Dillions of viewers watch the 2022 World Cup matches, but how many actually watched the ball? His name is Al Rihla, which means “the journey” in Arabic. Mathematicians prefer to evoke an icosidodecahedron… from the Greek ico for “twenty”, dodded for “twelve” and edre for “face”. Each Cup is an opportunity for 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 aged 6 to 12 to draw a Telstar balloon facing them. It’s not easy and some designs 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 most “round” is the icosahedron, with its twenty faces in the form of equilateral triangles. To make it even rounder, we truncate its twelve vertices, we inflate the whole thing, and we get… the Telstar.
Archimedes, meanwhile, looked for semi-regular polyhedra, whose faces are still regular polygons, but not necessarily with the same number of sides. He counted thirteen of them, including our icosidodecahedron, with twelve pentagons and twenty triangles. The Adidas engineers contented themselves with stitching each of the pentagons with a triangle to obtain Al Rihla. Beyond 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.
Resistance study
The engineer has many other concerns than aesthetics, even if he must not forget that. Symmetries are also important to prevent the ball from flying in uncontrolled directions. The physics of soccer ball flight is complex and requires both theoretical and experimental studies.
One of the pioneers was Gustave Eiffel, of course more interested in early aviation than football. 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 in which he did not believe at first, which is now called the drag crisis.
When a ball flies, the air exerts a resistance which tends to slow it down. It seems obvious that this force is all the weaker as the speed is small. And yet, when a ball gradually slows down and a certain speed is reached, we suddenly observe 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 balloon, it is around 10 meters per second. When a player hits the ball, his initial speed is often much higher than this value before gradually decreasing.
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