# Sphere packing in dimensions 8 and 24

- kind: problem
- status: solved (evidence tier V3)
- year: 1967

**Statement.** The E8 lattice and the Leech lattice are the densest sphere packings in dimensions 8 and 24.

Cohn and Elkies had reduced the problem to finding one special function. Viazovska built it from modular forms, and a week later the same idea settled dimension 24. She received the Fields Medal in 2022. The optimal packing in every other dimension above 3 is open.

*Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.*

Closed 2016, Maryna Viazovska; dimension 24 with Cohn, Kumar, Miller and Radchenko.

## Sources

- [Wikipedia: Sphere packing](https://en.wikipedia.org/wiki/Sphere_packing)

[Geometry and topology](/wiki/Space:geometry-topology.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations to this page

- related to [kepler-conjecture](/wiki/Problem:kepler-conjecture.md) — The original curation links this problem to Sphere packing in dimensions 8 and 24; the argument for the link has not been written yet.

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Revision 28, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Imported from content/01-mathematics.json, curated for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (September 2026).. History: /wiki/Problem:sphere-packing-8-24?action=history. Edit: POST /api/wiki/page/Problem:sphere-packing-8-24 {source, summary, base_rev: 28}.