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.
Highlights
Topology, broadly defined as the study of certain properties of geometric figures that do not change as these figures or spaces undergo continuous deformation, is a relatively young branch of mathematics, developed as a distinct field by Henri Poincare at the end of the 19th century
Topologists usually associate the foundation of their discipline with Leonhard Euler, whose famous formula relating the numbers of vertices, edges and faces of any three-dimensional polyhedron, VÀEþF 1⁄4 2, is of fundamental importance in topology
It was the consideration of the crystallographic unit cell and its minimal asymmetric part, the asymmetric unit (ASU), that some time ago made us realize that the Euler’s formula for such ‘incompletely bounded’ figures will be different, yielding a sum that is smaller by 1 (Dauter & Jaskolski, 2020)
Summary
Broadly defined as the study of certain properties of geometric figures (or spaces) that do not change as these figures or spaces undergo continuous deformation, is a relatively young branch of mathematics, developed as a distinct field by Henri Poincare (see the biographical notes in Appendix A) at the end of the 19th century. It was the consideration of the crystallographic unit cell (as an object sharing its bounding elements, or k-cells, with its neighbors) and its minimal asymmetric part, the asymmetric unit (ASU), that some time ago made us realize that the Euler’s formula for such ‘incompletely bounded’ figures will be different, yielding a sum that is smaller by 1 (Dauter & Jaskolski, 2020). We have designed a game ‘Let’s compute Euler’s number’ which offers an entertaining way of absorbing the concept of the Euler characteristic
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