The room a thirteenth sphere would need

Two stackings, one density counts the neighbours a sphere has in a close packing and finds twelve, twice, in two different arrangements. This essay asks the question that count invites and does not answer: whether twelve is all there is room for.

Newton thought it was. In 1694 David Gregory thought a thirteenth might fit, the two disagreed in writing, and the disagreement stood unresolved until Schütte and van der Waerden settled it in Newton’s favour in 1953. Two hundred and fifty-nine years is a long time for a question about balls, and the length of it is the interesting part.

The question with the spheres taken out #

The first move is to remove the spheres. A sphere touching a central one touches it at a single point, and that point is a direction from the centre. Two neighbours at directions separated by an angle θ have centres at distance 2 sin(θ/2) from each other, in units of the sphere radius, and they overlap exactly when that is less than 2 — which is exactly when θ is less than 60°.

So the question is not about spheres at all. It is: how many points can be put on a sphere with every pair at least 60° apart? The general form — put n points on a sphere and make the smallest angle between any two as large as possible — is Tammes’s problem, and writing θ(n) for the answer, the kissing number is the largest n for which θ(n) ≥ 60°.

That restatement is the whole of the mathematical content, and it is worth noticing what it throws away. The spheres had radii and volumes and could have been packed against each other in space; the points have only directions. Everything about the problem that sounded three-dimensional is gone, and what is left is a question about a two-dimensional surface. The kissing number is not really a fact about spheres. It is a fact about how much of a sphere’s surface a 60° cap covers, and how badly such caps tile.

The naive count is worth running, because it is what makes thirteen look plausible. A spherical cap of angular radius 30° — half of 60°, so that two such caps just touch when their centres are 60° apart — covers a fraction (1 − cos 30°)/2 of the sphere, which is about 6.7%. Fourteen of them would cover 93.8% and fifteen would overflow. So counting area alone permits fourteen, and the true answer is twelve. The gap between what the area allows and what the geometry allows is where two and a half centuries went.

It is worth doing the same count in the plane, because there it comes out right. A 30° arc on a circle is a twelfth of it, so six such arcs fill the circle exactly and a seventh has nowhere to go — the area count and the true answer agree, and they agree because arcs on a circle are intervals and intervals tile it without waste. Caps on a sphere do not tile it; three mutually touching caps leave a curved triangle between them that belongs to no cap, and it is those leftovers, summed over the whole surface, that account for the difference between fourteen and twelve. The failure of the naive count is exactly the failure of caps to tile.

What a search can and cannot say #

The search here anneals a repulsion energy and then hill-climbs on the minimum angle itself. It is worth being exact about what that produces, because the temptation is to over-read it.

A search of this kind produces a lower bound on θ(n). It finds an arrangement, measures it, and reports the number. Nothing it does rules out a better arrangement it failed to find. So the search can confirm that twelve points fit — it exhibits twelve at 63.43° apart, which is comfortably past 60° — and it cannot prove that thirteen do not. It can only report that it did not manage it.

That makes calibration the important part. Three of the small cases have exact answers that come from regular polyhedra: four points at the vertices of a tetrahedron, arccos(−1/3) or 109.47° apart; six at the vertices of an octahedron, 90° apart; twelve at the vertices of an icosahedron, arccos(1/√5) or 63.4349° apart. Each of those is computed here from the polyhedron itself, and the search is then run at the same n and its answer compared. It reproduces all three to within five thousandths of a degree, and it overshoots none of them.

The direction of the error matters as much as its size. A search that came back above a proved optimum would not have found something; it would have a bug, because the optimum is the maximum and nothing exceeds it. So an overshoot is a defect detector rather than a discovery, and it is one of the things this essay’s machinery is required to refuse.

With that calibration in hand the thirteen-point result can be read. The search finds thirteen points at 56.66° — not close to 60°, short by more than three degrees, which is more than half the slack that twelve enjoy. The value proved optimal in 2012 is 57.1367°, so the search falls about half a degree short of the truth even here, and the shortfall has a cause worth knowing: the optimal thirteen-point arrangement has a rattler, a point that no minimal pair involves and which is therefore free to move without changing the score. That makes the landscape flat around the optimum, and a hill-climb on a flat landscape stops.

