A football is a truncated icosahedron

2026-06-09

A traditional football is stitched from 12 pentagons and 20 hexagons. That is not a designer's choice - it is forced geometry. Take an icosahedron, a solid with 12 sharp corners where five triangles meet, and slice every corner off. Each corner becomes a pentagon (five edges arrived there), and each triangular face, with its three corners now cut, opens into a hexagon. Twelve pentagons, twenty hexagons. The football.

It has another famous name. In 1985 chemists found a molecule of 60 carbon atoms sitting at exactly these 60 corners - buckminsterfullerene, C₆₀, the "buckyball". The shape that makes a good football also makes a stable molecule, and it is the same shape for the same reason: it is the truncated icosahedron.

This is the other dual pair, sliding the way the cube and octahedron do. Drag from the dodecahedron to the icosahedron, or tap a named stop. The football is the truncated icosahedron stop, three-quarters of the way along.

The stops along the way

Start at the dodecahedron - 12 pentagons, 20 corners. Begin shaving its corners and each pentagon grows into a ten-sided face while triangles appear at the corners: the truncated dodecahedron. Keep going to the halfway point and the cuts reach the edge midpoints, giving the icosidodecahedron - 12 pentagons and 20 triangles, the exact balance of the two solids, just as the cuboctahedron balances the cube and octahedron.

Past halfway you are now shaving the icosahedron's corners instead. A little way in, the cut is just right: regular pentagons at the 12 corners, regular hexagons from the 20 faces. That is the football. Finish the slice and the pentagons close to points, leaving the bare icosahedron.

Why pentagons have to be there

You cannot tile a ball out of hexagons alone - hexagons tile a flat floor, and a floor has no curvature to wrap into a sphere. To close a surface you need some faces with fewer sides to pull in the curvature, and the bookkeeping (the same Euler count that limits the Platonic solids) says you need exactly twelve pentagons, no matter how many hexagons you add. A football has 20 hexagons; a bigger buckyball can have hundreds - but always 12 pentagons. They are the 12 corners of the icosahedron hiding inside, and they never go away.

Cut the 12 corners off an icosahedron and you hold a football - and the most famous molecule in chemistry. Twelve pentagons to bend it round, twenty hexagons to fill it in.