The $E_8$ lattice is an exceptional lattice inside $\mathbb{R}^8$.
The $E_8$-lattice may be presented as the linear span with integer coefficients of the following basis vectors in the real vector space $\mathbb{R}^8$:
$(1,-1,0,0,0,0,0,0)$
$(0,1,-1,0,0,0,0,0)$
$(0,0,1,-1,0,0,0,0)$
$(0,0,0,1,-1,0,0,0)$
$(0,0,0,0,1,-1,0,0)$
$(0,0,0,0,0,1,-1,0)$
$(0,0,0,0,0,1,1,0)$
$\big(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\big)$
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 $\frac{\pi^4}{384} \approx 0.2537$
See also:
Discussion in relation to sphere packing:
Review:
Last revised on January 16, 2023 at 11:32:55. See the history of this page for a list of all contributions to it.