Symmetries of a Soccer Ball

Question

The FIFA World Cup 2018 in Russia is played with an Adidas ball, called Telstar 18. It consists of 6 square panels and 12 panels with symmetry of a parallelepiped, but with curved sides. The ball has the octahedral symmetry group without reflections. This means that there exist (in theory) both left hand and right hand balls.

Questions:

1. How can I find (3,3) matrices, implementing this symmetry group?
2. Opposite squares are parallel but twisted a few degrees. Is the angle of the twisting a parameter in the matrices, so that this type of ball can be constructed with other angles and other forms of the four side panels?

Answer 1

Having looked at a Telstar 18 soccer ball for some hours, I will answer my own question.

The 12 larger panels have a slight resemblance of a parallelogram (NOT parallelepiped!!!), although their sides are curved and its angles are 90 and 120 degrees.

Let the soccer ball with radius 1 has its center of gravity in the origin, and let the squares be perpendicular against the x, y, and z axis, respectively, with their centers of gravity on the same axis. The symmetry group of the octahedron without reflexions is generated redundantly by the following transformations of the football on itself:

A Rotating 90 degrees around the x axis

B Rotating 90 degrees around the y axis

C Rotating 90 degrees around the z axis

D Rotating 120 degrees around one diagonal of an inscribed cube

The last lines go through the origin and one point (+/-1, +/-1, +/-1). The points of the symmetry axises of the rotations D are visible on the ball as a node between three panels, the borders of which meet in an angle of 120 degrees. It is easy to give the matrices of these transformations.

The ball has two parameters: The size of the squares, depending on distance of the planes from the origin, and the slope of the sides of one square. The last parameter gives the skewness of the ball, but reflexions are not included in the symmetry group anyhow, due to the placement of the four side panels. Given these parameters, the long side of a large panel must go through two corners of a square and a corner of the inscribed cube on the sphere, and this side is well defined by these conditions.

Adidas was free to choose these parameters. If the squares are too small, then the larger panels would become too wide and would not give a good approximation of the double curved surface of the sphere. It the squares are larger, the four side panels would become longer and more narrow and easier to bend, but the squares would give worse approximations of the sphere.

If there squares should grow still more, we could get an Archimedean solid, the snub cube, with the same symmetry group. This solid consists of 6 squares and 32 equilateral triangles with the same side, see Wikipedia. However, Telstar has only 18 panels and its design is new and far more elegant.