spherepacking

Sphere packing is the arrangement of non-overlapping equal spheres within a Euclidean space with the goal of maximizing the fraction of space covered by the spheres, known as the packing density. The problem is posed in various dimensions and has both theoretical and practical significance in coding theory, crystallography, and communications.

In three dimensions, the Kepler conjecture asserts that the densest possible packing density is π/√18 ≈ 0.74048, achieved by the face-centered cubic (fcc) and hexagonal close packing (hcp) arrangements. The conjecture was proven rigorously in the late 20th and early 21st centuries by Thomas Hales and collaborators, with a formal proof completed in 2014.

In higher dimensions, results vary. In eight dimensions, the E8 lattice yields the densest known packing, and in twenty-four dimensions, the Leech lattice provides the densest known lattice packing. Landmark results by independent researchers established the optimality of these packings in their respective dimensions, representing notable advances where complete general proofs exist only in those specific dimensions.

In four dimensions, the D4 lattice gives the densest known lattice packing, but a general proof of global optimality remains open. In many other dimensions, neither the exact maximum density nor the optimal arrangement is known.

Techniques for establishing bounds on packing density include linear programming bounds and the use of spherical designs, with substantial computer-aided verification contributing to the Kepler proof and to modern results in special dimensions.

Sphere packing also has applications in digital communications, where packing efficiency relates to error-correcting codes and channel capacity, as well as in materials science for modeling crystal structures.