How close the twelve must be

Twelve pentagons and no way round them settles how many. The twelve belongs to the vertex settles why it is twelve and not some other number. Everything except the hexagons settles how nearly the charge decides the rest of the face vector. Not one of the three says a word about where the pentagons are, and none of them can: the charge is a sum over faces, one term each, and a sum over faces cannot see which face touches which.

What it cannot see is a graph. Put a point for each of the twelve pentagons and join two points when those pentagons share a bond; that graph is a complete description of how crowded the pentagons are, it has between nought and thirty edges, and the charge takes the same value for every one of them.

The edge count of that graph is the quantity this page is about. It is not free — it is bounded below by an argument three lines long, and bounded below again by a better argument that the first one hides — and what the two bounds do not agree about is where the smallest cage with no contact at all can be.

Thirty at one end, and nothing at the other #

The two extremes are worth fixing before the middle.

The dodecahedron has twelve pentagons and nothing else, so every bond is shared by two pentagons and every pentagon touches five others. Its contact graph is the icosahedron’s — twelve points, five edges at each, thirty edges in all — and thirty is the most twelve pentagons can ever have, since a pentagon has only five bonds to offer.

At the other end sits a cage whose pentagons touch nothing: each of the twelve surrounded entirely by hexagons, contact graph with no edges at all. The chemistry calls that arrangement isolated pentagons, and the smallest cage with it is one of the most-drawn objects of the twentieth century.

Between the two ends the question is how fast the contact count can fall, and the honest way to ask it is not about a typical cage. Most cages have their pentagons considerably more crowded than they have to be. The quantity that means something is the least over all cages of a given size, and reading it requires every cage of that size.

Sixty edges, and where they can go #

The first bound is a count of edges and it takes four lines.

The twelve pentagons carry five bonds each, which is sixty bonds counted from the pentagon side. A bond between two pentagons is counted twice — once from each — and a bond between a pentagon and a hexagon once. So with c contacts, the number of bonds running from a pentagon to a hexagon is 60 − 2c. Each hexagon has six bonds altogether, so h hexagons can absorb at most 6h of them:

$$60 - 2c \le 6h, \qquad \text{that is} \qquad c \ge 30 - 3h.$$

The bound is not a formality. At no hexagons it gives thirty, which the dodecahedron attains. At two hexagons it gives twenty-four, and the one cage of twenty-four atoms has twenty-four contacts. At three it gives twenty-one and the one cage of twenty-six atoms has twenty-one; at four it gives eighteen and the better of the two cages of twenty-eight atoms has eighteen. For the first four sizes the pentagons are as far apart as the edge count permits, exactly, because with that few hexagons they have nowhere else to be.

At five hexagons the agreement stops. The bound says fifteen and the best of the three cages of thirty atoms has seventeen. From there the two separate steadily: at six hexagons twelve against fifteen, at eight six against twelve, at ten nothing against ten. The measured least falls by three for each of the first four hexagons and then by about one for each hexagon after — 17, 15, 14, 12, 11, 10, 9, 8 — so each extra hexagon buys three contacts while there are few of them and one when there are many.

A bound that is exact and then is not is more useful than one that is never exact, because the place it stops being exact is information. What changes at five hexagons is that the pentagons stop being forced against each other and start being arrangeable, and the constraint that takes over is not about how many bonds the hexagons have.

A hexagon can face three pentagons, not six #

The edge count lets a hexagon spend all six of its bonds on pentagons. It cannot, if the pentagons are to stay apart, and the reason is the one fact every count of this kind rests on.

Three faces meet at every atom of a trivalent net. Take a hexagon and two pentagons on neighbouring bonds of it: those two bonds share an atom, three faces meet at that atom, and the three are the hexagon and the two pentagons — so the two pentagons share a bond and are in contact. Pentagons that keep apart therefore occupy alternate bonds of any hexagon they touch, and a hexagon has three alternate bonds.

Redo the count with three in place of six. With no contacts at all the sixty pentagon bonds must all run to hexagons, and each hexagon absorbs at most three of them, so

$$60 \le 3h, \qquad h \ge 20,$$

and a cage with isolated pentagons has at least twenty hexagons, which is at least sixty atoms. The first bound put that at forty atoms and the second puts it at sixty, and the second is right.

