Question 20.43
13. If you “unbulge” an ordinary soccer ball so that each of its sewn pieces is flat, you get a polyhedron, but it is not a regular polyhedron. It is a truncated icosahedron, one of the semiregular polyhedra. Some of the faces are regular hexagons, and some are regular pentagons. Explain why not all of its faces can be regular hexagons. All the vertices have the same vertex type. What is it? How many pentagons are there and how many hexagons? How many vertices?
13.
Three hexagonal faces meeting at at point would form a solid angle of , hence would form a flat surface. The vertex type on an ordinary soccer ball is 5.6.6. The ball has 32 faces (of which 12 are pentagons and 20 are hexagons), 60 vertices, and 92 edges.