Soccer balls are usually covered with a design based on regular pentagons and hexagons.
How many pentagons/hexagons MUST there be, and why?
I'm picturing 3D structures that approach a sphere with an increasing number of smaller and smaller polygons. And I'm comparing it to the 2D analogy of making a closed regular polygon of increasing number of sides.
2D: the sum of the degrees of all the angles in an Ngon is 180*(N2), or 180N  360. Picture radii drawn from each vertex to the center and you have N isoceles triangles. Of course the N tiny angles in the center have to sum to 360.
3D: picture the 3D structure formed by connecting the sides of one of the faces to the center of the "soccer ball", and you have a sort of pyramid. Sum up all of the tiny 3D center angles of all the pyramids and you have a solid angle that is the same as the sum of the surface areas of all the outer faces. I seem to remember that the solid angles are measured in steradians, and that 4 pi steradians makes a sphere. So there is your 720 degrees. Not a proof, but ....

Posted by Larry
on 20040409 19:34:19 