Sphere Packing - Hypersphere Packing

Hypersphere Packing

The sphere packing problem is the three-dimensional version of a class of ball-packing problems in arbitrary dimensions. In two dimensions, the equivalent problem is packing circles on a plane.

In dimensions higher than three, the densest regular packings of hyperspheres are known up to 8 dimensions. Very little is known about irregular hypersphere packings; it is possible that in some dimensions the densest packing may be irregular. Some support for this conjecture comes from the fact that in certain dimensions (e.g. 10) the densest known irregular packing is denser than the densest known regular packing.

Dimension 24 is special due to the existence of the Leech lattice, which has the best kissing number and is the densest lattice packing. No better irregular packing is known, and at best an irregular packing could improve over the Leech lattice packing by only 2×10−30.

Another line of research in high dimensions is trying to find asymptotic bounds for the density of the densest packings. Currently the best known result is that there exists a lattice in dimension n with density bigger or equal to for some number c.