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 3-D structures that approach a sphere with an increasing number of smaller and smaller polygons.   And I'm comparing it to the 2-D analogy of making a closed regular polygon of increasing number of sides.

2-D:  the sum of the degrees of all the angles in an N-gon is 180*(N-2), 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.

3-D:   picture the 3-D 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 3-D 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 2004-04-09 19:34:19