{"id":"kepler-conjecture","domain":"named-results","type":"Theorem","title":"Kepler Conjecture","slug":"kepler-conjecture","url":"/mathematics/named-results/kepler-conjecture/","proposed":"1611, Johannes Kepler","proved":"1998, Thomas Hales (computer-assisted); 2014, Flyspeck formal verification","summary":"Cannonball stacking is the densest possible sphere packing, at pi/sqrt(18) ~= 74.05%. Proved by Hales 1998, fully formally verified 2014.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Hales, T.C. (2005). A proof of the Kepler conjecture. Annals of Mathematics, 162(3), 1065-1185."},{"tier":1,"citation":"Hales, T. et al. (2017). A formal proof of the Kepler conjecture. Forum of Mathematics, Pi, 5, e2."},{"tier":2,"citation":"Viazovska, M. (2017). The sphere packing problem in dimension 8. Annals of Mathematics, 185(3), 991-1015."},{"tier":3,"citation":"Szpiro, G.G. (2003). Kepler's Conjecture. Wiley."}],"relationships":[{"type":"related_to","target":"circle","label":"Circle — 2D packing analogue"},{"type":"similar_method","target":"four-colour-theorem","label":"Four Colour Theorem"},{"type":"referenced_in","target":"hilberts-23-problems","label":"Hilbert's 23 Problems — Problem 18"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"2.1"}