Fun fact, relevant to said XKCD: while it's not possible to construct a 3-dimensional convex polyhedron using only congruent, regular hexagons as faces, as the soccer ball example shows, it is possible to do so using a combination of congruent, regular hexagons and congruent, regular pentagons.
That's obviously not news, so, here's the fun part: such a polyhedron must always have exactly 12 pentagons. The proof of this isn't hard, and is based on Euler's formula, V - E + F = 2. See https://math.stackexchange.com/a/18347 for details.
That's obviously not news, so, here's the fun part: such a polyhedron must always have exactly 12 pentagons. The proof of this isn't hard, and is based on Euler's formula, V - E + F = 2. See https://math.stackexchange.com/a/18347 for details.