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