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.