nLab E8 lattice



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


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

  • (1,1,0,0,0,0,0,0)(1,-1,0,0,0,0,0,0)

  • (0,1,1,0,0,0,0,0)(0,1,-1,0,0,0,0,0)

  • (0,0,1,1,0,0,0,0)(0,0,1,-1,0,0,0,0)

  • (0,0,0,1,1,0,0,0)(0,0,0,1,-1,0,0,0)

  • (0,0,0,0,1,1,0,0)(0,0,0,0,1,-1,0,0)

  • (0,0,0,0,0,1,1,0)(0,0,0,0,0,1,-1,0)

  • (0,0,0,0,0,1,1,0)(0,0,0,0,0,1,1,0)

  • (12,12,12,12,12,12,12,12)\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)


Relation to sphere packing

If 7-spheres of radius 22\frac{\sqrt{2}}{2} are centered around every point of the E 8E_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 π 43840.2537\frac{\pi^4}{384} \approx 0.2537


See also:

Discussion in relation to sphere packing:


  • Ursula Whitcher, Eight-dimensional spheres and the exceptional E 8E_8, AMS MathVoices Feature Column (Sep. 2022) [web]

Last revised on January 16, 2023 at 11:32:55. See the history of this page for a list of all contributions to it.