Twelve, and no way to have eleven

The whole argument

Walk round a hub of a flat triangular grid and the six angles you pass add to 360°. Take one triangle away and only five are left: the sheet no longer lies flat, and closing the gap pulls it into a shallow cone. The 60° you removed is the curvature of that hub — its angular defect. Descartes' theorem says that for any closed convex polyhedron, however lumpy, the defects of all its vertices add to exactly 720°, which is the sphere's total curvature and nothing else. So if the flat hubs cost nothing and each pinched hub costs 60°:

720° ÷ 60° = 12

There is no arrangement of hexagons alone that closes. You may have twelve pentagons, or six pentagons and something worse, or four squares (that is a cube, and its defect is 8 × 90°), but you may not have eleven pentagons and a dome. The number is fixed before the design starts; only the positions are yours.

A dome, at any size you like

Below is a real one, built here from an icosahedron and a triangular grid. The two numbers h and k say how far you walk across that grid to get from one corner of an icosahedron face to the next: h steps one way, then k steps at sixty degrees. Everything else follows, including how many parts there are. The picture shows the panels — the dual of the strut grid, which is what a football is sewn from. The twelve are lit.

Count the lit panels at any setting on that slider and there are · of them. The hexagons go from 0 to hundreds; the pentagons do not move. What the grid does change is T = h² + hk + k², the number of grid triangles per icosahedron face, and every part count on the dome is that number times a constant: ·. Those three are not measured off the picture — they are counted off the built mesh, and Euler's V − E + F comes back · every time.

The curvature, weighed

The claim above is worth more if the 720° is measured rather than asserted, so it is: every face angle of the built mesh is computed, summed at each hub, subtracted from 360°, and the lot added up.

The last column is the interesting one and it does not move: 720.000000° for a twelve-strut icosahedron and for a dome with a thousand hubs alike. The distribution is what changes. At 1V the twelve hubs carry the whole of it, 60° each. By 6V a pentagon hub is down to · and the twelve of them hold only · of the curvature — the rest has leaked out into hexagon hubs that are no longer quite flat. The pentagons stop being where the curvature is and go on being where the curvature has to start.

A table printed before this page existed

A dome is sold as a bag of struts of a few lengths, and the lengths are published as chord factors: multiply by the radius and cut. They have been in print since the 1960s. Nothing on this page has ever seen that table — the numbers below come out of the mesh above, as the distances between adjacent hubs on a unit sphere.

Eleven chord factors over three domes, to the five decimals the tables print. One of them is worth stopping on. The long strut of a 2V dome — the one that runs between two six-way hubs — is not merely close to 1/φ, the reciprocal of the golden ratio. It is 1/φ, and the only reason the last digit below differs is that the page computes it through a square root and a projection in double precision:

That is the icosahedron's own edge geometry surviving one subdivision, and it means a builder cutting 0.61803 R is cutting a number the Greeks would have recognised. The identity is exact — the test file next door proves it again in fifty digits, where the last bit has nowhere to hide.

The short strut has no such pedigree, and it is instructive to see what it is instead. It runs from an icosahedron vertex to the midpoint of an edge after that midpoint has been pushed out to the sphere, so if θ is the angle an icosahedron subtends between two neighbouring vertices — cos θ = 1/√5 — the strut is 2 sin(θ/4). That is · against the table's 0.54653: a nested radical in √5, with a quarter-angle in it because the projection halved the arc and then the chord halved it again. Both struts come out of the same solid; only one of them has a name.

The football, and the molecule with the same shape

Set h = 1, k = 1 above. The panel count is 32 — twelve pentagons and twenty hexagons — which is a football, and its dual is a cage of 60 corners, which is C₆₀. The geometry says something sharper than the shape, though: it says the ninety bonds come in exactly two kinds, and how many of each.

Sixty bonds run between a pentagon and a hexagon; thirty run between two hexagons. That is a count, not a fit, and it is the count chemistry measures. What chemistry does not agree with is the length. On the ideal sphere the thirty are the · longer bonds. In the real molecule they are the shorter ones — 1.40 Å against 1.46 Å — because the hexagon–hexagon bonds carry the double-bond character and pull in. The shape gets the combinatorics exactly right and the metric exactly backwards, which is worth more as a lesson than another agreement would have been.

Which numbers are allowed

Not every dome exists. T = h² + hk + k² with whole h and k cannot make every integer — it makes · of the first hundred, and the gaps are hard: there is no T = 2, no 5, no 6, no 8. The list is the Loeschian numbers, and the reason for the gaps is a theorem about which numbers are norms in the ring of Eisenstein integers, which is a long way from roofing.

This is testable outside the room, because viruses use the same grid. A capsid is protein subunits arranged with icosahedral symmetry, and Caspar and Klug's 1962 observation is that they sit at exactly these positions, 60T subunits to a shell, with twelve pentamers wherever T goes. Every capsid number anyone has measured is on the list above:

Two of those rows are worth a second look. T = 169 can be built two ways — as the plain (13,0) and as the chiral (8,7) — because 169 is a square of a number that is itself a sum of this form, and the biggest capsid in the table is exactly where that ambiguity first bites. And one of them carries a handedness. When h ≠ k and neither is zero the grid is chiral — (2,1) and (1,2) are mirror images of each other and no rotation takes one to the other, which this page checks by mirroring the vertex set of one and looking for the other. T = 7 is the first such number, and the phage HK97 is T = 7 laevo: a virus with a fixed handedness because there was no achiral option at that size. T = 9, by contrast, is (3,0), and is its own mirror image.

Where you are allowed to cut it

A dome is not a sphere; it is a sphere with the bottom taken off, and the cut has to land on struts. Stand the dome on a five-way hub — the orientation every kit uses — and ask which horizontal planes are made entirely of struts already there. The answer has a parity in it:

Even frequencies have an equator and odd ones do not. A 2V, 4V or 6V dome has a closed ring of 5v struts exactly at the widest point, so it can be cut into a true hemisphere that stands on flat ground. A 3V or 5V dome has no such ring at any height, which is why they are sold as 3/8 and 5/8 spheres — cut above or below the equator, on a ring that does exist. The hemisphere keeps exactly six of the twelve pentagons, which is the one place on this page where six is a fact about symmetry rather than about hexagons.

Why the bag of struts is lettered

The last practical consequence. Projecting a flat grid onto a sphere cannot keep the pieces equal, so the struts come in classes, they get letters, and a builder sorts them into piles before starting. How many piles, and how unequal:

The spread grows with the frequency rather than shrinking — a 2V dome's struts differ by ·, an 8V dome's by · — which is the opposite of the intuition that finer means more uniform. Finer means flatter per panel and more different from each other, because the shortest struts stay huddled around the twelve pentagons while the ones out in the middle of a face relax towards the size a flat grid would have. The twelve are visible in the bill of materials, not just in the picture.

What could have gone wrong, and did not

Every line below is computed on this page from the built meshes. The numbers on the right of each are the outside world's — printed dome tables, a molecule's bond census, published capsid numbers — and none of them is used to compute anything above.

The mesh is built with no knowledge of what it should come out as: vertices are lattice points of one icosahedron face mapped onto the sphere and then spun through the sixty rotations of the icosahedral group, and struts are found by looking for the gap in the sorted list of distances between hubs. Nothing tells it to make twelve of anything.