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Abstract：
The simple Euler polyhedral formula, expressed as an alternating count of the bounding faces, edges and vertices of any polyhedron, V - E + F = 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 (χm) and value because the addends have to be weighted according to their symmetry. Although derived in geometry (in fact in crystallography), χm has an elegant topological interpretation through the concept of orbifolds. Alternatively, χm 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.
摘要：
简单的欧拉多面体公式，表示为边界面的交替计数，任何多面体的边和顶点，V—E+F=2，是数学若干分支中的一个根本概念。显然,它在几何学中很重要，但它在拓扑中也是众所周知的，其中类似的伸缩和被称为任何有限空间的欧拉特征χ。还可以计算单位多面体(例如单位单元，非对称单元或Dirichlet域)构建，以对称的方式，所有空间群中的无限晶格。在本申请中,χ具有修改的形式(χm)和值,因为加数必须根据它们的对称性来加权。尽管是从几何学（实际上是从晶体学中得出的），χm通过双模的概念具有优雅的拓扑解释。或者，χm可以用哈里奥和笛卡尔的定理来说明，早于欧拉的发现。那些历史定理,专注于多面体的角缺陷，在deGuadeMalves的公式中表达精美。在更一般的解释中，Gauss-Bonnet定理将欧拉特性与任何封闭空间的一般曲率联系起来。本文以欧拉公式为主题，概述了数学的这些有趣方面。最后,一个游戏的设计，允许读者以娱乐的方式吸收欧拉特征的概念。
