This question from Mark Saul <73047.3156@CompuServe.COM> came to me via Dick Askey and Dennis Stanton: > Speaking of mathematicians' help, can anyone tell me WHY the > soccer ball is constructed as it is? Why, for example, is it not > made of spherical triangles, sewn together to make an icosahedron? > Is the soccer ball the answer to some well-known minimal problem?: > Does it minimize stitching? or deviation from a sphere? or stresss > when stuffed? Or is it just pretty?
I'm not sure if there's any nice problem for which the soccer ball is minimal. One nice unsolved problem for polyhedra is to minimize the total edge length, with fixed enclosed volume. But it seems the solution to this one is probably a triangular pyramid. This probably corresponds to the minimal stitching question.
As a wild guess about soccer balls, I'd point out that this is the largest polyhedron with three-fold vertices. Perhaps it is harder to get something to stay together when there are more than three faces meeting at one point. So with this restriction, the soccer ball is the closest you can get to a sphere among the Archimedean solids.
Maybe someone else will have further ideas. Of course, it _is_ pretty.