A gap the sphere does not have

The surfaces a count by genus skips puts a bill on the projective plane. A closed net with three bonds at every atom, drawn on that surface with faces of five and six sides, must have six pentagons — half the sphere’s twelve — and may have any number of hexagons. The essay built the net with none, by halving a dodecahedron, and left the rest where the arithmetic leaves it: “an arithmetic that says what a net must charge does not say that a net exists.”

The sphere’s own version of the question has a known answer with a flaw in it. Twelve pentagons are compulsory on a closed cage and the hexagons are free, and a cage exists with every number of hexagons except one. No cage has twelve pentagons and a single hexagon, although the bill is paid.

So the projective plane’s version is not rhetorical. Six pentagons and h hexagons pays the bill for every h, and the question is which h a net actually reaches.

A net on the projective plane is a cage with a centre #

The projective plane is covered twice by the sphere: every point of it has two points of the sphere over it, and the two are swapped by a single symmetry of the sphere that reverses orientation and fixes nothing. That double cover is a standard construction and nothing here develops it; what matters is what it does to a net.

A net on the projective plane lifts. Every face of it becomes two faces of a net on the sphere, every bond two bonds, every atom two atoms. Three bonds still meet at every atom, pentagons stay pentagons and hexagons stay hexagons, so six pentagons and h hexagons below become twelve pentagons and 2h hexagons above — a closed cage of 20 + 4h atoms, of the kind chemistry calls a fullerene. And the lift carries the cover’s swap with it: the cage has a symmetry of order two that reverses orientation and fixes no face, no bond and no atom.

The argument runs backwards as well. A cage with such a symmetry can be halved — identify each face with its image, each bond with its image, each atom with its image — and because nothing is fixed, the result is a net on a surface, with half of every count. Half of Euler’s two is one, and a closed surface with characteristic one is the projective plane.

So a net with h hexagons exists on the projective plane exactly when some cage with 2h hexagons has a free symmetry of order two that reverses orientation. A question about a surface that cannot be oriented has become a question about the symmetry of an ordinary cage.

The move has been made once before on a flatter surface. Every net folds onto a torus divides a periodic net by its translations, and the two plane groups that fold into a surface are the two whose operations fix nothing — the translations alone, giving a torus, and a glide, giving a Klein bottle. The rule in both places is the one used here: a quotient by symmetries is a surface exactly where no symmetry fixes anything, and whether the surface can be oriented is decided by whether any of those symmetries reverses orientation. What changes on a cage is that the group is finite, so the quotient is closed, and that a single free reversal is enough.

For a cage built as a convex solid, that symmetry has a familiar name. A theorem of Mani’s says every symmetry group of a net of this kind is the group of symmetries of some convex realisation of it, and the only operation of order two that reverses orientation and fixes no point of the sphere around a solid is the inversion through its centre. Among the improper operations of order two a mirror fixes a whole plane and the centre fixes only itself. So the projective plane’s nets are exactly the halves of the centrosymmetric cages. Mani’s theorem is quoted; what is computed below is the combinatorial statement, which needs no solid at all.

Every cage to forty-four atoms, from its spiral #

To find which cages have a centre, first every cage has to be found.

The method is the one fullerene chemistry uses. Peel the faces of a cage off in a spiral — a first face, a ring round it, a ring round that, each face touching the one before — and write down the number of sides of each as it goes. The result is a string of fives and sixes with exactly twelve fives. Conversely, a string of twelve fives and some sixes can be wound up: each new face is attached to the one before it and to the oldest face still short of neighbours, the last face closes the lid, and the string either closes into a cage or fails at some step, because a face is given more neighbours than it has sides or the lid does not fit.

Nothing in a string records where an atom is. A cage here is only which faces touch which, a structure with the distances thrown away, and every question below is a question about that pattern of contacts rather than about a shape.

Winding up every string with twelve fives and up to twelve sixes, and then identifying the cages that arrive more than once — from different starting faces and directions — gives the complete list up to forty-four atoms. The identification reads each cage from every choice of starting face, neighbour and sense of turning, numbering the faces in the order a walk outwards meets them, and keeps the least of the resulting codes: two cages are the same cage exactly when their least codes agree.

Two checks are available, and both pass. The counts at each size — 1, 0, 1, 1, 2, 3, 6, 6, 15, 17, 40, 45 and 89, from twenty atoms to forty-four — agree with the published enumeration of fullerene isomers, which is compared only after the counting is done. And the zero at twenty-two atoms comes out of the winding itself: all thirteen strings of twelve fives and one six were tried, and not one of them closes. That is the sphere’s exception, reached by exhaustion rather than quoted.

