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

Journal Article · 31 January 2022 · Journal of Applied Crystallography (Online)

DOI: https://doi.org/10.1107/s160057672101205x · OSTI ID:2470008

Naskręcki, Bartosz  ^[1]; Jaskolski, Mariusz  ^[2]; Dauter, Zbigniew  ^[3]

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1. Adam Mickiewicz Univ., Poznan (Poland)
2. Adam Mickiewicz Univ., Poznan (Poland); Polish Academy of Sciences (PAS), Poznan (Poland)
3. Argonne National Laboratory (ANL), Argonne, IL (United States)

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 (χ_m) and value because the addends have to be weighted according to their symmetry. Although derived in geometry (in fact in crystallography), χ_m has an elegant topological interpretation through the concept of orbifolds. Alternatively, χ_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.

• Research Organization: Argonne National Laboratory (ANL), Argonne, IL (United States)
• Sponsoring Organization: USDOE
• Grant/Contract Number: AC02-06CH11357
• OSTI ID: 2470008
• Journal Information: Journal of Applied Crystallography (Online), Journal Name: Journal of Applied Crystallography (Online) Journal Issue: 1 Vol. 55; ISSN 1600-5767
• Publisher: International Union of CrystallographyCopyright Statement
• Country of Publication: United States
• Language: English