Kissing numbers, sphere packings, and some unexpected proofs FLORIAN PFENDER ∗ and G¨ UNTER M. ZIEGLER ∗∗ TU Berlin, MA 6-2, D-10623 Berlin, Germany {fpfender,ziegler}@math.tu-berlin.de April 19, 2004 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 inequali- ties 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 E 8 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 D 4 lattice, which are also the vertices of a regular 24-cell—it was proved that the bounds given by Delsarte’s method aren’t 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 [21, 22]. Independently, Delsarte’s by now classical approach has recently also 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] have now proved that the sphere packings in dimensions eight and twenty-four given by the E 8 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 we start with a brief review of the sphere packing and kissing number problems. Then we look at the remarkable kissing configurations in dimensions four, eight and twenty-four. We give a sketch of Delsarte’s method, and how it was applied for the kissing number problem in dimensions eight and twenty-four. Then Musin’s ideas kick in, which leads us to look at some non-linear optimization problems, as they occur as subproblems in his approach. Finally we sketch an elegant construction of the Leech lattice in dimension twenty-four, which starts from the graph of the icosahedron and very simple linear algebra. This is the lattice which Cohn and Kumar have now proved to be optimal in dimension twenty-four, by another extremely elegant and puzzling adaption of Delsarte’s method. A sketch for this will end our tour. * Supported by the DFG Research Center “Mathematics in the Key Technologies” (FZT86) ** Partially supported by Deutsche Forschungs-Gemeinschaft (DFG), via the DFG Research Center “Mathematics in the Key Technologies” (FZT86), the Research Group “Algorithms, Structure, Randomness” (Project ZI 475/3), and a Leibniz grant 1