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Sphere packing, lattice packing, and related problems Abhinav Kumar StonyBrook April 25, 2018 Sphere packings Definition A sphere packing in Rn is a collection of spheres/balls of equal size which do not overlap (except for touching). The density of a sphere packing is the volume fraction of space occupied by the balls. (cid:126) (cid:126) (cid:126) (cid:126) (cid:126) (cid:126) (cid:126) (cid:126) (cid:126) (cid:126) (cid:126) (cid:126) (cid:126) In dimension 1, we can achieve density 1 by laying intervals end to end. In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth 1940] (cid:15)(cid:15)(cid:12)(cid:15)(cid:12)(cid:12)(cid:15)(cid:15)(cid:12)(cid:12) (cid:14)(cid:15)(cid:14)(cid:13)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:15)(cid:13)(cid:14)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:13) (cid:15)(cid:14)(cid:15)(cid:12)(cid:14)(cid:13)(cid:15)(cid:12)(cid:13)(cid:14)(cid:12)(cid:15)(cid:14)(cid:13)(cid:15)(cid:12)(cid:13)(cid:12) (cid:14)(cid:15)(cid:14)(cid:13)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:15)(cid:13)(cid:14)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:13) (cid:15)(cid:14)(cid:15)(cid:12)(cid:14)(cid:13)(cid:15)(cid:12)(cid:13)(cid:14)(cid:12)(cid:15)(cid:14)(cid:13)(cid:15)(cid:12)(cid:13)(cid:12) (cid:14)(cid:14)(cid:13)(cid:14)(cid:13)(cid:13)(cid:14)(cid:14)(cid:13)(cid:13) Sphere packing problem Problem: Find a/the densest sphere packing(s) in Rn. In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth 1940] (cid:15)(cid:15)(cid:12)(cid:15)(cid:12)(cid:12)(cid:15)(cid:15)(cid:12)(cid:12) (cid:14)(cid:15)(cid:14)(cid:13)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:15)(cid:13)(cid:14)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:13) (cid:15)(cid:14)(cid:15)(cid:12)(cid:14)(cid:13)(cid:15)(cid:12)(cid:13)(cid:14)(cid:12)(cid:15)(cid:14)(cid:13)(cid:15)(cid:12)(cid:13)(cid:12) (cid:14)(cid:15)(cid:14)(cid:13)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:15)(cid:13)(cid:14)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:13) (cid:15)(cid:14)(cid:15)(cid:12)(cid:14)(cid:13)(cid:15)(cid:12)(cid:13)(cid:14)(cid:12)(cid:15)(cid:14)(cid:13)(cid:15)(cid:12)(cid:13)(cid:12) (cid:14)(cid:14)(cid:13)(cid:14)(cid:13)(cid:13)(cid:14)(cid:14)(cid:13)(cid:13) Sphere packing problem Problem: Find a/the densest sphere packing(s) in Rn. In dimension 1, we can achieve density 1 by laying intervals end to end. Sphere packing problem Problem: Find a/the densest sphere packing(s) in Rn. In dimension 1, we can achieve density 1 by laying intervals end to end. In dimension 2, the best possible is by using the hexagonal lattice. [Fejes To´th 1940] (cid:15)(cid:15)(cid:12)(cid:15)(cid:12)(cid:12)(cid:15)(cid:15)(cid:12)(cid:12) (cid:14)(cid:15)(cid:14)(cid:13)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:15)(cid:13)(cid:14)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:13) (cid:15)(cid:14)(cid:15)(cid:12)(cid:14)(cid:13)(cid:15)(cid:12)(cid:13)(cid:14)(cid:12)(cid:15)(cid:14)(cid:13)(cid:15)(cid:12)(cid:13)(cid:12) (cid:14)(cid:15)(cid:14)(cid:13)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:15)(cid:13)(cid:14)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:13) (cid:15)(cid:14)(cid:15)(cid:12)(cid:14)(cid:13)(cid:15)(cid:12)(cid:13)(cid:14)(cid:12)(cid:15)(cid:14)(cid:13)(cid:15)(cid:12)(cid:13)(cid:12) (cid:14)(cid:14)(cid:13)(cid:14)(cid:13)(cid:13)(cid:14)(cid:14)(cid:13)(cid:13) Sphere packing problem II In dimension 3, the best possible way is to stack layers of the solution in 2 dimensions. This is Kepler’s conjecture, now a theorem of Hales and collaborators. (cid:15)(cid:15)(cid:12)(cid:15)(cid:12)(cid:12)(cid:15)(cid:15)(cid:12)(cid:12) (cid:14)(cid:15)(cid:14)(cid:13)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:15)(cid:13)(cid:14)(cid:15)(cid:12)(cid:14)(cid:13)(cid:12)(cid:13) (cid:15)(cid:15)(cid:12)(cid:12)(cid:15)(cid:15)(cid:12)(cid:12) (cid:15)(cid:14)(cid:15)(cid:12)(cid:109)(cid:14)(cid:13)(cid:15)(cid:12)(cid:109)(cid:13)(cid:14)(cid:12)(cid:15)(cid:109)(cid:14)(cid:13)(cid:15)(cid:12)(cid:109)(cid:13)(cid:12) (cid:14)(cid:15)(cid:14)(cid:14)(cid:15)(cid:13)(cid:15)(cid:14)(cid:12)(cid:13)(cid:109)(cid:14)(cid:15)(cid:13)(cid:12)(cid:12)(cid:13)(cid:15)(cid:14)(cid:109)(cid:13)(cid:12)(cid:14)(cid:15)(cid:15)(cid:14)(cid:12)(cid:13)(cid:109)(cid:14)(cid:15)(cid:13)(cid:12)(cid:12)(cid:13)(cid:109)(cid:13)(cid:12) (cid:15)(cid:14)(cid:15)(cid:14)(cid:12)(cid:14)(cid:13)(cid:15)(cid:14)(cid:12)(cid:13)(cid:13)(cid:14)(cid:12)(cid:13)(cid:15)(cid:14)(cid:14)(cid:13)(cid:15)(cid:14)(cid:12)(cid:13)(cid:13)(cid:12)(cid:13) (cid:14)(cid:14)(cid:13)(cid:14)(cid:13)(cid:13)(cid:14)(cid:14)(cid:13)(cid:13) There are infinitely (in fact, uncountably) many ways of doing this! These are the Barlow packings. Face centered cubic packing Image:GregAL(Wikipedia),CCBY-SA3.0license But (until very recently!) no proofs. In very high dimensions (say ≥ 1000) densest packings are likely to be close to disordered. Higher dimensions In some higher dimensions, we have guesses for the densest sphere packings. Most of them arise from lattices. In very high dimensions (say ≥ 1000) densest packings are likely to be close to disordered. Higher dimensions In some higher dimensions, we have guesses for the densest sphere packings. Most of them arise from lattices. But (until very recently!) no proofs. Higher dimensions In some higher dimensions, we have guesses for the densest sphere packings. Most of them arise from lattices. But (until very recently!) no proofs. In very high dimensions (say ≥ 1000) densest packings are likely to be close to disordered.

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Problem: Find a/the densest sphere packing(s) in Rn. Face centered cubic packing .. Let S(Λ) = {u1,,uN} be the set of minimal vectors of Λ, i.e..
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Problem: Find a/the densest sphere packing(s) in Rn. Face centered cubic packing .. Let S(Λ) = {u1,,uN} be the set of minimal vectors of Λ, i.e..

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