soccer balls, pentagons, hexagons

Factual Questions
1

My daughter left her soccer ball on the floor of the bathroom last night, and as a result I was able to study the stitch pattern as I performed the morning ritual on the throne.

I noticed a pattern around each hexagonal patch of alternating equivalent hexagons and equilateral pentagons.

Is it mathematically assured that if I perfectly draw hexagons and pentagons on a sphere in the above pattern with equilateral sides, that I will “close” perfectly? Or is some fudging required to make a soccer ball?

2

No fudging required - this pattern also exists in nature in the form of Carbon-60: the buckminsterfullerene.

3

The mathematical solution is twelve pentagons and twenty hexagons. The proof is given here - Solution to the polyhedron football problem in issue 4.2.

I’ve heard that bees use the same kindof pattern for their hives.

4

It’s called a truncated icosahedron by some, and you can see how closely it resembles a soccerball here.