[Sphere packings in] higher dimensions
In no dimension above 3 is it known for certain what configuration of spheres yields the densest packing. Cases in which spheres are arranged on repetitive lattices are related to error-correcting codes and groups. Up to 8D, the densest packings of this type are known to be ones obtained by successively adding layers individually optimized in each dimension. And in fact up to 26D (with the exception of 11 through 13) all the densest packings known so far are lattices that work like this. In 8D and 24D these lattices are known to be ones in which each sphere touches the maximal number of others (240 and 196560 respectively). (In 8D the lattice also corresponds to the root vectors of the Lie group E8; in 24D it is the Leech lattice derived from a Golay code, and related to the Monster Group). In various dimensions above 10 packings in which successive layers are shifted give slightly higher densities than known lattices. In all examples found so far the densest packings can always be repetitive; most can also be highly symmetrical--though in high dimensions random lattices often do not yield much worse results.