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.
Whoa!
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* 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.
Surely you've seen the point group I-sub-h?
Well, if you keep chopping, you can expand those pentagonal faces until they meet as a dodecahedron. This in turn has 12 pentagonal faces, three of which meet at each of 20 vertices.
If you then cut away at those 20 vertices of the dodecahedron, expanding the triangular cross-sections until those meet, you get another icosahedron!
Note that the cube and octohedron have the same relationship, while the tetrahedron "cuts away" to another tetrahedron.
Actually, the balls being used this year for the World Cup are not polyhedral, having incorporated a curved base-ball like substructure. Also, shame on you :):) for not mentioning fullerene, the soccer-ball carbon molecule
p.s more details at
http://geomblog.blogspot.com/2006/06/breaking-news-soccer-balls-no-long…
I'll confess that the point-group part of my brain and the plane-geometry part of my brain are not always in close communication. (There were harsh words; we're working past it.) My mental images with point-groups are almost all of the models you build with the kits that contain multicolored plastic straw thingies that fit onto the metal centers that look like jacks -- not so much visions of Platonic solids.
Of course fullerene and soccer balls are kin; but remember that organic chemistry is what nearly convinced me to be a physics major, so I'm not as quick on the buckyball draw as others are.
I knew the balls in the World Cup were more spherical (and that there was speculation that this would make it harder for the players to judge the paths of incoming balls, etc.). My soccer ball is still old school.
And Dave, if you keep chopping as described, what game do you end up playing with what's left?
You're not alone in failing to recognise the symmetry of the truncated icosahedron that is an old-style soccer ball (actually just a football where I am! England) and the connection with fullerene symmetry. Indeed, when Harry Kroto and the late Rick Smalley were trying to figure out the structure of the new carbon allotrope they'd discovered, it took them quite some time to realise that you couldn't close a sphere with hexagons alone (apparently).
Incidentally, fullerenes are pure carbon, so in one sense they're not organic at all.