Hexagonal Sphere Packing

Kepler’s conjecture: the densest packing of equal spheres.

Packing density 74.05%

Theoretical max 74.05%

Circles shown 0

Coordination # 6

Circle radius 20 px

Gap between circles 0 px

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Kepler’s conjecture (1611) asked: what is the densest way to pack equal spheres? The cannonball stacking used by grocers for oranges looked optimal, but proving it took nearly four centuries.

Hexagonal close packing (HCP) and face-centred cubic (FCC) both achieve a density of π / (3√2) ≈ 74.05% — meaning 74.05% of space is filled with sphere, and 25.95% is void. In 2D, hexagonal circle packing reaches π / (2√3) ≈ 90.69%.

Each sphere touches exactly 12 neighbours in 3D HCP/FCC. In 2D hexagonal packing, each circle touches exactly 6.

Thomas Hales proved the conjecture in 1998 using a computer-assisted proof of 250+ cases. The result was formally verified in 2014 — the Flyspeck project.

Hexagonal packing appears in honeycomb cells, bubble rafts, colloidal crystals, and the Wigner-Seitz cells of metals.