There is one assumption in the method and it is worth naming. Not every fullerene has a face spiral; the first known counterexample has three hundred and eighty atoms. Below forty-four atoms agreement with the published counts, which were obtained by a method that does not assume a spiral, is the evidence that none was missed.

One hexagon, two hexagons, and no centre anywhere #

With the cages in hand, each one’s symmetries can be listed. A symmetry of a cage is a permutation of its faces carrying neighbours to neighbours, and the codes above supply every one of them at once: two starting choices that read the same code are related by a symmetry, and it reverses orientation exactly when the two choices turn in opposite senses.

A projective net with one hexagon would halve a cage of twenty-four atoms, and there is exactly one such cage. A net with two hexagons would halve a cage of twenty-eight atoms, and there are two.

The cage of twenty-four atoms has twenty-four symmetries, and six of them are of order two and reverse orientation. Every one of the six fixes four faces, four bonds and four atoms. The first cage of twenty-eight atoms has four symmetries and none reverses orientation at all. The second has twenty-four, six of order two that reverse orientation, and every one fixes four faces, four bonds and six atoms.

Not one of the twelve reversing involutions is free, so no cage of twenty-four or twenty-eight atoms halves, and the projective plane has no net with one hexagon and none with two.

The reason every one of them fixes something is a fact about the sphere that the counting never uses and that explains what it finds. A symmetry of the sphere of order two that reverses orientation is either a reflection, which fixes a whole great circle, or the antipodal map, which fixes nothing. A reflection’s circle cannot avoid the cage: it crosses faces and bonds and passes through atoms, and those are fixed. So a cage halves exactly when one of its symmetries is the antipodal kind rather than a mirror, and these three cages have only mirrors.

The two gaps are therefore a different kind of gap from the sphere’s. The sphere at one hexagon has no cage at all. The projective plane at one and two hexagons has cages — they simply lack a centre. One failure is of existence and the other of symmetry, and the bill cannot see either.

Three hexagons, and the first net that has any #

At thirty-two atoms there are six cages, and one of them has a centre.

The drawing places one face outside and every other atom at the average of its three neighbours, the placement nobody chose, and then spreads the inner rings so that their numbers can be read. Every number from one to nine appears twice. The two faces carrying a number are opposite each other on the cage, and the outside face has a partner inside the drawing like every other.

Halving gives sixteen atoms, twenty-four bonds and nine faces, and 16 − 24 + 9 = 1. That is the smallest net on the projective plane with a hexagon in it, and it is the only one with three: the other five cages of thirty-two atoms have no centre.

It is worth seeing what the pairs are doing in the flat drawing. A face and its partner are never neighbours — a symmetry that swapped two neighbours would fix the bond between them — and the pentagons pair with pentagons and hexagons with hexagons, since a symmetry preserves the number of sides. Six pairs of pentagons and three pairs of hexagons is the face vector of the halved net read directly off the picture.

A centre is rare, and rarer as the cage grows #

Taking the same computation to forty-four atoms gives the rest of the bottom row.

One cage of thirty-six atoms halves, three of forty and three of forty-four, and every one of those cages has exactly one free reversing symmetry up to relabelling, so each gives exactly one projective net. The bottom row of the census reads 1, 0, 0, 1, 1, 3, 3.

The cages that halve are among the most symmetric of their size. Most cages of forty or forty-four atoms have one, two or four symmetries; the ones that halve have twenty, eight and twenty, and twelve each. That is not a coincidence of small numbers. A centre composes with every other symmetry a cage has, so a group containing it contains a partner for every rotation, and a cage with very few symmetries has very little room to hold a centre among them. The fraction that halve falls — a sixth, a fifteenth, three fortieths, three eighty-ninths — because the number of cages grows much faster than the number of highly symmetric ones. At forty-four atoms forty-two of the eighty-nine cages have no symmetry but the identity. That is the ordinary state of a large enumeration, and it is the fact counting what a group cannot tell apart is built on: almost every arrangement is alone in its orbit, and the symmetric ones are the few that make the counts come out uneven.

How much symmetry, against which kind #

The carriers invite a wrong conclusion, which is that a cage halves when it has enough symmetry. It does not, and the most symmetric cages of each size show why.

The two cages of thirty-two atoms with twelve symmetries have the same number of symmetries and the same number reversing orientation, six each. The first has four reversing involutions, three that fix faces and one that fixes nothing, and it halves. The second has four as well, and all four fix faces. Nothing about the count separates them.

