Sphere and Circle Packing: How Densely Can Space Be Filled?

The question behind every crystal and every crate of oranges

Packing fraction φ is simply the volume (or area) taken up by the objects divided by the total volume of the space they fill. It sounds like a puzzle-book curiosity, but it governs how atoms arrange themselves into metal crystals, how tightly a greengrocer can stack oranges, and how densely powders and granular materials can be poured. The answer depends entirely on whether the packing is ordered (a repeating lattice) or disordered (poured in randomly), and the gap between the two is large and well measured.

2D: square packing loses to hexagonal

Lay equal circles on a square grid, each touching its four neighbours, and they cover π/4 ≈ 0.7854 of the plane — the rest is the four little curved gaps at every grid cell's corners. Shift alternate rows so each circle nests into the notch between two circles below it (hexagonal packing, six neighbours per circle) and the fraction rises to π/(2√3) ≈ 0.9069. Hexagonal packing is provably optimal for the plane — proved by Axel Thue in 1892 and rigorously by László Fejes Tóth in 1940 — which is also why bees build honeycomb from hexagonal cells rather than squares: it is the most material-efficient way to tile a plane with equal-size cells.

3D: five packings worth knowing

Simple cubic (SC)             φ = π/6            ≈ 0.5236   (6 neighbours)
Body-centred cubic (BCC)      φ = π√3/8         ≈ 0.6802   (8 neighbours)
Random close packing (RCP)    φ                    ≈ 0.64     (disordered, jammed)
Face-centred cubic (FCC)      φ = π/(3√2)       ≈ 0.7405   (12 neighbours)
Hexagonal close-packed (HCP)  φ = π/(3√2)       ≈ 0.7405   (12 neighbours)

FCC and HCP tie for the densest lattice packing, and the tie is not a coincidence — both are built by stacking hexagonal layers of spheres, each new layer nestling into the hollows of the one below; the only difference is whether the third layer repeats the position of the first (HCP, sequence ABAB…) or offsets it again (FCC, sequence ABCABC…). Many real metals — copper, aluminium and gold form FCC crystals; magnesium and zinc form HCP — sit at exactly this density for the same geometric reason a fruit seller stacks oranges in a pyramid.

The Kepler conjecture

Johannes Kepler conjectured in 1611 that no arrangement of equal spheres — ordered or disordered — can exceed π/(3√2) ≈ 0.7405, the FCC/HCP density. It looks obvious from a stack of cannonballs, but proving that no cleverer, irregular arrangement could possibly do better resisted rigorous proof for almost four centuries. Thomas Hales announced a proof in 1998 using exhaustive computer verification of thousands of geometric cases; the community's difficulty checking such a large case analysis by hand led Hales to launch the Flyspeck project, which formally verified every step in a proof assistant and confirmed the result beyond doubt in 2014.

Random packing never reaches the lattice maximum

Pour identical spheres into a container and shake them and they jam at a packing fraction of roughly 0.64 — random close packing — well short of the 0.7405 ordered maximum. The gap exists because disordered jamming gets trapped in a local mechanical equilibrium: no small rearrangement can squeeze the spheres tighter without first pulling some of them apart to make room, and physically shaking a real container rarely provides enough energy to escape that local trap and reorganise into a crystal. This 0.64 figure recurs constantly in materials science, from powder metallurgy to how tightly ball bearings settle in a bag.

Kissing numbers

A related question asks not how much space is filled but how many neighbours a single sphere can simultaneously touch — the kissing number. In 2D it is exactly 6, easy to see and to prove. In 3D it is 12, achieved by FCC and HCP alike, but proving that a 13th sphere cannot be squeezed in took until 1953: 12 touching spheres leave just enough slack room around the central sphere that the impossibility of a 13th is genuinely subtle, unlike the tight 2D case.

Frequently asked questions

Try it live

Everything above runs in your browser — open Sphere and Circle Packing and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.