30/11/2022

In Folha, Viana talks about the mathematics of the World Cup ball.

Reproduction of Marcelo Viana's column in Folha de S. Paulo.

Marcelo Duarte wrote a beautiful column about the technological evolution of the World Cup ball. My interest in this story lies in its mathematical aspect. In the early days, soccer balls were made of leather , and the basic question was how to produce a round object from a material that is essentially flat. The same problem occurs in other situations. In haute couture: how to use flat fabrics to elegantly cover the curves of the human body? In cartography: what is the best way to represent the globe (round!) using flat maps? Important advances in mathematics have resulted from the study of these practical questions.

In theory, the solution would be to manufacture the ball with a large number of small pieces of leather, forming a reasonably "round" polyhedron (geometric solid). There are many ways to do this, but all must obey Euler 's formula : F – A + V = 2, where F is the number of faces (pieces of leather), A is the number of edges (seams), and V is the number of vertices (where the seams meet). In practice, a balance must be found, as sewing a ball with many faces is complicated.

The balls used in the early World Cups had irregularly shaped faces. The 1970 Mexico World Cup introduced a very elegant concept, which was used until 2002 and returned in 2010 and 2018: the Telstar has the shape of a truncated icosahedron, with 32 faces, 20 hexagons and 12 pentagons (check Euler's formula!). Originally, the hexagons were painted white and the pentagons black, and I believe that for many of us this is still the paradigm of what a soccer ball is.

Advances in materials engineering have freed us from the limitations of leather, allowing us to experiment with other concepts. The Brazuca, the ball used in the 2014 World Cup , holds the record for the fewest faces, only 6. In fact, as mathematicians pointed out at the time, the Brazuca is actually a cube (with rounded faces)!

To read the full text, visit the newspaper's website.