Every football has twelve pentagons, and it could not have any other number
A football, a carbon-60 molecule and a virus shell have nothing obvious in common. All three are solving the same constraint, and it forces the same number every time.
Pick up a traditional football. It is made of pentagons and hexagons, with three panels meeting at every corner. Count the pentagons and you will find twelve.
Now look at buckminsterfullerene, the C₆₀ molecule discovered in 1985: sixty carbon atoms arranged in pentagons and hexagons, three bonds at every atom. Twelve pentagons.
Look at a geodesic dome. Its panels are triangles rather than pentagons, so the count shows up in the other place it can: among all those six-way junctions there are exactly twelve where only five triangles meet. Same twelve, wearing a different hat.
It is forced, and here is the forcing.
The four lines
Let the cage have V corners, E edges and F faces, and let every face be a pentagon or a hexagon, so that F = F₅ + F₆ with nothing else allowed. (Admit a square or a heptagon and the answer changes; that is not a loophole, it is the hypothesis doing its job.)
Three faces meet at every corner, and every edge is shared by exactly two faces. Counting edge-ends two different ways gives 3V = 2E. Counting face-edges the same way gives 2E = 5F₅ + 6F₆.
Now apply Euler's formula, V − E + F = 2, which holds for any convex polyhedron. Substitute, clear the fractions, and everything cancels except one term:
F₅ = 12
The number of hexagons has vanished entirely. It does not appear in the answer, which means it cannot affect the answer. You may use twenty hexagons or twenty thousand; you will use twelve pentagons.
The same fact, told as curvature
There is a second way to see it that explains why the hexagons drop out.
A cube is flat everywhere except at eight points. Walk across a face and nothing curves. Walk around a corner and you find 270° of material where a flat surface would have offered 360°. The missing 90° is the curvature, and on a polyhedron there is nowhere else for curvature to be: it is concentrated entirely at the vertices.
Add up that deficit over all eight corners of a cube: 8 × 90° = 720°. Do it for an icosahedron: 12 × 60° = 720°. For a dodecahedron: 20 × 36° = 720°. The total never moves. It is 720° for every convex polyhedron that exists, regular or not, which Descartes noticed around 1630 and which is the Gauss–Bonnet theorem with the curvature squeezed into points instead of spread smoothly over a surface.
Now: three regular hexagons at a corner come to exactly 3 × 120° = 360°. They lie flat, and contribute nothing to the 720° that has to be there.
On a football every corner is two hexagons and one pentagon: 120° + 120° + 108° = 348°, a deficit of 12°. There are sixty such corners — one per carbon atom, in the C₆₀ version — and 60 × 12° = 720°, exactly. The pentagons are carrying the entire curvature of the object and the hexagons are along for the ride, which is why you can add as many hexagons as you like and the pentagon count will not move.
You can watch the total hold still in the Regular Tilings Lab: switch between the five Platonic solids and the per-vertex defect changes every time while the product with the vertex count does not.
Where the constraint bites
In 1962 Donald Caspar and Aaron Klug asked why so many viruses are icosahedral. The pressure on a virus is brutal: its genome is tiny, so it can encode very few distinct proteins, and it must build a closed container out of them. The cheapest possible container is one protein repeated many times.
But a closed shell assembled from identical repeated units has very few options, and for a related reason: you cannot tile a closed surface with a single repeated unit in identical surroundings more than sixty times. Caspar and Klug's answer was the triangulation numbers — a family of icosahedral arrangements in which subunits sit in nearly equivalent positions rather than identical ones. The bookkeeping differs from the football's (their lattices are triangulated, and the twelve special sites are five-fold vertices rather than pentagonal panels) but the pressure is the same one, and it is why an electron micrograph of an adenovirus looks like a twenty-sided die. The virus is not imitating a Platonic solid. It is running out of alternatives.
The same constraint is why Buckminster Fuller's domes need their pentagons, why C₆₀ is stable at exactly sixty atoms, and why a chemist looking at a proposed fullerene structure can reject it at a glance if it has thirteen pentagons.
What this is an example of
The useful generalisation is not about pentagons. It is that some properties of an object are determined by so little information that you can read them off without inspecting the object.
You do not need to see a football to know it has twelve pentagons. You need to know that it is closed, that its faces are pentagons and hexagons, and that three meet at each corner. Everything else — the size, the material, the number of hexagons, whether it is inflated — is irrelevant to that particular question.
Quantities like this survive deformation and ignore measurement error, which is why they are the ones worth learning.