Sphere Packing — Kepler Conjecture

Packing Type

Lattice: View layer: 0 Sphere radius: 1.0 Rotation: 30°

Packing Density

FCC (= HCP)π/(3√2) ≈ 74.05%

BCCπ√3/8 ≈ 68.02%

Simple Cubicπ/6 ≈ 52.36%

Random (Bernal)≈ 64%

Current lattice74.05%

Kepler (1611) conjectured FCC/HCP achieve the maximum packing density π/(3√2) ≈ 74.05%. Hales (1998) proved it in a 250-page proof + 3GB of computer verification (Flyspeck project fully formalized in Lean 2014). In dimensions 8 and 24, the E8 and Leech lattices achieve even higher-efficiency packings (Viazovska 2017, Fields Medal 2022).