Kissing Numbers

Hans Mittelmann (Arizona State University) and Frank Vallentin (Centrum voor Wiskunde en Informatica, Amsterdam), have found new upper bounds for 'kissing' numbers in higher dimensions. In geometry the kissing number is the maximum number of non-overlapping unit balls that can simultaneously touch a central unit ball. In two dimensions the kissing number is six. One can clearly see this if one groups coins around one central identical coin. The kissing number is only known for the dimensions 1, 2, 3, 4, 8 and 24. Mittelmann and Vallentin used sophisticated semidefinite programming methods with very high precision to compute upper bounds for all dimensions up to 24. In all cases they determined the now best known upper bounds. The improvements over the previously best bounds many of which dated from 1979 were substantial. For example, the bound for dimension 16, just found in 2007, was improved by more than 10 percent. It also allowed to prove a conjecture by Conway and Sloane stated in their famous book on sphere packings. Here is the paper. You can see the current bounds and additional information at Wikipedia

The kissing problem has a rich history. In 1694 Isaac Newton and David Gregory had a famous discussion about the kissing number in three dimensions. Gregory thought thirteen balls could fit while Newton believed the limit was twelve. Only in 1953, Schütte and Van der Waerden proved Newton right. In the seventies Delsarte developed a method to determine upper bounds for the kissing number based on linear programming. For four dimensions, for instance, the Delsarte bound is 25, while the exact kissing number is 24. Musin proved this in 2003.

Vallentin and Bachoc had developed a new method to determine upper bounds for the kissing number, based on representation theory and semidefinite programming. In 2008 they found new best bounds up to dimension 10 but numerical problems prevented further results. Mittelmann and Vallentin could now overcome these difficulties. Research on kissing numbers has applications in geometry, error correcting codes in telecommunications, and string theory.

Additional Links

http://mathworld.wolfram.com/KissingNumber.html
http://local.wasp.uwa.edu.au/~pbourke/geometry/kissing/