This tool builds three families of packings on a canvas: exact 2D lattices (square, hexagonal), a physically simulated random pack (gravity plus positional collision-relaxation until the pile jams), and projected 3D lattices (simple cubic, BCC, FCC, HCP) rendered with depth-sorted, shaded spheres. Each mode measures or reports the real packing fraction — the proportion of space filled by the circles or spheres — so you can directly compare it against the mathematically proven optimum for that lattice type.
How packing efficiency depends on arrangement: hexagonal is the densest possible circle packing in 2D (≈90.69%, Thue/Fejes Tóth), while FCC and HCP are the densest sphere packings in 3D (≈74.05%, the proven Kepler conjecture) — far above simple cubic's ≈52.36%.
Choose a Mode (2D lattice, Random pack, or 3D lattices), pick a lattice type from the button row, drag the Circle/sphere count slider, and click Drop & relax to regenerate or run the physics-based jamming simulation; toggle Show contact neighbours to see which circles are actually touching.
Random close packing of spheres reliably jams at about 64% density — well below the 74.05% of an ordered FCC/HCP lattice — because disordered piles get geometrically stuck before reaching the crystalline optimum, a gap that puzzled physicists for decades.
The hexagonal lattice, where each circle touches 6 neighbours, fills about 90.69% of the plane — proven to be the mathematical optimum for equal circles by Thue and later Fejes Tóth.
It states that no arrangement of equal spheres in 3D space can exceed the packing density of FCC or HCP, about 74.05% (π/3√2); proposed by Johannes Kepler in 1611, it was finally proven rigorously in a computer-assisted proof completed in 2014.
Both stack identical close-packed layers of spheres, each achieving the same local density and kissing number of 12, but they differ in the third-layer stacking sequence (ABCABC for FCC vs ABAB for HCP), which changes the crystal symmetry without changing the overall packing fraction.
It's the number of neighbouring circles or spheres that touch a given one in the packing — 6 for a 2D hexagonal lattice and 12 for 3D FCC/HCP, versus only 4 for a square lattice or 6 for simple cubic.
As circles fall and collide under gravity, they lock into disordered local configurations that geometrically block further compaction well before reaching crystalline order, which is why random close packing typically settles around 82-84% in 2D or roughly 64% for spheres in 3D — both short of the ordered lattice maximum.