Three hundred and forty ways for twelve pentagons to touch

A closed cage in which three bonds meet at every atom and every ring is a pentagon or a hexagon must have exactly twelve pentagons, whatever its size. Twelve pentagons, and no way round them derived that from three lines of Euler’s arithmetic, and the twelve belongs to the vertex traced it to the three bonds rather than to the closing of the surface. The arithmetic is a sum over faces, and a sum cannot see which face touches which. How close the twelve must be named what it cannot see: a graph whose twelve points are the pentagons and whose edges join the pairs that share a bond. That essay and the last three contacts go at once counted the graph’s edges, the pentagon contacts, and found how few a cage of each size can have, down to none at sixty atoms.

The first of those essays ended by noticing that the count is one number read off an object with much more in it, and asked which graphs on twelve points actually occur. A graph on twelve points with every degree at most five can take an astronomical number of forms, and the planarity the sphere imposes cuts that down only somewhat. The census answers the question for every cage to sixty atoms, and the answer has three parts: there are very few graphs, they do not identify their cages, and they are nearly all pieces of one graph.

Every cage, and its graph #

The census winds up every face spiral with twelve pentagons for each size from twenty atoms to sixty, recognises each new cage by its canonical code, and checks the number of cages against the published sequence at every size: 1,812 at sixty atoms, 5,770 in all. For each cage it reads off the contact graph. Two contact graphs are the same when the twelve pentagons of one can be relabelled to give the other, so the graphs are sorted by an invariant, a colouring of the twelve points refined repeatedly by the colours of their neighbours, and then compared exactly within each colour class by a search for a relabelling that preserves every contact.

The exact comparison matters. A first version of the census used a canonical form for the graphs, the least labelling over every way of breaking ties among the points, and it stalled at sixty atoms for nearly two hours on the cage with no contacts at all. A graph with no edges has every relabelling as a symmetry, twelve factorial of them, and a canonical form that tries ties one by one tries them all. The search for one relabelling between two graphs stops at the first that works, and for a symmetric graph the first works at once.

Three hundred and forty graphs #

The 5,770 cages have 340 distinct contact graphs. The figure shows how the two counts part company. To about forty atoms nearly every cage has a graph of its own. Then the cages multiply, from 45 at forty-two atoms to 1,812 at sixty, and the graphs do not keep up: 71 at forty-four atoms, 138 at fifty, 241 at sixty. The arrangement of the pentagons is a much coarser description of a cage than the cage itself.

The third line in the figure says the same thing from the other side. A graph carried by only one cage identifies that cage: knowing which pentagons touch which, the cage is determined. Among small cages that is the rule, and from forty-two atoms on it almost never happens: one such graph at forty-two, one at forty-four, none at forty-six, fifty or fifty-four, two at fifty-six and fifty-eight, seven at sixty. Over the whole census 46 of the 340 graphs belong to a single cage. The contact graph is not a fingerprint. Cages that share it differ in where their hexagons sit relative to their pentagons, which the graph does not record. The small cages are identified for the opposite reason: with at most a few hexagons there is so little room that the pentagons’ arrangement leaves no freedom for the hexagons, and two cages with the same contacts would have to be the same cage. The sixty-atom cages that keep a graph of their own are seven, with 0, 2, 6, 10, 11, 12 and 12 contacts: the truncated icosahedron, whose pentagons touch nothing and which is the only cage of its size so spread out, one with two contacts near it, and a few arrangements in the middle of the range that happen to admit only one placement of the hexagons. Elsewhere, where the bulk of cages lie, a graph is shared by dozens.

The commonest arrangements #

The most shared graph has six contacts arranged as two chains of three pentagons, two pairs and two isolated pentagons, and it belongs to 179 cages, from fifty atoms to sixty: three at fifty, eighty-seven at sixty. The next has seven contacts, a chain of four, a chain of three and two pairs, and belongs to 170. Every graph in the top ten has between six and eight contacts and appears at five to eight different sizes, which is where the bulk of cages lie: the contact counts at each size are concentrated a few above the least possible, and the arrangements that achieve them are reused.

One graph recurs at thirteen different sizes, every size from thirty-six atoms to sixty: sixteen contacts, in two clusters of six pentagons each. That is the arrangement of two caps, six pentagons gathered at each end of an elongated cage, with hexagons between them, and the hexagons between the caps can be added in many ways without changing which pentagon touches which. The capped tubes that a centre at every other ring lengthened one ring at a time are one family of such cages. Read through the contact graph, many different cages at many sizes are one object.

