Naskręcki Bartosz, Jaskolski Mariusz, Dauter Zbigniew
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznan, Poland.
Department of Crystallography, Faculty of Chemistry, A. Mickiewicz University, Poznan, Poland.
J Appl Crystallogr. 2022 Feb 1;55(Pt 1):154-167. doi: 10.1107/S160057672101205X.
The simple Euler polyhedral formula, expressed as an alternating count of the bounding faces, edges and vertices of any polyhedron, - + = 2, is a fundamental concept in several branches of mathematics. Obviously, it is important in geometry, but it is also well known in topology, where a similar telescoping sum is known as the Euler characteristic χ of any finite space. The value of χ can also be computed for the unit polyhedra (such as the unit cell, the asymmetric unit or Dirichlet domain) which build, in a symmetric fashion, the infinite crystal lattices in all space groups. In this application χ has a modified form (χ) and value because the addends have to be weighted according to their symmetry. Although derived in geometry (in fact in crystallography), χ has an elegant topological interpretation through the concept of orbifolds. Alternatively, χ can be illustrated using the theorems of Harriot and Descartes, which predate the discovery made by Euler. Those historical theorems, which focus on angular defects of polyhedra, are beautifully expressed in the formula of de Gua de Malves. In a still more general interpretation, the theorem of Gauss-Bonnet links the Euler characteristic with the general curvature of any closed space. This article presents an overview of these interesting aspects of mathematics with Euler's formula as the leitmotif. Finally, a game is designed, allowing readers to absorb the concept of the Euler characteristic in an entertaining way.
简单的欧拉多面体公式，表示为对任何多面体的边界面、边和顶点进行交替计数，即面数 - 边数 + 顶点数 = 2，是数学多个分支中的一个基本概念。显然，它在几何学中很重要，但在拓扑学中也广为人知，在拓扑学中，类似的叠缩和被称为任何有限空间的欧拉示性数χ。χ的值也可以针对以对称方式构建所有空间群中的无限晶格的单位多面体（如晶胞、不对称单元或狄利克雷域）进行计算。在这种应用中，χ具有修改后的形式(χ)和值，因为加数必须根据它们的对称性进行加权。尽管χ是在几何学（实际上是在晶体学）中推导出来的，但通过orbifolds的概念，它有一个优雅的拓扑解释。或者，可以使用哈里奥特和笛卡尔的定理来说明χ，这些定理早于欧拉的发现。那些关注多面体角缺陷的历史定理，在德瓜·德马尔韦斯的公式中得到了优美的表达。在更一般的解释中，高斯 - 博内定理将欧拉示性数与任何封闭空间的总曲率联系起来。本文以欧拉公式为主线，概述了这些有趣的数学方面。最后，设计了一个游戏，让读者以一种有趣的方式理解欧拉示性数的概念。
