E8 Lattice - Geometry

Geometry

See 5₂₁ honeycomb

The E₈ lattice points are the vertices of the 5₂₁ honeycomb, which is composed of regular 8-simplex and 8-orthoplex facets. This honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure (Gosset regarded honeycombs in n dimensions as degenerate n+1 polytopes). In Coxeter's notation, Gosset's honeycomb is denoted by 5₂₁ and has the Coxeter-Dynkin diagram:

This honeycomb is highly regular in the sense that its symmetry group (the affine Weyl group) acts transitively on the k-faces for k ≤ 6. All of the k-faces for k ≤ 7 are simplices.

The vertex figure of Gosset's honeycomb is the semiregular E₈ polytope (4₂₁ in Coxeter's notation) given by the convex hull of the 240 roots of the E₈ lattice.

Each point of the E₈ lattice is surrounded by 2160 8-orthoplexes and 17280 8-simplices. The 2160 deep holes near the origin are exactly the halves of the norm 4 lattice points. The 17520 norm 8 lattice points fall into two classes (two orbits under the action of the E₈ automorphism group): 240 are twice the norm 2 lattice points while 17280 are 3 times the shallow holes surrounding the origin.

A hole in a lattice is a point in the ambient Euclidean space whose distance to the nearest lattice point is a local maximum. (In a lattice defined as a uniform honeycomb these points correspond to the centers of the facets volumes.) A deep hole is one whose distance to the lattice is a global maximum. There are two types of holes in the E₈ lattice:

• Deep holes such as the point (1,0,0,0,0,0,0,0) are at a distance of 1 from the nearest lattice points. There are 16 lattice points at this distance which form the vertices of an 8-orthoplex centered at the hole (the Delaunay cell of the hole).
• Shallow holes such as the point are at a distance of from the nearest lattice points. There are 9 lattice points at this distance forming the vertices of an 8-simplex centered at the hole.