How much of space can you fill with equal spheres? Kepler guessed the answer in 1611 and it took almost four centuries to prove — and in every dimension except 1, 2, 3, 8 and 24 nobody knows to this day. Follow the question through 42 interactive demonstrations: pour thousands of spheres into a box, orbit the 240 roots of E₈, watch a random ellipsoid inflate its way to a 2025 record, and see an AI move a bound that had not shifted since 1978.
See also: Lie Groups for the root systems behind E₈, Information Theory for the codes that are secretly packings, and Fourier Analysis for the transforms that certify the upper bounds.
How much of space can spheres fill? Start flat, with circles on a page
Stack cannonballs, pour spheres into a box, and watch order beat disorder
How many spheres can touch one sphere? Newton said twelve, Gregory said thirteen
Build your own lattice, watch its Voronoi cells, and hunt for the densest one
Volume collapses, corners grow spikes, and intuition stops working entirely
Two impossibly perfect arrangements in dimensions 8 and 24 — and the proofs that they win
Every error-correcting code is a sphere packing — and the Golay code builds Leech
Proving no packing can beat a number — with a magic function and its Fourier transform
Klartag's 2025 breakthrough — a random ellipsoid grows inside a lattice and breaks a 78-year record
An AI moves a bound untouched since 1978 — and what is still wide open