E8 Lattice - Sphere Packings and Kissing Numbers

Sphere Packings and Kissing Numbers

The E₈ lattice is remarkable in that it gives solutions to the lattice packing problem and the kissing number problem in 8 dimensions.

The general sphere packing problem asks what is the densest way to pack n-dimensional (solid) spheres in Rn so that no two spheres overlap. Lattice packings are special types of sphere packings where the spheres are centered at the points of a lattice. Placing spheres of radius 1/√2 at the points of the E₈ lattice gives a lattice packing in R8 with a density of

It is known that this is the maximum density that can be achieved by a lattice packing in 8 dimensions. Furthermore, the E₈ lattice is the unique lattice (up to isometries and rescalings) with this density. It is conjectured this density is, in fact, optimal (even among irregular packings). Researchers have recently shown that no irregular packing density can exceed that of the E₈ lattice by a factor of more than 1 + 10−14.

The kissing number problem asks what is the maximum number of spheres of a fixed radius that can touch (or "kiss") a central sphere of the same radius. In the E₈ lattice packing mentioned above any given sphere touches 240 neighboring spheres. This is because there are 240 lattice vectors of minimum nonzero norm (the roots of the E₈ lattice). It was shown in 1979 that this is the maximum possible number in 8-dimensions.

The kissing number problem is remarkably difficult and solutions are only known in 1, 2, 3, 4, 8, and 24 dimensions. Perhaps surprisingly, it is easier to find the solution in 8 (and 24) dimensions than in 3 or 4. This follows from the special properties of the E₈ lattice (and its 24-dimensional cousin, the Leech lattice).