Soccer balls are usually covered with a design based on regular pentagons and hexagons.
How many pentagons/hexagons MUST there be, and why?
I don't really understand what the question is asking: how many MUST there be?
I'll just add my thoughts to the fray.
The plane may be tiled with regular hexagons. They meet three to a corner at 120 degree angles.
If you were to replace some of the hexagons with pentagons but keep the sides joined, the figure would no longer remain flat. The way the hexagons are replaced is one at each vertex and no pentagons sharing an edge.
Each hexagon the borders 3 pentagons and each pentagon borders 5 hexagons. The ratio of hexagons to pentagons, then is 5:3.
Why does the soccer ball (truncated icosahedron) have 20 hexagons and 12 petagons?
Besides conforming to the above ratio and resultion fron the trucation of the 12 verticies of a 20 sided figure, I'd say one could look into the dihedral angles (but I'm not gonna).
One could ask:
Of all the polyhedra to choose from to make a ball, why this one?
Lots of symmetry (all the arcimedian solids do)
All sides nearly the same size (no dodecagons and triangles together, etc)
Very close to a sphere (there are better, but it is really close)
Jer

Posted by Jer
on 20040409 09:37:29 