Football: mirror icosahedron

The surface of a classic soccer ball is composed of 12 slightly curved black regular pentagons and 20 white regular hexa­gons.

By the way, such a ball was not always consid­ered ”classic”: this cut and colouring were first used for the offi­cial world cup ball in 1970 in Mexico. The black-and-white colouring was then chosen from сontrast consid­er­a­tions – so the ball was more visible on then common black-and-white TVs. It was even named Telstar — after a TV satel­lite. In the years to come the offi­cial balls changed their colour­ings, but the cut remained unchanged until the 2002 cham­pi­onship in Germany.

From a math­e­mat­ical point of view, a classic soccer ball is a trun­cated icosa­he­dron. This fact and the theory of reflec­tion groups (in three-dimen­sional case — of Coxeter groups) allows one to make a simple yet beau­tiful model.

One should take a trihe­dral angle composed of same isosceles trian­gles. Given the base length $a$, the length of legs that are glued together to form the trihe­dral angle should be $r=\frac{1}{4}\sqrt{2(5+\sqrt{5})}\,a$ which with a good preci­sion is $r\approx0{.}95\,a$. (For example, if $a=10$ cm, then $r=9{.}5$ cm.) The mirror angle is very close to that of a regular tetra­he­dron, but yet differs.

Another impor­tant detail is a (plane) regular triangle coloured black-and-white in such a way that the white inte­rior is a regular hexagon. (To achieve this, the sides of black trian­gles should be taken three times less than the side of orig­inal regular triangle.)

If such a triangle is now put in the trihe­dral angle, a model of a classic soccer ball is seen inside! The image won't change if the angle is moved around the line of sight.

For the ”ball” to be seen completely, the triangle put in shouldn't be too large. One should not put it further than a third of the mirror angle's height from the vertex. (That is, with the base of the mirror triangle being $a=10$ cm, the side of triangle to put in can be taken $3$ cm, so the sides of small black trian­gles on it — $1$ cm.)

The simplest way to make the isosceles mirror trian­gles is to cut them out of plastic with mirror coating. They can be put together with duct tape or with wide elec­trical tape, gluing the legs of trian­gles — the trihe­dral angle's edges.

What kind of magic mirror angle is it that the mirror image forms a soccer ball? (In fact — an icosa­he­dron, which is even more clearly visible if one puts in a solid color triangle.)

The mirror angle is asso­ci­ated with the icosa­he­dron itself: its vertex is in the icosa­he­dron's center, and the mirrors cross the sides of one of its edges. That is where the condi­tions on the sides of isosceles trian­gles forming the mirror angle come from: if the triangle's base $a$ is the length of icosa­he­dron's edge, then the leg $r$ is the radius of its circum­scribed sphere.

And the fact that the image in this mirror triangle is an icosa­he­dron is guar­an­teed by the theory of reflec­tion groups.