nLab sphere packing

Idea

A sphere packing in a Euclidean space of dimension $n+1$ 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 $n+1 = 1$ the problem is trivial.

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

For $n+1 = 3$ 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 $n+1 = 8$ the denest packing is that of spheres centered on the E8-lattice (Viazovska 16).

For $n+1 = 24$ 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 $n$ 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

Textbook accounts:

• Peter Engel, Chapter 10 of: Geometric Crystallography – An Axiomatic Introduction to Crystallography, D. Reidel Publishing (1986) $[$doi:10.1007/978-94-009-4760-3$]$

  (which is otherwise concerned with crystallography via crystallographic groups)

See also:

• Wikipedia, Sphere packing

Solution to the sphere packing problem in dimensions 8 and 24:

• 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]

• 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, doi:10.4007/annals.2017.185.3.8]

Exposition:

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

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

• Thomas Hartman, Dalimil Mazáč, Leonardo Rastelli, Sphere Packing and Quantum Gravity (arXiv:1905.01319)