One more property of the search is worth stating because it is what makes the table monotone. Any arrangement of n + 1 points contains an arrangement of n points, obtained by deleting one, and deleting a point cannot bring the survivors closer together. So θ(n) is non-increasing, and a table that showed it rising anywhere would be reporting a failed search at the smaller count rather than a discovery at the larger. That is checked here across the whole range, which turns eleven entries into eleven cross-checks on each other rather than eleven independent measurements.

The shape of that curve says something the single answer does not. Eleven and twelve have essentially the same optimum, because the eleven-point optimum is the icosahedral twelve with one vertex deleted, and deleting a vertex from an optimal arrangement cannot make the remaining points closer. Then thirteen loses almost seven degrees at once. The sequence is not a smooth decline that happens to cross 60° between twelve and thirteen; it is flat and then it collapses, and the collapse is what makes twelve a natural stopping point rather than an arbitrary one.

The slack that kept the question open #

The reason the disagreement was possible is measurable, and it is the difference between two arrangements of the same twelve points.

In a close packing the twelve contacts are at exactly 60° from their nearest partners, and there are twenty-four such pairs, forming the edges of a cuboctahedron. Exactly 60° means exactly touching: every one of those twenty-four pairs of neighbours is in contact with the other, and none of them can move at all without pushing something. That arrangement is rigid and it is completely full.

The icosahedral arrangement is not. Its twelve points are 63.4349° apart at their closest, in thirty pairs, and 63.4349° is 3.4349° more than the 60° required. Every neighbour has room. Rotate one and it does not disturb the others; the whole shell can be jostled, and the twelve spheres in it rattle. That slack is not an artefact of choosing the icosahedron — it is the most separated twelve points there are, so 3.4349° is the largest slack any twelve can have.

That is the room Gregory was counting on, and it is why no picture settles the question. A drawing of twelve spheres round one shows visible gaps, and the eye reads a gap as a place something might go. Adding the gaps up gives an area that would hold two more spheres, and adding them up is exactly the wrong operation, because the free area is in twelve separate slivers and a thirteenth sphere needs it in one piece. The impossibility is a statement about how the spare room is distributed, not about how much of it there is.

There is a second thing the two arrangements say, and it is the more surprising one. The icosahedral twelve are the better spherical code and the close-packed twelve are the better packing. The best local arrangement is not the one that extends: an icosahedron has five-fold symmetry, and the crystallographic restriction forbids a lattice from having it, so the shell that keeps its neighbours furthest apart cannot be repeated periodically. Local optimality and global optimality want different things, which is the same lesson covering and packing want different lattices draws from three other quantities.

It is worth being precise about what “rattle” means, because it is the physical form of the slack. Twelve spheres in the icosahedral shell each touch the central one and none touches another; the shell has twelve independent degrees of freedom in the tangential directions, bounded only by the 3.4349° of clearance. Twelve spheres in the close-packed shell touch the central one and each touches four others, and the contact network is rigid. So the two arrangements differ not only in an angle but in whether the outer shell is a mechanism or a structure — and the first is a shell somebody could plausibly hope to squeeze another sphere into, which is what makes the historical disagreement a reasonable one rather than a blunder.

The plane, where nobody argued #

The contrast that explains the two and a half centuries is with the same question one dimension down.

In the plane the argument is one line. Put n points on a circle; they cut it into n arcs whose lengths sum to 360°, so the smallest arc is at most 360/n, and the regular arrangement makes every arc equal and attains the bound. So θ₂(n) = 360/n exactly, θ₂(6) = 60°, θ₂(7) = 51.43°, and six circles touch one with nothing to spare.

Nothing about that argument survives to the sphere. It works because the arcs between consecutive points partition the circle and their lengths add up, so the smallest is at most the average. On a sphere the analogous regions — the cells of the points, in the sense the cell nobody chose builds them — also partition the surface and their areas also add up, but the quantity being bounded is an angle rather than an area, and no relation forces a point with a large cell to be far from its neighbours. The averaging argument has nothing to average.

The plane’s ease extends past the local question to the global one, which is worth noting because in space the two came apart. The densest lattice in the plane shows the hexagonal lattice is the densest planar lattice packing, and its kissing number is the same six the local argument gives — the arrangement that is best around one circle is also the arrangement that repeats. In space the two answers are the icosahedral twelve and the close-packed twelve, and they are different arrangements of the same count. So the plane is not merely easier to prove things about; it is a case where fewer distinct things are true.