At forty atoms the most symmetric cage does not halve. It has twenty-four symmetries and twelve reverse orientation, and its six reversing involutions are all mirrors. A cage of the same size with only eight symmetries has four reversing involutions, one of which fixes nothing, and it halves. At forty-four atoms six cages have twelve symmetries each: three halve, two have reversing involutions that are all mirrors, and one has no symmetry reversing orientation at all.

This is the distinction eleven, eleven and ten draws for point groups, arriving from the other side. A group of order twelve may contain the inversion, may contain improper operations without it, or may be rotations only, and which it is cannot be read off twelve. A cage halves exactly when its group is of the first kind. So the projective plane does not ask for a symmetric cage; it asks for a cage whose symmetry includes the one operation that turns the sphere inside out without fixing anything, and a cage with twenty-four symmetries and six mirrors among them has everything except that.

The same effect separates the projective plane’s gaps from the sphere’s. On the sphere, any cage at all will do, and past one hexagon there is always at least one. On the projective plane a cage has to be special, and at twenty-four and twenty-eight atoms there are only three cages to choose from.

What the count does not settle #

Past forty-four atoms. The census covers hexagon counts up to six on the projective plane, and after the two gaps every count has a net. Whether that continues for every larger count is not settled here. Some larger counts are settled one at a time: the cage of sixty atoms built by truncating the dodecahedron’s dual halves, since its antipodal map fixes nothing, into thirty atoms, forty-five bonds and sixteen faces — six pentagons and ten hexagons — and the cage of a hundred and eighty atoms built the same way halves into six pentagons and forty hexagons. Both are checked, and neither is part of the census. A construction producing a cage with a centre for every even number of hexagons from six up would settle the whole question; nothing above is such a construction.

The rows below characteristic one. A closed surface of characteristic minus one charges minus six, which a net of pentagons and hexagons cannot pay: a heptagon pays for a pentagon on the torus, and below the torus heptagons are compulsory. Whether every face vector permitted there is realised is the same question on a surface the sphere does not cover twice, and the lift used here is special to the projective plane.

Other numbers of bonds at an atom. Everything above has three bonds meeting at every atom. The charge depends on that number — four bonds charge the sphere eight triangles and the projective plane four — and the census of cages, and so of their centres, would have to be redone for each.

A chemical cage. The projective plane cannot sit in space without passing through itself, so a halved cage is a combinatorial object and not a molecule. What the census says about molecules is only the part that concerns the sphere: which cages have a centre.

What was computed, and how. Every string of twelve pentagons and up to twelve hexagons, wound up; the cages that close, identified by their least code; the counts, compared with the published ones; every cage’s symmetries as permutations of its faces, each checked to carry neighbours to neighbours; among them the involutions that reverse orientation, each tested for fixed faces, fixed bonds and fixed atoms; and the free ones, sorted up to relabelling by the cage’s own symmetries. Two inputs are refused: the thirteen strings with a single hexagon, none of which closes, and a string with the right faces in an order that cannot close.

Who counted the cages #

Branko Grünbaum and Theodore Motzkin proved in 1963 that a convex solid with three edges at every corner and faces that are pentagons and hexagons exists for every number of hexagons except one, which is the sphere’s row above with its single gap, established long before anyone was looking for such cages in a mass spectrometer.

The spiral is David Manolopoulos’s, with John May and Stephen Down, from 1991, and it became the standard way of naming and counting fullerene isomers: An Atlas of Fullerenes, by Patrick Fowler and Manolopoulos in 1995, lists the isomers by their spirals. The flat drawing is W. T. Tutte’s of 1963, and the theorem that every symmetry group of such a net is realised by a convex solid is Peter Mani’s of 1971.

The projective plane’s row does not seem to have been of interest to anybody who counted cages, because a molecule cannot be one; and the question of which surfaces carry which nets was being asked by topologists who were not counting cages. The lift is where the two meet.

Where this goes: whether the projective plane has another gap #

Two gaps appear in the first seven counts and none after them. The obvious conjecture is that there are no more, and the obvious obstacle to checking it by the method above is growth: the number of strings to wind up more than doubles with every two atoms, and the cages with a centre stay a handful while the cages without one run into hundreds. A proof would not count. It would build, for every number of hexagons from three upwards, one cage with a centre — and the natural candidates are tubes capped at both ends by halves of one symmetric cap, lengthened a ring at a time, which is a construction this essay has not carried out.