On Fri, 26 Sep 2003, David Wilson wrote:
Not being well-versed in the area, I looked up "Kissing Number" at Mathworld.
In the table, I notice that the "NL" (non lattice kissing number) values are missing for dimensions 1 thru 8. Don't we know that NL(1,2,3) = (2,6,12)?
Yes, we do. We also know the values 240, 196560 for n = 8, 24. But those are all the values we know. And certainly at worst NL(d) >= L(d) for d = 4 thru. True. I think the values for L are known precisely for n < 10 and n = 24.
I gather [...] that we know 24 = L(4) <= NL(4) <= 25,
Yes, that's all we knew (before the recent claim of Musin).
and are discussing means to establish or eliminate NL(4) = 25 (though I've been following the thread only cursorily).
Yes, you've got it, babe!
Do we know of any lower bounds better than NL(d) for d <= 8 that we might put in the table?
I think you meant "better than L(d)". Anyway, L(d) is the best lower bound we know (and almost certainly the correct answer) for d <= 8.
Also, bear with me here, I'm going to rehash some elementary stuff. In 2 dimensions, we easily see that a unit circle kissing unit circle S "uses up" pi/3 radians of the circumference of S, which cannot be used by any other kissing circle. This bounds NL(2) <= (2pi)/(pi/3) = 6.
True, O King.
Similarly, a unit sphere kissing unit sphere S "uses up" an area A of the surface of C subsumed by a circle with a 60-degree diameter, so that NL(3) <= (4pi)/A. Does this observation generalize to higher dimensions?
Yes of course, but it only gives the bound 13 in 3 dimensions.
Does it result in an upper bound formula for NL(d)?
Yes, but better bounds are known.
Is there a nice bounding asymptotic? Does the formula grow relatively worse or better for increasing d? I'm sure this is textbook, but I've never seen it developed. If there is an accessible online development, that would be great.
The details of that bound were worked out by John Leech, but are not of much interest because, as I said, better bounds are known. My book with Sloane, "Sphere Packings, Lattices, and Groups" ("SPLAG") will tell you most of what's known about them. (Not much has happened since the latest edition in print.) Regards, John Conway