It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8.
The norm[1] of the E8 lattice (divided by 2) is a positive definite even unimodular quadratic form in 8 variables, and conversely such a quadratic form can be used to construct a positive-definite, even, unimodular lattice of rank 8.
An alternative description of the E8 lattice which is sometimes convenient is the set of all points in Γ′8 ⊂ R8 such that In symbols, The lattices Γ8 and Γ′8 are isomorphic and one may pass from one to the other by changing the signs of any odd number of half-integer coordinates.
One possible basis for Γ8 is given by the columns of the (upper triangular) matrix Γ8 is then the integral span of these vectors.
This is the group generated by reflections in the hyperplanes orthogonal to the 240 roots of the lattice.
The full E8 Weyl group is generated by this subgroup and the block diagonal matrix H4⊕H4 where H4 is the Hadamard matrix The E8 lattice points are the vertices of the 521 honeycomb, which is composed of regular 8-simplex and 8-orthoplex facets.
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.
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.
(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.
[7] Maryna Viazovska proved in 2016 that this density is, in fact, optimal even among irregular packings.
[8][9] 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.
Furthermore, the theta function of an even unimodular lattice of rank n is actually a modular form of weight n/2.
The theta function of an integral lattice is often written as a power series in
so that the coefficient of qn gives the number of lattice vectors of norm n. Up to normalization, there is a unique modular form of weight 4 and level 1: the Eisenstein series G4(τ).
It follows that the number of E8 lattice vectors of norm 2n is 240 times the sum of the cubes of the divisors of n. The first few terms of this series are given by (sequence A004009 in the OEIS): The E8 theta function may be written in terms of the Jacobi theta functions as follows: where Note that the j-function can be expressed as, The E8 lattice is very closely related to the (extended) Hamming code H(8,4) and can, in fact, be constructed from it.
It has a minimum nonzero Hamming weight 4, meaning that any two codewords differ by at least 4 bits.
One can construct a lattice Λ from a binary code C of length n by taking the set of all vectors x in Zn such that x is congruent (modulo 2) to a codeword of C.[12] It is often convenient to rescale Λ by a factor of 1/√2, Applying this construction a Type II self-dual code gives an even, unimodular lattice.
The E8 lattice is also closely related to the nonassociative algebra of real octonions O.
Embedded in the octonions in this manner the E8 lattice takes on the structure of a nonassociative ring.
(One must, of course, extend the definitions of order and ring to include the nonassociative case).
This amounts to finding the largest subring of O containing the units on which the expressions x*x (the norm of x) and x + x* (twice the real part of x) are integer-valued.
A detailed account of the integral octonions and their relation to the E8 lattice can be found in Conway and Smith (2003).
Imaginary octonions in this set, namely 14 from 1) and 7*16=112 from 3), form the roots of the Lie algebra
[13] In 1982 Michael Freedman produced an example of a topological 4-manifold, called the E8 manifold, whose intersection form is given by the E8 lattice.
In order for the theory to work correctly, the 16 mismatched dimensions must be compactified on an even, unimodular lattice of rank 16.