The extrapolation of the measured numbers agrees with the second bound and not the first, which is the part worth noticing. The least contact count at twelve hexagons is eight and it is falling by one per hexagon; running that on reaches nought at twenty hexagons, which is sixty atoms. The census stops at forty-four atoms and knows nothing of the truncated icosahedron, and the rate it is falling at points straight at it.

Most cages are nowhere near the least #

The least is an extreme and the extreme is rare, which is a statement about the cages rather than about the bound.

At forty-four atoms the distribution peaks at twelve contacts and the mode is fifty per cent above the minimum. Two of the eighty-nine cages achieve eight, nine achieve nine, and two are as crowded as eighteen — five-sixths of the way back to the dodecahedron’s thirty on a cage more than twice its size.

This is the same shape as everything a large enumeration produces and it is the fact counting what a group cannot tell apart is built on: the symmetric and the extreme cases are a handful and the ordinary ones are almost everything. It also says why the minimum is the interesting statistic. An average contact count would be a statement about how many ways there are to build a crowded cage, which is a combinatorial fact about the enumeration; the minimum is a statement about what a cage of that size is permitted to do, which is a fact about the surface.

Sixty atoms, and the cage that meets the bound exactly #

The bound says twenty hexagons at least. The cage that realises it has exactly twenty.

Take the dodecahedron, take its dual, and truncate it: every corner of the icosahedron becomes a pentagon and every triangle becomes a hexagon, giving twelve pentagons, twenty hexagons and sixty atoms. Each pentagon’s five neighbours are the five hexagons that were the triangles round its corner, so no two pentagons meet, and the sixty pentagon bonds are distributed three to each of the twenty hexagons — the bound satisfied with nothing to spare.

That exactness is the reason the cage is the smallest rather than merely the smallest known. A cage with fewer hexagons cannot absorb sixty pentagon bonds three at a time, and a cage with exactly twenty must absorb them exactly three at a time, so every hexagon in it has pentagons on alternate bonds and hexagons on the others. The arrangement is forced once the counts are, which is why the construction has icosahedral symmetry rather than being one of many.

Leapfrogging again gives a hundred and eighty atoms with eighty hexagons, and the pentagons stay apart there with room over: sixty bonds among eighty hexagons needs only three quarters of them to carry any. Past the exact case the constraint stops binding, which is the usual fate of a bound that a construction meets.

Six pentagons, and the same arithmetic one surface down #

The alternate-bond count uses nothing about the sphere. It uses three faces at an atom, five bonds on a pentagon and three usable bonds on a hexagon, and every one of those survives a change of surface — so the same two lines run on the projective plane, where the charge is six and a net carries six pentagons rather than twelve.

Six pentagons carry thirty bonds. With no contacts all thirty run to hexagons, three to a hexagon at most, so a projective net with isolated pentagons needs at least ten hexagons — and a net of six pentagons and ten hexagons has thirty atoms.

That number has already been met. The cage of sixty atoms halves, because its antipodal map fixes no face, no bond and no atom, and what it halves to is thirty atoms with six pentagons and ten hexagons. Its pentagons are isolated on the sphere, and a quotient cannot bring two faces into contact that were not in contact above — the map is a bijection on bonds — so they are isolated below. The smallest projective net with isolated pentagons is the half of the smallest cage with them, and the two arithmetics agree without either being told about the other.

The agreement is not a coincidence and it is not quite a theorem either. Halving divides every count by two, so the bound on the sphere — sixty bonds, three a hexagon, twenty hexagons — divides to thirty bonds, three a hexagon, ten hexagons, and the two inequalities are the same inequality. What the halving does not guarantee is that the minimal case on one surface halves onto the minimal case on the other; it does here because the cage of sixty happens to have a centre, and a cage having a centre is the rare condition that the count of centres turns on.

Four bonds an atom, where the argument stops working #

The charge depends on how many bonds meet, and it is worth transposing this page one column along that argument, because the transposition fails and the failure is the point.

At four bonds an atom the neutral face is the square and the sphere charges eight triangles. Eight triangles carry twenty-four bonds between them; a contact spends two; a square has four bonds. The edge bound transposes without trouble and says a cage with isolated triangles needs at least six squares.

The alternate-bond argument does not transpose at all. It rested on three faces meeting at an atom: two pentagons on neighbouring bonds of a hexagon meet at the atom those bonds share, and there is no room at that atom for a fourth face to come between them. At four bonds an atom there is room. Two triangles on neighbouring bonds of a square meet at a vertex where four faces meet, and the fourth face separates them — so a square may spend all four of its bonds on triangles and keep them apart, and the argument that improved the bound on the sphere has nothing to improve it with here.

