Two impossibly perfect arrangements in dimensions 8 and 24 — and the proofs that they win
Above three dimensions, sphere packing is almost entirely unsolved. Nobody knows the densest arrangement in four dimensions, or five, or any of the hundreds that follow — with two spectacular exceptions. In dimension 8 the answer is a lattice called E₈, and in dimension 24 it is the Leech lattice. Both were proven optimal in 2016, within a week of each other, after thirteen years of knowing almost exactly what the proof would have to look like and being unable to write it down.
These are not merely dimensions where someone happened to try harder. They are structurally different. The lattices that win there are so symmetric, and their distances so rigid, that there is simply nowhere left for a competitor to hide.
You can reach E₈ by being greedy. Start with points on a line, spaced to touch. Stack copies of that line as tightly as they will go and you get the hexagonal packing. Stack copies of that into its hollows and you get the grocer’s stack. Keep going — each time, take the best arrangement you have and nestle copies of it in a new direction. These are the laminated lattices, and the recipe is so naive it has no business working. It produces the optimal packing in dimensions 1, 2 and 3, and it keeps producing the best arrangement anyone knows all the way to 8.
The greedy climb lands on a lattice so symmetric it is its own dual. Proven optimal in 2016.
The bars show centre density — the density divided by the volume of the unit ball — because raw density collapses toward zero in every dimension no matter how good the lattice is, which would make any comparison across dimensions meaningless. Note that it does not climb steadily: it sags through the middle dimensions, bottoming out around K₁₂, and then rises to exactly 1 at the Leech lattice. That the summit lands on a round number is not a coincidence — Leech is unimodular, so its fundamental cell has volume 1, and its minimum distance is exactly 2.
Try it: Click through the rungs and watch the kissing number climb from 2 to 240 to 196,560. Notice that centre density does not improve steadily — it sags through the middle dimensions before rising to exactly 1 at the summit.
E₈’s shortest vectors number exactly 240, and they come in two families that look nothing alike. There are 112 with ±1 in two coordinates and zero elsewhere, and 128 with ±½ in every coordinate and an even number of minus signs. That the union of these two families is a lattice at all is the first surprise; that it is the densest packing in its dimension is the second. Each of the 240 touches the sphere at the origin, so the kissing number in dimension 8 is 240 — a fact proven back in 1979, decades before the packing problem itself fell.
Try it: The points are rotating in eight dimensions, not three — the projection down to your screen is recomputed every frame. Switch to the Coxeter plane for the famous view in which all 240 roots sort themselves into eight perfect rings of 30. Turn on the edges to see the 6,720 pairs at minimum distance, 56 through every root.
Here is what makes E₈ impossible to beat. Pick any root and measure the distance to all 240. You do not get a spread of values — you get five, all with even squared length, and the counts are the same no matter which root you started from. A packing with that much rigidity has no slack anywhere, and a proof of optimality has to exploit exactly that: the certificate function must touch zero at every distance that occurs and stay negative everywhere between.
Distances from one root to all 240, including itself. Only five values occur, every squared distance is an even integer, and the counts are identical no matter which root you measure from — drag the slider and nothing changes. The 56 roots at distance √2 are the nearest neighbours; the single root at √8 is the antipode. A function proving E₈ optimal has to touch zero at every one of these radii and stay negative between them, which is why such functions were so hard to find.
Sixteen dimensions further up sits the Leech lattice, discovered by John Leech in 1967 while he was thinking about error-correcting codes rather than geometry. Its minimal vectors number 196,560, and they are built directly out of the Golay code — 1,104 of one shape, 97,152 from the code’s 759 octads, and 98,304 from its 4,096 codewords. The packing and the code are not analogous. They are the same object.
Try it: Every one of these points is generated from the Golay code when the page loads, not read from a table. Toggle the three shapes to see how the 196,560 decomposes, and drag to orbit — the cloud itself is turning in twenty-four dimensions.
It would be tidier if 8 and 24 were simply the dimensions that got the most attention. They are not. Several unrelated-looking facts all single them out, and every one of those facts fails everywhere else.
A lattice is unimodular when its fundamental cell has volume 1, and even when every vector has even squared length. Both conditions at once is an extraordinarily rigid demand: such lattices exist only when the dimension is a multiple of 8. In dimension 8 there is exactly one, and it is E₈. In dimension 24 there are exactly 24 of them, and the Leech lattice is the only one with no vectors of squared length 2 — no "short" vectors at all — which is precisely what lets its spheres be so large.
Try it: Open each panel and compare the third column. The pattern is not that 8 and 24 are better studied — it is that the tools which work there have no counterpart anywhere else.
In 2003 Henry Cohn and Noam Elkies showed that a single well-chosen function could bound every packing in a dimension at once, and computed numerically that in dimensions 8 and 24 the perfect such function had to exist. Nobody could construct it. Thirteen years later Maryna Viazovska built the dimension-8 function explicitly out of modular forms, engineering it to vanish to second order at precisely the distances the previous demo lists. Her paper is twenty-three pages. Within a week she and four collaborators had done dimension 24. She received the Fields Medal in 2022, the second woman ever to do so. The machinery behind that certificate is worth seeing in operation.