Kepler

In 1591 Walter Raleigh asked his assistant Thomas Harriot how many cannonballs sat in the stacks on his ship decks. Harriot answered with a formula, then wrote to Johannes Kepler with the harder question underneath it: is the stack the best possible way to pack spheres?

Kepler answered in 1611, in a New Year's pamphlet on six-cornered snowflakes: pack a layer in a honeycomb, nest the next layer in its hollows, repeat — and no arrangement in the universe does better. Every grocer piling oranges already knew the move. Nobody could prove it.

η = π / √18 = 0.740480…

That number is the fraction of space the spheres fill. Just under three quarters. The other 25.95 per cent is air, and it is irreducible.

Gauss settled the easy half in 1831: among regular lattices, face-centred cubic wins. The hard half — that no irregular, disordered, adversarial arrangement sneaks past 0.7405 — resisted everyone. Hilbert put it on his list of the century's problems in 1900.

In 1998 Thomas Hales and Samuel Ferguson finished it: 250 pages of mathematics driving three gigabytes of code that checked thousands of sphere configurations by exhaustion. The referees at the Annals worked for four years and declared themselves “99 per cent certain” — the proof was too large for humans to fully audit.

So Hales removed the humans. The Flyspeck project rebuilt the entire argument inside formal proof assistants, every inference machine-checked back to the axioms. It finished on 10 August 2014. The greengrocer was right all along.

The crate above stacks spheres in face-centred cubic order; the sack beside it takes the same spheres dumped and jostled. The gap between their two readouts is the whole theorem.

Arrangement	Structure	Density η
Simple cubic	spheres on a grid, π/6	0.5236
Random loose pour	dropped, no shaking	≈ 0.55–0.60
Random close pack	shaken and jammed	≈ 0.64
Body-centred cubic	π√3/8	0.6802
Face-centred cubic	the grocer's pyramid, π/√18	0.7405

Watch the copper plane sweep the crate: every layer is a honeycomb, and each layer sits in the hollows of the one below. A honeycomb has two families of hollows, so after layers A and B you choose: return over A — that repetition, ABAB, is hexagonal close packing — or take the third position, C. The cycle ABCABC is face-centred cubic.

Both choices score exactly π/√18. Kepler's conjecture doesn't crown one crystal; it crowns the move — layer, nest, repeat.