What that leaves is the weaker bound, exact. Twenty-four bonds at four a square needs six squares, and a cage with eight triangles and six squares at four bonds an atom is the cuboctahedron: twelve atoms, twenty-four bonds, triangle and square alternating round every one of them, no two triangles touching. The bound is met with nothing to spare at the smallest size it permits.

So the gap between the two bounds that this page is about — forty atoms against sixty — exists only at three bonds an atom, and it exists because three is the degree at which a face has no room to sit between two others at a shared atom. That is the same sentence the charge at a vertex makes about the charge itself, arriving here about an arrangement rather than about a count.

What the count says and what it does not #

Computed here. Every cage of pentagons and hexagons from twenty to forty-four atoms, two hundred and twenty-six of them, each one’s contact graph as a graph on twelve points, the least and the most at each size, the whole distribution at each size, and the contact count of the dodecahedron and of two leapfrogs of it computed from coordinates by a route that shares nothing with the census.

The contact count is not the whole arrangement. Two cages with eight contacts each can have quite different contact graphs — different degree sequences, different shapes — and nothing above distinguishes them. The edge count is one number off a graph with a great deal more in it, and which graphs on twelve points occur at all is a question this page does not ask.

The census reaches forty-four atoms and the interesting cage has sixty. The rate of one contact per hexagon is measured over eight sizes and extended over another eight by nothing but arithmetic. That it lands on sixty is evidence for the second bound and is not a proof of anything; the proof that sixty is the smallest is the alternate-bond count, which needs no census at all.

And this is combinatorics, not chemistry. A cage with no two pentagons adjacent is the stable one, and the reason is strain in real bonds rather than anything counted here. The bound above says such a cage cannot exist below sixty atoms; it says nothing about whether one that could exist would.

The second refusal is the one the essay turns on. A cage of forty atoms with isolated pentagons is exactly what the edge bound permits — at ten hexagons it demands nothing — and all forty cages of that size carry at least ten contacts. The refusal is what makes the gap between the two bounds a measurement rather than an observation about which inequality looks stronger.

Who wanted the pentagons apart #

The rule that a stable carbon cage has no two pentagons adjacent was stated by Patrick Fowler and others in the years after 1985, as an empirical regularity with a strain argument behind it: two pentagons sharing a bond force more curvature into a smaller region than the bonding will comfortably take. That it first becomes satisfiable at sixty atoms is the observation the cage of sixty is famous for, and it is a combinatorial fact rather than a chemical one — the alternate-bond count above is the whole of it, and it was in the literature as a remark before it was used as an explanation.

The enumeration of cages by face spiral is Manolopoulos’s, with May and Down, from 1991, and counting pentagon adjacencies became a standard way of ranking isomers shortly after; the quantity is usually called the pentagon adjacency index. The minima at small sizes are known and the ones computed above agree with them.

The reason the count belongs beside the charge rather than in that literature is what it says about the charge. Every count before it establishes that a quantity is forced — twelve pentagons, six on the projective plane, a face vector realised at some hexagon count. This one establishes that a quantity is not forced, and then finds the weaker thing that is.

Still open: which graphs on twelve points occur #

The least contact count is one number read off a graph, and the graph is the object. A natural next question is which graphs occur at all: a cage’s contact graph has twelve points, every degree at most five, and the drawings above show two of them, but nothing here says which of the many graphs on twelve points with those properties belong to some cage.

Two constraints are visible and neither is checked. The contact graph is drawn on the sphere along with everything else, so it is planar, and its faces are regions of the cage — which ought to restrict it considerably more than the degree bound does. And the pentagons in contact form connected clusters whose shapes are limited: a cluster of five pentagons all touching one another would need a sixth face they all surround, and what that face can be is decided by the same arithmetic the charge at a vertex works with.

The other direction is the one the numbers point at. The least count falls by one a hexagon for eight consecutive sizes and then, at twenty hexagons, is nought and stays nought. Whether it falls by exactly one at every intermediate size — so that the sequence runs 8, 7, 6, 5, 4, 3, 2, 1, 0 from forty-four atoms to sixty — is a prediction that the census cannot reach and that one more size would begin to test.