How many contacts the graphs have #

The distinct graphs pile up between eight and thirteen contacts, fifty-one of them at ten, and thin out in both directions: a handful above twenty, where only small cages can force so much touching, and a handful below five, where only cages near sixty atoms can spread the pentagons so far apart. There is one graph with no contacts, belonging to one cage, the truncated icosahedron at sixty atoms, and one with thirty, the dodecahedron. The spread in the middle is wide because at mid sizes the pentagons can be grouped into clusters in many ways: every partition of twelve into chains, triangles and small patches that the hexagons can separate. Twenty-five of the graphs are connected, all twelve pentagons in one cluster; the rest split into two to twelve clusters.

Why there are so few #

A graph on twelve points with every degree at most five can be drawn in an enormous number of ways, and planarity, which the sphere imposes, removes only some of them. The census finds 340. Most of the reduction comes from the geometry of a single pentagon, and it can be proved in a line.

Round a pentagon its five neighbouring faces sit in a cycle, and two faces that are consecutive round it share a bond, because three faces meet at every atom. So if a pentagon touches three other pentagons, two of the three sit in consecutive places among its five sides, since three places among five cannot all be separated, and those two touch each other. In general, a pentagon touching $d$ others has at least $2d - 5$ contacts among them: one if it touches three, three if it touches four, and all five, a closed ring, if it touches five. This is the pentagon’s version of the count round a hexagon that the last three contacts go at once used to bound the least contact count, with five sides in place of six. It is a proof, and the checks the census is held to confirm it on every graph to forty-four atoms.

It removes whole families at once. A pentagon touching three pentagons that do not touch one another, a three-pointed star of contacts, never occurs; neither does any graph in which a point of degree four has an independent set of three among its neighbours. The clusters of touching pentagons that do occur are correspondingly few. Counted by their number of pentagons, number of contacts and degrees, there are about sixty kinds across the census. Every cluster of four pentagons with three contacts is a chain; every pentagon at the junction of a cluster carries a triangle of contacts with it. The clusters look like pieces of the dodecahedron’s surface, patches of mutually adjacent pentagons, because the rule that forces neighbours of a pentagon to touch is exactly the rule that holds on the dodecahedron, where it is satisfied with equality everywhere.

The second reduction is that clusters must be separable by hexagons. Two clusters of pentagons are apart only if a band of hexagons runs between them, and a cage of a given size has only so many hexagons to spend. That is why small cages have one large cluster and the cages near sixty atoms have many small ones, and why the number of distinct graphs at each size grows so much more slowly than the number of cages: most new cages at a size rearrange the hexagons around an existing arrangement of clusters rather than inventing a new one.

The icosahedron holds them #

The dodecahedron is the cage with no hexagons, and each of its twelve pentagons touches five others. Its contact graph is the icosahedron’s: twelve points, thirty edges, five at each point, the dual of the dodecahedron itself. Every other cage has hexagons and so fewer contacts, and a natural question about any contact graph is whether it is simply part of that one: whether its twelve pentagons can be placed on the icosahedron’s twelve vertices so that every contact lies along an icosahedral edge.

For a graph with few edges that might seem easy, but it is a real constraint, because the icosahedron lacks some small configurations entirely. It has no four mutually adjacent vertices, since its faces are triangles that never close into a tetrahedron, and no two vertices sharing three neighbours, since two vertices of the icosahedron share at most two. A cage in which four pentagons all touch one another, or in which two pentagons both touch the same three others, could not have an icosahedral contact graph however few contacts it had.

No cage to sixty atoms has either configuration. And beyond the absence of those two, the census asks the whole question of every graph: a backtracking search for a placement on the icosahedron, which either returns one or proves there is none. Of the 340 graphs, 338 fit. The figure at the head of this essay draws three of the placements: the icosahedron itself, the commonest arrangement with its six contacts on six icosahedral edges, and the arrangement with none. Every cage from thirty-four atoms to sixty has a contact graph that is a piece of the icosahedron.

