Soccer balls are usually covered with a design based on regular pentagons and hexagons.

How many pentagons/hexagons MUST there be, and why?

(In reply to re(2): The 720 degree deficit by Tristan)

I noticed a minor mistake in my thinking there, but still reach the same conclusion.

I was picturing a solid with which each vertex connected two pentagons and a hexagon.  A hexagon would be surrounded by six pentagons.  Then there would be a jagged edge that goes up and down around in a circle.  The vertices of the concave angles on the jagged edge would already each connect two pentagons.  Therefore, the last must be a hexagon.  The problem here is that the convex angles on the jagged edge now each connect two hexagons and a pentagon, which means this theoretical solid can't exist.

Posted by Tristan on 2004-04-13 16:36:27