The 23-Page Proof That Solved Sphere Packing in 8 Dimensions

Maryna Viazovska posted her paper to arXiv on March 14, 2016. Twenty-three pages. The problem it closed had been open since at least 1611, when Kepler guessed how cannonballs stack densest in three dimensions. Thomas Hales finally proved the 3D case in 1998, but his argument ran 250 pages and leaned on heavy computer search. The 8-dimensional version had been considered hopelessly out of reach.

Viazovska's idea was to find a single "magic function" — a function whose Fourier behavior would force the E8 lattice to be optimal. Henry Cohn and Noam Elkies had set up this approach back in 2003, but nobody knew how to construct the function. Viazovska built it out of modular forms, an ancient corner of number theory that nobody had thought to point at sphere packing.

Peter Sarnak read it and said it was "stunningly simple, as all great things are." Hales, the man who had ground out the 3D case, said she had "pulled a Ramanujan."

A week later, on March 21, Viazovska and four collaborators — Cohn, Abhinav Kumar, Stephen Miller, Danylo Radchenko — posted a 12-page sequel. Same method, dimension 24, the Leech lattice. Two of mathematics' famous lattices, two papers, eight days apart.

Dimensions 8 and 24 are special; nothing similar is expected to work in dimension 5 or 17. But Viazovska had shown the locked door was the right shape after all. In 2022 she became the second woman to win the Fields Medal, and the first laureate with a degree from a Ukrainian university.