This is exciting!

This is exciting!

Gil's post doesn't actually say what the densest sphere packings in dimensions 8 and 24 are, but if you click through to the paper abstracts, they are the packings based on the E8 lattice and the Leech lattice.

I wonder if this was the result mathematicians expected?

Thanks David Eppstein for the reshare.

Originally shared by Gil Kalai

https://gilkalai.wordpress.com/2016/03/23/a-breakthrough-by-maryna-viazovska-lead-to-the-long-awaited-solutions-for-the-densest-packing-problem-in-dimensions-8-and-24/
https://gilkalai.wordpress.com/2016/03/23/a-breakthrough-by-maryna-viazovska-lead-to-the-long-awaited-solutions-for-the-densest-packing-problem-in-dimensions-8-and-24

Comments

  1. David EppsteinMarch 23, 2016 at 7:19 PM

    I think it was expected that these would be the densest packings, yes, but actually being able to prove it is a big surprise.

    ReplyDelete
  2. John BaezMarch 23, 2016 at 10:39 PM

    Yay! I'm so sick and tired of saying that E8 and the Leech lattice are the densest lattice packings of sphere; it was always expected that they were the densest sphere packings.

    Cohn and Kumar had shown that the E8 lattice was the densest of sphere packings in 8 dimensions up to an error of at most 10^{-14}, and the Leech was the densest in 24 dimension up to an error of at most 1.65 · 10^{−30}. Later work showed that E8, too, could only be off by a factor of at most about 10^{-30}. So the world have been a mean, ugly place if these beautiful lattices didn't wind up winning.

    ReplyDelete
  3. Allen KnutsonMarch 24, 2016 at 5:37 AM

    So... how about 16? The favorites there are E_8 + E_8 and D_16^+, as I recall. Also, are there dimensions where it's known that lattice packings aren't the densest?

    ReplyDelete
  4. John BaezMarch 24, 2016 at 8:28 AM

    I don't think anybody knows the densest sphere packings except in dimensions 1,2,3,8,24. In 2012, Abhinav Kumar wrote:

    Even starting in R^9, interesting phenomena emerge. For example, the best packings known in dimension 9 are a continuous family, one of which is a lattice, and the others are obtained by moving half the spheres relative to the other half (the fluid diamond packings).

    In dimension 10, the current record is held by the Best packing (40 translates of a lattice, obtained as the inverse image of a non-linear binary code in F_2^{10} under the reduction Z^{10} → F_2^{10}).

    It is believed that for large enough n, the maximum density will be attained by a non-lattice packing and in fact, by a periodic packing (Zassenhaus conjecture). Both these conjectures are currently out of reach.

    ReplyDelete
  5. Allen KnutsonMarch 24, 2016 at 8:34 AM

    "by a non-lattice packing and in fact, by a periodic packing"? That's really weird.

    ReplyDelete
  6. John BaezMarch 24, 2016 at 8:49 AM

    Well, that "Best" packing in 10d is periodic but non-lattice. 

     (I wouldn't have used the connective "and, in fact" to describe the relation between those two conjectures.)

    ReplyDelete

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