Sphere packings

The 3D face-centered cubic (fcc) packing shown in the main text has presumably been known since antiquity, and has been used extensively for packing fruit, cannon balls, etc. It fills space with a density π/Sqrt[18] ≃ 0.74, which Johannes Kepler suggested in 1609 might be the maximum possible. This was proved for periodic packings by Carl Friedrich Gauss in 1831, and for any packing by Thomas Hales in 1998. (By offsetting successive layers hexagonal close packing (hcp) can be obtained; this has the same density as fcc, but has a trapezoid-rhombic dodecahedron Voronoi diagram—see note below and page 929—rather than an ordinary rhombic dodecahedron.)

Random packings of spheres typically have densities around 0.64 (compared to 0.74 for fcc). Many of their large pores appear to be associated with poor packing of tetrahedral clusters of 4 spheres. (Note that individual such clusters—as well as for example 13-sphere approximate icosahedra—represent locally dense packings.)

It is common for shaking to cause granular materials (such as coffee or sand grains) to settle and pack at least a few percent better. Larger objects normally come to the top (as with mixed nuts, popcorn or pebbles and sand), essentially because the smaller ones more easily fall through interstices.