Klartag's 2025 breakthrough — a random ellipsoid grows inside a lattice and breaks a 78-year record
In 1905 Hermann Minkowski proved that in every dimension n there exists a lattice packing of density about 2⁻ⁿ. The proof is a counting argument: average over a family of lattices, show the average is good enough, conclude that at least one member must be. It produces no lattice. It names no construction. It simply rules out the possibility that every lattice is bad.
Forty-two years later Claude Ambrose Rogers found something better. He took a lattice and grew an ellipsoid inside it, stopping when lattice points blocked further growth, and read a packing off the result. The gain was a factor of n — density c·n·2⁻ⁿ instead of 2⁻ⁿ. In 1992 Keith Ball sharpened the argument and improved the constant, but the shape of the bound, c·n·2⁻ⁿ, did not move. For 78 years nobody could win another factor of anything.
Then in 2025 Bo’az Klartag — a convex geometer who had never worked on sphere packing — went back to the abandoned ellipsoid method and changed one thing. Instead of growing the ellipsoid along a fixed schedule, he let it grow in random directions, freezing each direction the moment a lattice point landed on the boundary. That single edit bought a second factor of n: c·n²·2⁻ⁿ, the largest improvement to the lower bound since 1947. In dimension d it packs roughly d times more spheres than any earlier method — about a hundredfold in 100 dimensions, a millionfold in a million.
Everything that follows rests on one observation, and it is worth slowing down for because it is the reason growing a blob has anything to do with packing balls. Take a symmetric convex body E centred at the origin, and suppose no nonzero lattice point lies inside it. Then the translates of E/2, one centred at each lattice point, never overlap.
The reason is almost a triviality. Two copies of E/2 centred at u and w meet exactly when the difference u − w lies in E. When u and w are both lattice points, that difference is itself a lattice point. So overlapping translates and interior lattice points are not two related conditions — they are the same condition, stated twice. An empty interior is a packing certificate, and the only remaining job is to make E as large as it can possibly be.
Drag either violet handle to stretch and turn E. The green ovals are the translates of E/2, one at every lattice point — half the size of the dashed body, and disjoint for exactly as long as its interior stays empty. Push one lattice point inside and the pair of translates whose centres differ by that point turns red and visibly overlaps. Nothing about this is approximate: two translates meet precisely when their centres differ by a point of E. Inflating the circle to its limit reaches π/4 ≈ 78.54%, the density Rogers’ deterministic method starts from; the best empty ellipse of all manages π/√12 ≈ 90.69%, and it is not round. The whole argument from 1947 onward is about which shape to grow, and how.
Try it: Press Inflate until it touches and the starting circle grows to radius 1, resting on its four nearest lattice points at π/4 ≈ 78.54% — exactly where the deterministic method of 1947 begins. Then press Push past the limit: a point slips inside the dashed body, and the two translates whose centres differ by that point turn red and visibly bleed into each other, with the offending difference drawn as a line between their centres. Drag either handle to stretch and turn the body and inflate again to see whether your shape beats the circle, or press The best ellipse there is for the one that hits π/√12 ≈ 90.69%, the proven ceiling for the plane.
So: inflate an ellipsoid until the lattice stops it. The subtlety is that an ellipsoid does not have one size, it has n of them — a semi-axis in every direction — and they do not have to grow at the same rate. Rogers picked the growth schedule in advance. Klartag picks it by coin flip.
The process runs like this. Choose a random direction and stretch the ellipsoid along it, as far as the nearest lattice point allows. That point now rests on the boundary, and from then on the shape may only grow in directions that leave it there — the captured point is frozen into the geometry. Pick another random direction among those still free, stretch again, capture another point. Continue until no free direction remains.
In the plane this is over quickly: the first capture kills all but one direction, and the second capture ends it. Two points, and the ellipse is locked. Watch the run below and check the invariant that matters — points land on the boundary, never inside it. The final ellipse is therefore a legal E, and by the previous demo its half-scale translates are a packing.
Yellow points are the ones that stopped the growth and now rest on the boundary; the dashed line is the direction the ellipse is still free to grow in. Nothing ever ends up strictly inside — that emptiness is what makes the arrangement a packing. Run it a few times: the random direction chosen at the start changes where it lands, and it beats the circle every time.
Try it: Press Grow another ellipse several times. The only thing that changes between runs is the first random direction, and yet the final ellipse comes out differently oriented, differently stretched, and differently dense every time. Then switch on Show the packing it certifies to see the arrangement the run has just proved exists. Some runs land near the deterministic circle, some land within a whisker of the hexagonal optimum, and none ever puts a point inside.