A reading, offered as a reading and not derived: the twelve pentagons of a cage are its twelve points of curvature, and on a closed surface twelve points kept apart as far as the hexagons allow arrange themselves the way twelve points on a sphere do, each with five nearest neighbours, as at the vertices of an icosahedron. A contact is a pair of nearest neighbours close enough to share a bond, so the contact graph is part of the nearest-neighbour graph, and the nearest-neighbour graph of twelve well-spread points is the icosahedron. The reading explains the rule and also predicts where it should fail: in cages too small for the pentagons to spread out, where some must crowd more closely than icosahedral neighbours.

The two exceptions #

That is what the exceptions are. The cage of twenty-four atoms has only two hexagons, one at each end, and its twelve pentagons form two rings of six round them; each pentagon touches its two neighbours in its own ring and two in the other, twenty-four contacts in the pattern of a hexagonal antiprism. The icosahedron is a pentagonal antiprism with a cap on each end, and the six-fold pattern does not fit into the five-fold one: the search proves no placement exists. The other exception is a cage of thirty-two atoms with six hexagons, in which every pentagon touches exactly three others, eighteen contacts arranged in a pattern the icosahedron again cannot accommodate.

Both are small, both have pentagons touching evenly, and both owe their failure to a global shape rather than to a local configuration, since neither contains four mutually touching pentagons or two touching the same three. Above thirty-two atoms the census finds no further exception, and with 1,812 cages at sixty atoms alone, the rule is not an accident of small numbers. Whether it holds at every size beyond sixty is not settled by any census, and nothing here proves it.

What the graph does not see #

Where the hexagons are. The contact graph records which pentagons touch, and nothing about the hexagons between those that do not. That is why it does not identify cages, and why one graph can belong to 179 of them. A finer description would record, for each pair of pentagons, how far apart they are through the hexagons, a distance matrix on twelve points instead of an adjacency, and it would separate many of the cages the graph merges.

The other surfaces. A trivalent net on the projective plane carries six pentagons instead of twelve, as a gap the sphere does not have found by lifting each such net to a centrosymmetric cage on the sphere. The analogue of the icosahedron there is the hemi-icosahedron, the icosahedron with opposite vertices identified, and its graph joins every one of its six points to every other. Every graph on six points is a piece of it, so the projective plane’s version of the rule is empty: it says nothing. The rule on the sphere has content because twelve points with five neighbours each is a strict constraint, which six points with five neighbours each is not.

Chemistry. As the earlier essays said, a cage is a combinatorial object here and the stability of real carbon cages, which favours separated pentagons, is a matter of strain the census does not measure.

What the pictures cannot show is the cage. The figures draw twelve points and their contacts, placed on an icosahedron or laid out by a spring rule, and none of them draws the polyhedron the graph belongs to. The step from a cage to its graph throws away the hexagons, and the step back cannot be taken from the drawing: the commonest graph belongs to 179 different cages.

Who counted what #

The face-spiral enumeration of fullerenes is Manolopoulos’s, with May and Down, from 1991, and Brinkmann and Dress’s later generator made the counts routine far beyond sixty atoms. Pentagon adjacency has been tallied as a single number, the pentagon adjacency index, since the stability rule for carbon cages was stated in the late 1980s. The duality of the dodecahedron and the icosahedron is Euclid’s, and five solids from one inequality is where it sits in the arithmetic of regular polyhedra. What the census adds is the full list of contact graphs to sixty atoms, the measurement of how badly they fail to identify cages, and the finding that all but two are pieces of the icosahedron.

What the census must refuse #

The refused claim is the one the first two counts invite: that the contact graph, the richest object the arithmetic of pentagons reaches, identifies its cage. At forty atoms forty cages share thirty-seven graphs, and at sixty, 1,812 share 241.

Still open: whether every large cage is icosahedral #

The census establishes the icosahedral rule from thirty-four atoms to sixty and proves nothing beyond. A proof would have to show that once a cage has enough hexagons its pentagons can always be assigned to the icosahedron’s vertices so that touching pentagons are icosahedral neighbours. The natural route is through the cage’s dual, a triangulation of the sphere with twelve points of degree five and the rest of degree six. The arithmetic of the hexagons fixes everything except where they go, and the pentagons’ contact graph is the part of that triangulation joining degree-five points. Whether the degree-five points of every large such triangulation inherit an icosahedral neighbourhood structure, and whether the exceptions stop at thirty-two atoms or reappear at some larger size, has not been worked out.