nLab E₈ lattice

Idea

The $E_8$ lattice is an exceptional lattice inside $\mathbb{R}^8$.

Definition

The $E_8$-lattice may be presented as the linear span with integer coefficients of the following basis vectors in the real vector space $\mathbb{R}^8$:

• $(1,-1,0,0,0,0,0,0)$

• $(0,1,-1,0,0,0,0,0)$

• $(0,0,1,-1,0,0,0,0)$

• $(0,0,0,1,-1,0,0,0)$

• $(0,0,0,0,1,-1,0,0)$

• $(0,0,0,0,0,1,-1,0)$

• $(0,0,0,0,0,1,1,0)$

• $\big(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\big)$

Properties

Relation to sphere packing

If 7-spheres of radius $\frac{\sqrt{2}}{2}$ are centered around every point of the $E_8$ lattice, then each sphere touches 240 other spheres.

A proof that this arrangement gives the optimal sphere packing in 8 dimensions was given in Viazovska (2016). (The analogous statement holds in 24 dimensions with respect to the Leech lattice [CKMRV17].)

The optimal sphere packing constant in 8 dimensions is $\frac{\pi^4}{384} \approx 0.2537$

References

See also:

• Wikipedia, E₈ lattice

Discussion in relation to sphere packing:

• Maryna Viazovska, The sphere packing problem in dimension 8, Annals of Mathematics, 185 3 (2017) 991-1015 [arXiv:1603.04246, doi:10.4007/annals.2017.185.3.7]

Review:

• Ursula Whitcher, Eight-dimensional spheres and the exceptional $E_8$, AMS MathVoices Feature Column (Sep. 2022) [web]