nLab
sphere packing

Contents

Contents

Idea

A sphere packing in a Euclidean space of dimension n+1n+1 is an embedding of congruent n-spheres of some fixed radius, that do not intersect but may touch.

Sphere packing problem

The sphere packing problem is the problem of finding the densest possible sphere packings in any dimension.

Low dimensional cases

For n+1=1n+1 = 1 the problem is trivial.

For n+1=2n+1 = 2 it is relatively easy to show that the densest packing is given by spheres centered on the vertices of the honeycomb lattice?.

For n+1=3n+1 = 3 it used to Kepler's conjecture that the densest packing is that given by spheres centered on the cubical lattice. This was proven only with ample computer assistance by the Flyspeck project.

Exceptional cases

For n+1=8n+1 = 8 the denest packing is that of spheres centered on the E8-lattice? (Viazovska 16).

For n+1=24n+1 = 24 the densists packing is that of spheres centered on the Leech lattice (CKMRV 17).

An interpretation of the ingredients entering these proofs in terms of bounds on scaling dimensions of fields in conformal field theories, and interpretation of the result, via AdS/CFT as related to the weak gravity conjecture in AdS-quantum gravity, is due to (HMR 19).

Irregular cases

For other values of nn exact solutions remain unknown, but numerlical simulation shows that the densest sphere packings in the remaining dimensions will not be controlled by lattices, but be highly irregular.

References

General

See also

The solution to the sphere packing problem in dimensions 8 and 24 are originally due to

  • Maryna Viazovska, The sphere packing problem in dimension 8, Annals of Mathematics, Pages 991-1015 from Volume 185 (2017), Issue 3 (arXiv:1603.04246)

  • Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska, The sphere packing problem in dimension 24, Annals of Mathematics 185 (2017), 1017-1033 (arXiv:1603.06518)

In string theory

An interpretation of the ingredients entering these proofs in terms of bounds on scaling dimensions of fields in conformal field theories, and interpretation of the result, via AdS/CFT as related to the weak gravity conjecture in AdS-quantum gravity, is due to

Created on July 10, 2019 at 06:28:02. See the history of this page for a list of all contributions to it.