E8 lattice

Remarks

It is closed under addition and negation, so it is a subgroup of $\R^8$, and it contains $8$ independent vectors. It is even: $x \cdot x \in 2\Z$ for all $x \in E_8$. It is unimodular: its covolume is $1$. Up to isometry, it is the unique even unimodular lattice in $\R^8$. Its shortest nonzero vectors have length $\sqrt 2$, and there are $240$ of them. They are the $112$ vectors with two coordinates $\pm 1$ and six coordinates $0$. The other $128$ are the vectors with all coordinates $\pm\tfrac12$ and an even number of minus signs. These $240$ vectors form the root system of type $E_8$. Balls of radius $\sqrt 2/2$ centred at the points of $E_8$ form the densest sphere packing in $\R^8$, with density $\pi^4/384$, as proved by Viazovska. Each ball touches $240$ others, which is the kissing number in dimension $8$.