Kissing Numbers, Sphere Packings, and Some Unexpected Proofs, vol. 51, number 8
نویسندگان
چکیده
T he “kissing number problem” asks for the maximal number of blue spheres that can touch a red sphere of the same size in n-dimensional space. The answers in dimensions one, two, and three are classical, but the answers in dimensions eight and twenty-four were a big surprise in 1979, based on an extremely elegant method initiated by Philippe Delsarte in the early seventies which concerns inequalities for the distance distributions of kissing configurations. Delsarte’s approach led to especially striking results in cases where there are exceptionally symmetric, dense, and unique configurations of spheres: In dimensions eight and twenty-four these are given by the shortest vectors in two remarkable lattices, known as the E8 and the Leech lattice. However, despite the fact that in dimension four there is a special configuration which is conjectured to be optimal and unique—the shortest vectors in the D4 lattice, which are also the vertices of a regular 24-cell—it was proved that the bounds given by Delsarte’s method are not good enough to solve the problem in dimension four. This may explain the astonishment even to experts when in the fall of 2003 Oleg Musin announced a solution of the problem, based on a clever modification of Delsarte’s method [22], [23]. Independently, Delsarte’s by now classical approach has also recently been adapted by Henry Cohn and Noam Elkies [5] to deal with optimal sphere packings more directly and more effectively than had been possible before. Based on this, Henry Cohn and Abhinav Kumar [6] [7] have now proved that the sphere packings in dimensions eight and twenty-four given by the E8 and Leech lattices are optimal lattice packings (for dimension eight this had been shown before) and that they are optimal sphere packings, up to an error of not more than 10−28 percent. Here we try to sketch the setting, to explain some of the ideas, and to tell the story. For this
منابع مشابه
Kissing numbers, sphere packings, and some unexpected proofs
The “kissing number problem” asks for the maximal number of white spheres that can touch a black sphere of the same size in n-dimensional space. The answers in dimensions one, two and three are classical, but the answers in dimensions eight and twenty-four were a big surprise in 1979, based on an extremely elegant method initiated by Philippe Delsarte in the early seventies, which concerns ineq...
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