The football (soccer ball) is a unique-looking piece of sporting equipment. Instantly recognizable even to someone who don't know what a yellow card is, the football itself is a black-and-white pattern of geometric shapes inflated into a sphere. Upon closer inspection, one can see it's a mixture of white hexagons surrounding black pentagons. Why not just use all the same shape? Ask Plato.

The Greek philosopher Plato gave his name to the platonic solids, familiar to gamers as the standard set of gaming dice. They are the only three-dimensional shapes that have sides made of regular polygons, and are therefore useful as "fair" dice, that is, dice that have an equal chance of landing on any side. These shapes are the 4-sided tetrahedron (triangles), the 6-sided cube (squares), the 8-sided octahedron (triangles), the 12-sided dodecahedron (pentagons), and the 20-sided icosahedron (triangles). Hexagons, sadly, cannot form a platonic solid, and 10-sided dice are not platonic solids.

While it might make sense to build a football out of a dodecahedron, the more sides the polyhedron has, the closer it represents a true sphere and the less strain there is on the stitching and leather when inflated. A football is actually an icosahedron. Granted, it doesn't look like it's made of triangles, but that's because the triangles have been combined into groups of five and six: pentagons and hexagons.

If you unfold an icosahedron, the twenty triangles that comprise it look like this:

/\      /\      /\      /\      /\
/  \    /  \    /  \    /  \    /  \
/    \  /    \  /    \  /    \  /    \     unfolded
/______\/______\/______\/______\/______\    icosahedron
\      /\      /\      /\      /\      /\   20 equilateral
\    /  \    /  \    /  \    /  \    /  \  triangles
\  /    \  /    \  /    \  /    \  /    \
\/______\/______\/______\/______\/______\
\      /\      /\      /\      /\      /
\    /  \    /  \    /  \    /  \    /
\  /    \  /    \  /    \  /    \  /
\/      \/      \/      \/      \/

Notice that each vertex of each triangle, when folded up, is connected to exactly five triangles. This is obvious in the middle rows. The top and bottom rows are made of five triangles each, so when they are folded together, these tips will join five triangles together.

If we split each of these triangles up into 9 triangles, we get the following:

/\
/__\
/\  /\   triangle split
/__\/__\  into more triangles
/\  /\  /\
/__\/__\/__\

Now erase the six lines in the middle to combine those six triangles into a hexagon:

/\
/__\
/    \   hexagon inscribed
/      \  in a triangle
/\      /\
/__\____/__\

This leaves three small triangles at the corners. Since each corner connects five triangles, when the icosahedron is folded up, these triangles will form pentagons. It's interesting to note that the hexagons and pentagons are therefore all made of equally sized triangles, which is why the pentagons are slightly smaller.

There are twenty triangles, so there are twenty hexagons around a football. It's a bit trickier to count the pentagons, but there's one at the top, one at the bottom, and ten across the middle (two rows of five, and remember the left and right sides join to make two, not four, pentagons) for a total of twelve pentagons. Twenty hexagons plus twelve pentagons is thirty-two total panels.

Paint the pentagons black, and the hexagons white, and you've got a football!