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!