Sphere packing

Summary

A sphere packing is an arrangement of balls in space that do not overlap. The classical question asks for the largest fraction of space that a packing of equal balls can fill, its packing density. The answer is known in dimensions $1$, $2$, $3$, $8$ and $24$. In the plane it is the hexagonal packing, and in space the face-centred cubic packing of the Kepler conjecture. In dimensions $8$ and $24$ it is given by the E8 lattice and the Leech lattice. Sphere packings are linked with error-correcting codes, lattices, number theory and crystallography.