[math-fun] Kissing number in 4 dimensions claimed to be 24
I've just heard that Oleg Musin is circulating a paper in which he purports to prove that the kissing number in 4D is 24. ( For those unfamiliar with this problem: The kissing number K(n) in n-space is the maximum number of unit n-balls that can be simultaneously tangent to ("kissing") a given unit n-ball in n-space, such that when any two of these K(n) balls intersect, they are merely tangent to each other. Clearly K(2) = 6, and K(3) was a matter of much debate between Newton and Gregory (Gregory said it was 13, and Newton was right that K(3) = 12). It's amazing that this wasn't proved, however, until about 1875. It has been known for some time that K(4) must be 24 or 25. An arrangement for 24 has been known for a long time. The most symmetrical such arrangment of 24 4-balls all touching a 25th is described by assuming the 25th 4-ball to be centered at the origin, and the 24 "kissing" 4-balls to be centered at the 24 points having [2 coordinates = ±sqrt(1/2) and the other 2 coordinates = 0]. ) --Dan
On Wed, 24 Sep 2003 asimovd@aol.com wrote:
I've just heard that Oleg Musin is circulating a paper in which he purports to prove that the kissing number in 4D is 24.
Funsters should be aware that Wu-Yi Hsiang claimed to have proved this a few years ago, and still hasn't retracted, as far as I know, so he can be expected to dispute Musin's priority. Do you know anything about Musin's method? I ask because there are two plausible styles of proof. If it's combinatorial consideration of cases combined with elementary geometrical proofs of various numerical estimates, then I'll have strong doubts as to its correctness. But another style is possible, namely to deduce some geometrical consequence from an analytical property of spherical functions or some such, and then to use this to determine the geometrical configuration. This is what Sloane and Bannai did in their discussion of the kissing number in 24 dimensions, and if Musin's proof is in this style, I'll be much more likely to believe it. The arrangement you describe is almost certainly the only arrangement that gives 24; another way to describe it is as the vertices of the regular polytope {3,4,3}, the "16-cell" or "polyoctahedron". It extends to give the almost certainly best-possible sphere-packing in 4 dimensions, in which the centers are easily described as being the centers of the "black" cells of a 4-dimensional checkerboard. John Conway
John, could you elaborate on why
... If it's combinatorial consideration of cases combined with elementary geometrical proofs of various numerical estimates, then I'll have strong doubts as to its correctness.
I mean, surely a proof along these lines exists -- the space of configurations (of 25 balls, possibly intersecting, all kissing a central one) is compact, so it can in principle be completely checked with a finite number of cases of the form "this configuration has at least epsilon overlap, so every configuration within distance epsilon of it is ruled out." --Michael Kleber kleber@brandeis.edu
On Thu, 25 Sep 2003, Michael Kleber wrote:
John, could you elaborate on why
... If it's combinatorial consideration of cases combined with elementary geometrical proofs of various numerical estimates, then I'll have strong doubts as to its correctness.
I mean, surely a proof along these lines exists -- the space of configurations (of 25 balls, possibly intersecting, all kissing a central one) is compact, so it can in principle be completely checked with a finite number of cases of the form "this configuration has at least epsilon overlap, so every configuration within distance epsilon of it is ruled out." k Yes, of course. The same argument applies to solving the game of Chess. If next week someone claims to have done that, I won't believe them for the same reason, namely that the problem is so big that their argument is more likely to be wrong than right.
This has very little to do with whether a computer is used, although that would of course improve the reliability. The trouble is that the problem is so big that a computer won't make much difference to its size, and I just don't believe that at the present time there's any real hope of finding a valid proof of this kind. On the other hand, I know full well that finding an invalid one is much easier, so, essentially for Bayesian reasons, that's what I'd confidently expect a claimed proof of this kind to be. The question is moot because fortunately Dan tells us that Musin's proof is of the second kind, which I expect to be short, simple, readily checkable, and therefore more probably correct. John conway
participants (3)
-
asimovd＠aol.com -
John Conway -
Michael Kleber