# World Cup 2022: geometry and speed of the tournament ball

https://www.lemonde.fr/sciences/article/2022/12/07/coupe-du-monde-2022-geometrie-et-vitesse-du-ballon-du-tournoi_6153349_1650684.html

Millions of viewers are watching the 2022 World Cup matches, but how many have actually watched the ball? It is called Al Rihla, which means “the journey” in Arabic. Mathematicians prefer to call it an icosidodecahedron… from the Greek ico for “twenty”, dode for “twelve” and èdre for “face”. Each Cup is the occasion of a new geometry. In Mexico in 1970, the ball was called Telstar. We all know it with its white and black leather panels.

I asked children from 6 to 12 years old to draw a Telstar ball that was facing them. It’s not easy and some of the drawings are… imaginative. The twelve black pieces are pentagons and the twenty white pieces are hexagons.

Since Greek antiquity, five regular polyhedra have been known, all of whose faces are identical: they play an important role in Plato’s philosophy, each being associated with an “element”. The one that is the most “round” is the icosahedron, with its twenty faces in the shape of equilateral triangles. To make it even rounder, we truncate its twelve vertices, we inflate the whole, and we obtain… the Telstar.

Archimedes, for his part, looked for semi-regular polyhedra, whose faces are still regular polygons, but not necessarily with the same number of sides. He found thirteen, including our icosidodecahedron, with twelve pentagons and twenty triangles. The Adidas engineers simply sewed each of the pentagons with a triangle to obtain Al Rihla. Beyond the aesthetics, it is the symmetries of the object that interest the mathematician and we no longer count the appearances of the icosahedron in contemporary mathematics.

Study of the resistance

The engineer has many other concerns than aesthetics, even if he must not forget it. Symmetries are also important to prevent the balloon from going in uncontrolled directions. The physics of soccer ball flight is complex and requires theoretical and experimental studies.

One of the pioneers was Gustave Eiffel, who was of course more interested in early aviation than in soccer. He began by observing the fall of balls of various sizes from the second floor of “his” tower, before continuing his research in one of the first wind tunnels. In 1912, he discovered a phenomenon he did not believe in at first, which is now called the drag crisis.

When a ball flies, the air exerts a resistance that tends to slow it down. It seems obvious that this force is weaker the lower the speed. Yet, when a ball slows down gradually and a certain speed is reached, there is suddenly a significant increase in resistance. This critical speed depends on the size of the balloon but also on the roughness of its surface. For a Telstar or Al Rihla ball, it is about 10 meters per second. When a player strikes the ball, the initial speed is often much higher than this value before gradually decreasing.

When the ball reaches the critical speed, the resistance suddenly increases and the trajectory seems to break: a phenomenon that goalkeepers know well. For smoother balls, such as the Jabulani of the World Cup in South Africa, the crisis occurs around 14 meters per second: this is one of the reasons why players did not like this ball at all. The Brazilian goalkeeper Julio Cesar said: “It’s terrible, horrible, it looks like one of those balls you buy in the supermarket.” According to striker Robinho: “The guy who designed this ball has certainly never played soccer.”

What will the Blues think of Al Rihla? Their opinion will prevail over that of the physicists.