Proved by Hales (1998): hexagonal packing is densest — π/√12 ≈ 90.69%
Packing Mode:Square
Density:78.54%
Theoretical max:90.69%
Circles placed:0
Key Results
Square lattice: π/4 ≈ 78.54% Hex lattice: π/(2√3) ≈ 90.69% Kepler (1611): conjectured hex is optimal Gauss (1831): proved for lattice packings Hales–Ferguson (1998): computer-assisted proof for all packings Formal proof (2014): Flyspeck project verified it
3D Kepler Conjecture
In 3D: FCC/HCP lattice packs spheres at π/(3√2) ≈ 74.05%. Kepler (1611) conjectured this is optimal. Proved by Hales (1998) — the proof was 250 pages + 3GB of computer code!