Twelve pentagons, and no way round them

A repeating pattern in the plane may have rotations of order two, three, four or six and nothing else. The five-fold case is the one everybody remembers, and it is the one this collection has proved twice — once by arithmetic on a trace, once by constructing a lattice vector shorter than the shortest there is.

Both proofs need a lattice. Take the lattice away and the prohibition goes with it, and something more surprising than mere permission takes its place.

On a closed surface, a net of hexagons and pentagons must have exactly twelve pentagons. Not at most twelve, not typically twelve — exactly twelve, whatever the size of the cage and however many hexagons it has. The argument is three lines long and it never mentions symmetry.

Three lines #

Take any net drawn on a closed surface of the sphere’s kind, with three edges meeting at every vertex. Write V, E and F for the counts of vertices, edges and faces, and pₙ for the number of faces with n sides.

Euler’s relation says V − E + F = 2. Three edges at every vertex says 3V = 2E, because each edge has two ends and each vertex uses three of them. Counting edge-sides face by face says Σ n·pₙ = 2E, because each edge borders exactly two faces. Substituting the second and third into the first and clearing denominators leaves

$$\sum_n (6 - n)\, p_n = 12.$$

Read that sum. A hexagon contributes 6 − 6 = 0, so hexagons are free: a net may have any number of them and the total does not move. A pentagon contributes 1. So a net made of pentagons and hexagons alone has twelve pentagons, and the hexagon count is unconstrained.

The other rows of that chart are the rest of the classification. Six squares and no pentagons is a cube; four triangles is a tetrahedron; twelve pentagons and no hexagons is a dodecahedron. Every closed trivalent net is a way of paying twelve, and the five regular solids are the ways of paying it with faces all alike.

The same twelve, in degrees #

Euler’s relation is from the 1750s. The result above is older, and in its first form it is about angles rather than counts.

Descartes, a century earlier, measured the angular defect at each corner of a convex polyhedron: take the angles of the faces meeting there, add them, and subtract from a full turn. A flat sheet has no defect anywhere, because the angles at any interior vertex add to exactly a full turn. A corner has a positive defect, and the defect is how much has been cut away to make it close.

Descartes’ theorem is that the defects of all the corners of a convex polyhedron add to exactly two full turns — 720°, and never anything else. On a dodecahedron the three pentagons meeting at each of the twenty vertices give 3 × 108° = 324°, a defect of 36°, and twenty of those is 720°. On the sixty-vertex cage the vertices are of two kinds, two hexagons and a pentagon giving a defect of 12°, and sixty of those is again 720°.

The two statements are one statement. Descartes’ 720° and the sum coming to twelve are the same quantity expressed in different units — a full turn against a hexagon’s worth — and the conversion is the observation that a pentagon is a sixth of a turn short. That the older version is about angles and the newer about counts is why the counting version survives on a net with no angles in it at all, which is the version this collection can compute.

The plane pays zero #

The same three lines run on a flat repeating net, and one number changes.

A plane has Euler characteristic zero rather than two — the difference between a surface that closes up and one that does not — so the sum comes to zero rather than twelve. A sum of terms 6 − n, over faces with at least three sides each, can only vanish if no face has fewer than six sides and none has more. Every face is a hexagon.

That is the crystallographic restriction arriving by a completely different route, and it is worth being exact about the sense in which it is the same statement. The restriction proper is about rotations: no repeating pattern has a five-fold axis. What the accounting gives is about faces: no flat trivalent net has a five-sided cell. Neither implies the other in one step — a pattern can have five-sided faces without a five-fold rotation, and three of the eleven Laves tilings are made of pentagons with no five-fold symmetry anywhere in them.

What the two share is the quantity doing the forbidding. In the restriction it is the integrality of a trace in a lattice basis; here it is the Euler characteristic of the plane. Both are zero-ness of a kind, and both are removed by curving the surface.

The picture this collection usually draws for the restriction is the local one: regular pentagons set around a point in the plane leave a gap, because three of them cover 324° and four overlap. That gap is the same refusal the accounting above delivers, seen at one vertex instead of over a whole surface, and the two versions are worth keeping apart. The local picture is about angles and needs the pentagons to be regular; the accounting is about counts and needs nothing of the sort. Bent into any five-sided shapes whatever, the pentagons make the local argument evaporate while the accounting stands, which is why the accounting is the one that reaches a crumpled carbon cage.

One detail of that reading repays care, because it is where the analogy between the two arguments stops. The restriction is a statement about a lattice and holds for any pattern on it, however the pattern’s cells are shaped. The accounting is a statement about a net — a division of the surface into cells with edges and vertices — and says nothing about a pattern with no cells in it. A wallpaper of scattered dots has no faces to count and the sum has no subject. The two arguments overlap on the honeycomb, agree there, and are about different objects everywhere else.

