Can someone (Prof. Conway?) provide an update on the sphere packing problem. Has Hsiang's proof held up or is this still an open question?
Similarly, has the 4-D kissing number problem (24 or 25) been settled?
I plan to discuss this with a group of teachers and would like to have accurate information about the status of these problems.
Ed Dickey Ed.Dickey@SCarolina.edu
For the curious who might not recognize the problems by name, the sphere packing problem is to find the greatest proportion of a fixed space filled by identical spheres. To quote Conway and Sloane who quote Rogers "many mathematicians believe, and all physicists know" that the correct proportion is pi/sqrt(18) or 0.7405 ... . Hsiang offered a proof in 1991.
The kissing number problem is to find the greatest number of "spheres" all the same size, that can be arranged around another sphere. In two dimensions, the answer is six (six circles around another circle). In three dimensions, it's 12 (try it with tennis or billiard balls) proving it is another matter. In 4-D, it's 24 or 25 (unless the question has been settled).
_Sphere Packings, Lattices and Groups_ by Conway and Sloane (Springer-Verlag, 1993) provides an excellent discussion as well as figures.