That is the whole difference. The plane’s answer is forced by a conservation law and the sphere’s is not, and a question that no conservation law settles is a question that has to be settled case by case. Schütte and van der Waerden’s proof, and every proof since, works by an exhaustive analysis of how thirteen caps could be arranged, which is the kind of argument that only becomes possible when somebody organises the cases.

There is a partial substitute for the averaging argument, and knowing what it achieves is knowing why it is not enough. Modern proofs bound the kissing number using functions on the sphere with prescribed positivity — the linear programming bound of Delsarte and its semidefinite refinements — which is a genuine analogue of the averaging step: a quantity is summed over all pairs and bounded two ways. In three dimensions that method gives 13 rather than 12, which is close and not close enough, and closing the last unit takes a separate geometric argument. In dimensions eight and twenty-four the same method gives exactly the right answer, which is the reason those two dimensions are the ones where everything is known. So the sphere is not beyond averaging; it is beyond averaging at the precision this particular question needs.

What it says about a structure #

Three things, and the third is the one that shows up in a real diffraction pattern.

The zeroth thing, before those, is that the count is exactly the first coefficient of a series this collection already computes. How many vectors of each length reads a lattice’s theta series off its Gram matrix, and the first non-zero coefficient is the number of vectors at minimum distance — the kissing number of that lattice. So for a lattice packing the count needs no geometry at all: it is an enumeration inside a radius, and the face-centred cubic answer of twelve falls out of the same shell count that produces every other coefficient. What this essay adds is the part the series cannot see, which is whether a non-lattice arrangement could do better.

The first is that a coordination number of twelve is a ceiling and not a coincidence. Metallic structures reach twelve, and the reason no metal has a coordination number of thirteen with equal atoms is not chemistry — it is that thirteen contacts cannot be made at all. Any reported coordination number above twelve is a statement about atoms of different sizes, and the arithmetic here says so before the structure is looked at.

The second is that the ceiling depends on the radius ratio, immediately and steeply. The twelve is for equal spheres; a smaller central atom admits fewer neighbours and a larger one admits more, and the count is set by the same spherical-code question with the cap radius changed. So a coordination number is a measurement of a ratio as much as of a structure, which is what a net is a choice of what counts as a bond says from the other direction.

The third is the icosahedral shell itself. It is the best local arrangement and it cannot tile space, so a liquid or a glass of equal spheres, which is free to take the local optimum everywhere and never has to repeat, is under pressure to form icosahedral clusters — and that frustration between the best local order and the possible global order is the standard account of why metallic liquids resist crystallising, and part of why the most of an icosahedron a crystal can keep is a question worth asking at all. The 3.4349° of slack is where that account starts.

The second of those is the one doing structural work. A search that reported an angle above a proved optimum would be reporting a bug, and there is no other way for that to happen — so it is a test the machinery can fail for exactly one reason, which is the most useful kind. The sixth is a guard against a different error: body-centred cubic touches eight, so twelve is not what a kissing number is, it is what this particular arrangement achieves.

Where this stops #

The impossibility is not proved here and cannot be. Everything above is a search, and a search that fails to find thirteen points at 60° is evidence about the search as much as about the geometry. The proof is Schütte and van der Waerden’s, and the modern short one is Musin’s; both work by bounding what a hypothetical thirteen-point arrangement would have to look like and eliminating the possibilities, which is a different kind of work from anything computed here.

Nor is the exact value of θ(13) reached. The search gets 56.66° against the 57.1367° proved optimal in 2012, and that essay’s worth of a gap is caused by a single free point in the optimal arrangement. It is reported rather than hidden because the size of the gap is the honest measure of how much harder a max-min problem is than the count it decides — the count is settled to certainty and the angle is not.

And the dimensions above three are barely touched. The kissing number is known in dimensions one, two, three, four, eight and twenty-four and in no others, with dimension four settled only in 2003 and dimensions eight and twenty-four following from the same exceptional lattices that settle the packing problem there. Between them the gap between the best known arrangement and the best known bound is wide. A lattice cannot have all its vectors long meets the same wall from the other side: the quantities are easy to define, easy to bound crudely, and exactly known almost nowhere.