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So I'm really interested in building a mathematical model for how powerful computer chips could be given extra spatial dimensions. Obviously this is a squishy problem, since "computer chips" is not a rigorously defined entity. But I want to at least characterize the shape of the curve - i.e. linear, polynomial, exponential, super-exponential.

I think the starting place to start thinking about this is how many neighbors a sphere has in an optimal packing in N dimensions (because chips want to pack logic circuits in this way). To my surprise, there's not a closed form solution to this problem, and the optimal packing in higher dimensions in general is not even known.

That's crazy and surprising! Why is this the case? Looking for some intuition here. I'm not a real mathematician, please add color. Is there a connection with this packing and the sporadic groups? I browsed a couple of papers and some common characters appear, like the Leech Lattice.

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  • $\begingroup$ Why is this the case? Well, for context, we didn't know(1) until the late 90s the optimal packing in 3d. That we now know it in 8 and 24 dimensions is mildly astounding, and due to properties extra special to those dimensions. [(1) depending if you believe that Hales' original proof was enough or not; the referees of his paper couldn't certify his code, hence the more recent formalised proof] $\endgroup$
    – David Roberts ♦
    Commented Sep 8, 2020 at 6:34
  • $\begingroup$ I don't know what you mean by "this packing" in the last paragraph, but I guess you mean that we know the densest packing in 8 and 24 dimensions, as I said. The connections to sporadic (I guess you mean finite simple) groups are not obvious. The symmetries of the Leech lattice form a finite group that maps two-to-one onto the sporadic simple group $Co_1$. It also connects with the other sporadics in the "2nd generation of the so-called Happy Family". $\endgroup$
    – David Roberts ♦
    Commented Sep 8, 2020 at 6:42
  • $\begingroup$ If you can get a copy of Sphere packing, lattices and groups (doi.org/10.1007/978-1-4757-6568-7) it has a lot of info, not at the right level, perhaps, but with lots of references. Cohn has a nice 2014 survey arxiv.org/abs/1603.05202, predating the work Viazovska (+et al) establishing the d=8,24 cases. See Fig 1 on p53 $\endgroup$
    – David Roberts ♦
    Commented Sep 8, 2020 at 6:44

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