quaternionic projective line$\,\mathbb{H}P^1$
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.
The sphere packing problem is the problem of finding the densest possible sphere packings in any dimension.
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.
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).
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.
Textbook accounts:
Peter Engel, Chapter 10 of: Geometric Crystallography โ 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:
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:
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
Last revised on January 16, 2023 at 05:20:01. See the history of this page for a list of all contributions to it.