The pentagons are not where the symmetry is #

The dodecahedron pays its twelve with pentagons and nothing else, and its symmetry group is the icosahedral one — sixty rotations, six five-fold axes, and no lattice in three dimensions that can hold any of them.

But the twelve does not come from the symmetry, and this is the point most easily missed. The sixty-vertex cage above also has twelve pentagons and its symmetry group is the same icosahedral one. A cage with no symmetry at all — pentagons scattered anywhere the chemistry likes — still has twelve of them, because the derivation used only the vertex degree and the Euler characteristic and asked nothing about what maps the net to itself.

The right way round is the reverse of the intuitive one. Twelve pentagons is the constraint; five-fold symmetry is one of the ways of satisfying it tidily. A cage that puts its twelve pentagons in the twelve most symmetric places acquires icosahedral symmetry as a consequence, in the same way that a pattern acquires accidental symmetry when its motif is placed at a special position.

One more consequence is worth extracting before the cages are grown. The sum says nothing about where the twelve go, and that silence is the whole of the chemistry. Two pentagons sharing an edge produce a sharp local bend and a strained bond; twelve pentagons spread as far apart as the cage allows produce a shape that is nearly a sphere. Both satisfy the arithmetic exactly. So the counting fixes the budget and leaves the design entirely open, which is the same division of labour Neumann’s principle makes for physical properties: symmetry decides what is permitted and never what happens.

Growing a cage #

The claim is about every size, so it is worth producing a range of sizes and counting rather than trusting the algebra.

Two operations on a net do the growing, and both are defined on any polyhedron whatever. Dual puts one vertex in each face and one face at each vertex. Truncate cuts every vertex off, turning a vertex of degree d into a face with d sides and a face of n sides into one of 2n. Their composition — truncating the dual, which is called leapfrogging — takes a trivalent net to a larger trivalent net and triples the vertex count.

Twenty vertices, then sixty, then a hundred and eighty, then five hundred and forty. The hexagon count runs 0, 20, 80, 260 and has no ceiling. The pentagon count is twelve at every step, and it is counted from the faces after each net is built rather than carried along.

This is why carbon cages exist in the shapes they do. A sheet of graphite is a flat trivalent net of hexagons, obeying the plane’s answer of zero. Close it up and it must find twelve pentagons somewhere, and each pentagon is a place where the sheet bends. The sixty-atom molecule is the smallest cage in which no two pentagons touch, which is a stability condition rather than a counting one — the counting has already fixed the number and only their arrangement is left to argue about.

Twenty T vertices, and where the arithmetic has been before #

The four cages above have 20, 60, 180 and 540 vertices, and the ratios are not an accident of the construction. Write the vertex count as 20T. Then T runs 1, 3, 9, 27, and the face count in each case is 10T + 2 — twelve pentagons plus 10(T − 1) hexagons.

T is not free. The cages that carry the full icosahedral symmetry are those built by placing the twelve pentagons at the vertices of an icosahedron and filling the twenty triangular faces between them with a patch of honeycomb, and a triangular patch of honeycomb is specified by two whole numbers — how far along each of two lattice directions its corner sits. The number of hexagons in it is then

$$T = h^2 + hk + k^2,$$

which this collection has met before under another name. Those are the values a hexagonal lattice’s quadratic form takes — the same integers that count the vectors of each length in a triangular lattice, and the same ones that decide which sublattices of a hexagonal lattice are themselves hexagonal. The leapfrog operation multiplies T by three, which is why the four sizes above are the powers of three.

So the arrangement of the twelve is a lattice question after all — but a question about a lattice on the triangular faces of an icosahedron, not about a lattice in the plane the cage is drawn on. Biology found the same arithmetic in the 1960s, when the protein shells of icosahedral viruses turned out to be built of exactly this construction: twelve five-fold clusters, 10(T − 1) six-fold ones, and shell sizes that go 1, 3, 4, 7, 9, 13 — the same list of integers, in the same order.

What the accounting does not decide #

It does not say a cage exists. The sum is a necessary condition and nothing more. It says that if a closed trivalent net of pentagons and hexagons exists then it has twelve pentagons; whether one exists with a given hexagon count is a separate question, and the answer is no for exactly one value — there is no such cage with a single hexagon, for reasons the accounting cannot see. That single exception is the sharpest illustration of what a necessary condition is worth: the arithmetic is satisfied for every hexagon count from zero upwards, and the geometry supplies a net for all of them but one. A reader who took the sum as a construction would be wrong exactly once, which is the hardest kind of wrong to notice.

