The Euler characteristic as a basis for teaching topology concepts to crystallographers

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 chi of any finite space. The value of chi 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 chi has a modified form (chi(m)) and value because the addends have to be weighted according to their symmetry. Although derived in geometry (in fact in crystallography), chi(m) has an elegant topological interpretation through the concept of orbifolds. Alternatively, chi(m) 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.

Automated summary

The Euler polyhedral formula, V - E + F = 2, is a fundamental concept in mathematics with applications in geometry and topology. The Euler characteristic χ can be computed for symmetric unit polyhedra and has a topological interpretation. The Gauss-Bonnet theorem links the Euler characteristic with the general curvature of closed spaces.