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