth functions is represented graphically and analyzed in orphic diagrams, so named because of their similarity to orphic art. Several examples of 3D space groups are included, to emphasize the complexity of the growth functions in higher dimensions. A freely available Python library is presented to facilitate the discovery of the growth functions and the generation of orphic diagrams.
这项工作分析了二维欧几里得空间中所有可能的周期性镶嵌中顶点、边和面数量增长所遵循的规则，并将这些规则编码为几种类型的多项式增长函数。这些编码将镶嵌的几何、组合和拓扑性质映射到整数系数集。给出了关于这些编码的几个一般性陈述，并给出了严格的数学证明。增长函数的变化以图形方式表示，并在因与俄耳甫斯艺术相似而得名的俄耳甫斯图中进行分析。文中包含了几个三维空间群的例子，以强调高维中增长函数的复杂性。还提供了一个免费的Python库，以促进增长函数的发现和俄耳甫斯图的生成。