It does not need three edges at a vertex. That assumption was used and can be dropped, at the cost of a longer sum with vertex degrees in it. What cannot be dropped is the Euler characteristic, which is the whole content: the two on the right-hand side is the sphere’s, and any surface with a different one gives a different total. On a torus it is zero, and a trivalent net on a torus is all hexagons again — which is the flat surface the plane group p1 folds into, carrying its flatness and its answer with it.

Nor does it need the faces to be regular. Nothing above assumed a pentagon was a regular pentagon or that the cage was convex. A crumpled net of five- and six-sided cells, with no two alike, obeys the same sum — which is what makes the result usable on objects nobody would call polyhedra. The five regular solids need the regularity and are a much smaller answer; this is the general one.

Negative curvature reverses the sign, and the site has already priced it. A surface with handles has an Euler characteristic below zero, so the sum comes out negative and the net has to pay with faces of more than six sides. Seven-sided cells in a carbon sheet do exactly that, and the resulting surfaces are saddles rather than cages. That is the same three-way split Conway’s accounting makes for wallpaper: spend less than two dollars and the answer is finite, spend exactly two and it is flat, spend more and the list of possibilities never ends.

And it decides nothing about a crystal. The cages above are finite objects with no translational symmetry at all, so no lattice is involved and the restriction has no purchase on them. C₆₀ sits in crystals and does not stop being icosahedral when it does; what the crystal keeps of that symmetry is a subgroup of order twelve, at index five, and the five-fold axes are simply not among the operations the lattice has. The molecule’s twelve pentagons and the crystal’s permitted rotations are two different accountings that happen to sit inside one another, and running them together is the commonest way of getting this subject wrong.

The arithmetic of that index is worth one more sentence, because it is the whole of what “the crystal keeps” means. The icosahedral rotation group has order sixty. Its subgroups that a lattice can hold are the ones built from two-, three- and four-fold axes only, and the largest of them is the tetrahedral group of order twelve — so the index is five and a crystal keeps a fifth of the molecule’s symmetry. Five is not a coincidence there and it is not the five-fold axis either. It is the number of inscribed tetrahedra a dodecahedron’s twenty vertices split into, and the five-fold rotation is precisely the operation that permutes them cyclically. A lattice cannot hold that permutation, so it holds one tetrahedron and loses the cycle: the operations survive and the thing that relates them does not.

What none of the accounting says is that a cage must be nearly spherical, and it is worth naming the assumption because every picture here quietly makes it. A long carbon nanotube capped at both ends is a closed trivalent net of pentagons and hexagons with Euler characteristic two, so it has exactly twelve pentagons — six in each cap — and a cylinder of hexagons of any length between them. The twelve are all at the ends because the curvature is all at the ends, and the tube itself is a rolled-up flat sheet paying nothing. Length is free; closure is what costs twelve.

Where the twelve may not go #

The accounting fixes the number of pentagons and says nothing about their arrangement, and the arrangement is where every chemical fact about these cages lives.

Two pentagons sharing an edge concentrate the curvature: the vertex where they meet is more sharply bent than any vertex of an all-hexagon region, and in a carbon cage that bending strains the bonds. The isolated-pentagon rule is the observation that a cage in which no two pentagons touch is far more stable than one in which some do — and it is a statement about arrangement, not about count.

That rule explains a sequence a reader will have met without its reason. The smallest closed trivalent net is the dodecahedron, at twenty vertices, and its twelve pentagons all touch. Cages grow from there, and the smallest one whose twelve pentagons are all isolated has sixty vertices — which is why buckminsterfullerene is the first fullerene to be found, and why the next is at seventy rather than at sixty-two.

So the arithmetic supplies twelve at every size and the arrangement supplies a threshold. Below sixty vertices the pentagons cannot be kept apart — there is not enough hexagon to separate them — and every cage is strained. From sixty upwards there is room, and the ones that use it are the ones that exist.

How many cages there are #

The other thing the accounting does not decide is how many arrangements there are at a given size, and the answer is much larger than the sequence of stable molecules suggests.

Counting combinatorially distinct closed trivalent nets with a given vertex count is an enumeration over graphs, it has been done exhaustively, and the numbers grow fast. At sixty vertices there are 1,812 distinct cages satisfying every condition on this page — twelve pentagons, the rest hexagons, three edges at every vertex — and exactly one of them has its pentagons isolated.

That ratio is the essay’s point stated as a count. The Euler accounting narrows an infinite space to a finite one and leaves nearly two thousand candidates at one size; a rule about where the curvature sits narrows it to one. Neither step is chemistry and neither is enough by itself, and the second is a statement about strain that no counting argument produces.

The enumeration also puts a bound on what a search has to do. A structure proposed for a carbon cage of sixty atoms is one of 1,812 things, which is a list rather than a space — so the question which cage is this? is decidable by comparing against the list, and that is how a proposed fullerene structure is actually identified.