Packings

Packings are arrangements of non-overlapping objects placed inside a space with the aim of using as much space as possible. When the objects are congruent, the problem is called a packing problem; when the centers lie on a regular grid, the packing is called a lattice packing. A key measure is the density, the proportion of space occupied by the objects.

In the plane, circles arranged in a hexagonal lattice achieve the maximal density π/√12 ≈ 0.9069; this is the optimal circle packing in the plane.

In three dimensions, the densest arrangement of equal spheres is the face-centered cubic and hexagonal close-packed structures, with density π/√18 ≈ 0.74048. The Kepler conjecture, proposed in 1611, asserted optimality among all packings; proven by Thomas Hales and collaborators in 1998 with a computer-assisted proof, subsequently formalized in 2014.

In eight and twenty-four dimensions, the densest known packings arise from the E8 lattice and the Leech lattice, and these have been shown to be optimal. The proofs were completed by Viazovska and collaborators (eight dimensions, 2016) and by Cohn, Kumar, Miller, Radchenko, and Viazovska (24 dimensions, 2017).

Beyond equal spheres, packings consider unequal sizes and irregular shapes; related problems include coverings and tilings. Packings appear in coding theory as sphere packings in high-dimensional metric spaces, relating to error-correcting codes. Applications include crystallography, materials science, logistics, and data transmission.