Naskręcki Bartosz, Dauter Zbigniew, Jaskolski Mariusz
Faculty of Mathematics and Computer Science, A. Mickiewicz University, Poznań, Poland.
Macromolecular Crystallography Laboratory, NCI, Argonne National Laboratory, Argonne, Illinois, USA.
Acta Crystallogr A Found Adv. 2021 Jul 1;77(Pt 4):317-326. doi: 10.1107/S2053273321004320. Epub 2021 Jun 21.
The notion of the Euler characteristic of a polyhedron or tessellation has been the subject of in-depth investigations by many authors. Two previous papers worked to explain the phenomenon of the vanishing (or zeroing) of the modified Euler characteristic of a polyhedron that underlies a periodic tessellation of a space under a crystallographic space group. The present paper formally expresses this phenomenon as a theorem about the vanishing of the Euler characteristic of certain topological spaces called topological orbifolds. In this new approach, it is explained that the theorem in question follows from the fundamental properties of the orbifold Euler characteristic. As a side effect of these considerations, a theorem due to Coxeter about the vanishing Euler characteristic of a honeycomb tessellation is re-proved in a context which frees the calculations from the assumptions made by Coxeter in his proof. The abstract mathematical concepts are visualized with down-to-earth examples motivated by concrete situations illustrating wallpaper and 3D crystallographic space groups. In a way analogous to the application of the classic Euler equation to completely bounded solids, the formula proven in this paper is applicable to such important crystallographic objects as asymmetric units and Dirichlet domains.
多面体或镶嵌的欧拉示性数概念一直是许多作者深入研究的主题。之前的两篇论文致力于解释在晶体学空间群下空间的周期性镶嵌所基于的多面体的修正欧拉示性数消失（或归零）的现象。本文将此现象正式表述为一个关于某些称为拓扑轨形的拓扑空间的欧拉示性数消失的定理。在这种新方法中，解释了所讨论的定理源于轨形欧拉示性数的基本性质。作为这些考虑的一个附带结果，在一个使计算摆脱考克斯特在其证明中所做假设的背景下，重新证明了考克斯特关于蜂巢镶嵌的欧拉示性数消失的一个定理。通过由说明壁纸和三维晶体学空间群的具体情况所激发的实际例子，将抽象的数学概念可视化。以类似于将经典欧拉方程应用于完全有界固体的方式，本文证明的公式适用于诸如不对称单元和狄利克雷域等重要的晶体学对象。
