I’ve always liked soccer balls — and not just because you can play soccer with them. The arrangement of pentagons and hexagons to form a surfaces that’s reasonable spherical always seemed outstandingly clever. Who was the genius who first realized you could do that?
Well, my world has been rocked. I still think the soccer ball is clever, but in and entirely different kind of way. Tonight I was flipping through the copy of American Scientist that arrived with today’s mail, looking at the pictures*, and I came across this article on the topology (and combinatorics) of soccer balls. (The full article online requires a subscription.) Figure 8 revealed that a soccer ball is nothing but an icosohedron (that is, a Platonic solid with 20 faces, each of which is an equilateral triangle), each of whose 12 vertices is truncated. Chopping off each vertex leave a pentagonal face (so chopping off 12 vertices yields 12 pentagonal faces). And, each of these truncations chops off the corners of the faces that met to make the vertices you chopped off — turning the 20 equilateral triangles into 20 regular hexagons.
* I know some people claim to get American Scientist for the articles, I freely admit that the first night it’s in the house, I just look at the pictures.