E₈ lattice

In mathematics, the E₈ lattice is a special lattice in R⁸. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E₈ root system.

The norm 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. The existence of such a form was first shown by H. J. S. Smith in 1867, and the first explicit construction of this quadratic form was given by Korkin and Zolotarev in 1873. The E₈ lattice is also called the Gosset lattice after Thorold Gosset who was one of the first to study the geometry of the lattice itself around 1900.