One run proves nothing; it is luck. The argument is about the average. If the expected density of a random run exceeds what the deterministic construction gives, then some run must beat it — and that run is a genuine packing, because every run is a genuine packing. This is Minkowski’s averaging trick from 1905, applied not to a family of lattices but to a family of growth histories.
Run the plane version a few thousand times and the distribution has a striking property: the worst outcome is exactly the deterministic circle at π/4. Randomising the growth direction cannot lose — it can only decline to win. Everything above that floor is pure profit, the mean lands around 82%, and the luckiest runs come within a fiftieth of a per cent of π/√12, the proven optimum for the plane, which nothing can exceed.
Every run lands at or above the deterministic circle, and the worst possible outcome is the circle — randomising the growth direction cannot lose. In the plane the gain is a modest 1.05×. The reason the 2025 paper matters is that in dimension n the same mechanism gains a factor proportional to n: roughly a hundredfold in 100 dimensions, and a millionfold in a million.
Try it: Push the run count up and press Resample a few times. The histogram is stable, the left edge never crosses the circle’s mark, and the right edge never crosses the hexagonal optimum — the two facts that make the argument valid rather than merely encouraging. In two dimensions the average gain is a few per cent. The reason the 2025 paper is a landmark is that the same mechanism, run in dimension n, gains a factor proportional to n.
Every lower bound in this history has the same skeleton: poly(n)·2⁻ⁿ. Nobody has ever improved the exponential part — the 2⁻ⁿ is untouched since 1905 — so the whole 120-year contest has been fought over the polynomial in front of it. Minkowski: a constant. Rogers and Ball: one factor of n. Klartag: two.
That makes the improvements almost impossible to draw honestly, because against 2⁻ⁿ a factor of n² is nothing at all. Plot the raw densities and all four curves lie on top of one another, four generations of work hidden inside the width of a line. Divide the shared 2⁻ⁿ out and the real content appears: on a logarithmic dimension axis the curves become straight lines whose slope counts the powers of n — zero, one, one, two.
In dimension 1,000, and at the constants plotted here, the 2025 bound sits about 680× above the method of 1947 — that many more spheres in the same volume. The middle number is the one to trust: divide the multiplier by n and it does not move at all, whatever dimension you pick. The gain is not a better constant, it is one more whole factor of the dimension, and that is the part the theorem asserts.
What this chart is and is not. Every bound here has the form poly(n)·2⁻ⁿ, so on the raw axis they lie on top of one another: the exponential collapse dwarfs everything. Dividing 2⁻ⁿ out is what makes 120 years visible at all, and on a logarithmic dimension axis the surviving curves are straight lines whose slope counts the powers of n — 0 for Minkowski, 1 for Rogers and Ball, 2 for Klartag. That slope is a theorem. The vertical positions are not: Rogers’ constant is quoted here as 2/e, and Klartag’s theorem states its bound with an unspecified absolute constant c > 0, so the value 0.5 plotted for it is a display choice made to give the curve somewhere to sit. Read the shape of the improvement, never its absolute height. The axis also starts at n = 100 because all of these are asymptotic statements that say nothing in low dimensions.
Growing an ellipsoid inside a lattice gains a factor of n. This record stands, in essence, for 78 years.
A randomly growing ellipsoid gains another factor of n — the largest improvement to the lower bound since 1947.
Try it: Start on raw log₂ density to see what 120 years of progress looks like at the scale of the exponential — a single line, with three more hidden beneath it. Switch to 2⁻ⁿ divided out and drag along the chart to pick a dimension. The headline multiplier climbs with the dimension, but the number underneath it barely moves: divide the gain by n and you get the same constant everywhere, which is exactly what “one more factor of n” means.
Klartag’s theorem is an asymptotic statement in dimension n, proved by writing the growing ellipsoid as the solution of a stochastic differential equation and estimating its expected volume. What runs on this page is the discrete, low-dimensional process that the differential equation is a continuous limit of. It reproduces the mechanism — grow, capture, freeze, repeat — and the qualitative payoff that randomising cannot lose and usually wins. It does not reproduce the theorem, and no number on this page is a value the paper claims.
The bound itself is stated with an unspecified absolute constant, so the vertical position of the 2025 curve in the last demo is a display choice, not a result. What the chart shows honestly is the shape of the improvement: one extra power of the dimension. Read the slope, not the height.
The paper is “Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid,” published in Inventiones Mathematicae. Henry Cohn has written an expository account of the argument for readers who want the analysis rather than the animation, and Quanta covered the result in July 2025.