New non-isometric kissing configurations are constructed in dimensions 5 and 9, and the uniform 5D packings containing the known kissing configurations are classified.
The kissing number in four dimensions
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abstract
The kissing number problem asks for the maximal number k(n) of equal size nonoverlapping spheres in n-dimensional space that can touch another sphere of the same size. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimensions the problem was finally solved only in 1953 by Sch\"utte and van der Waerden. In this paper we present a solution of a long-standing problem about the kissing number in four dimensions. Namely, the equality k(4)=24 is proved. The proof is based on a modification of Delsarte's method.
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Variations on five-dimensional sphere packings
New non-isometric kissing configurations are constructed in dimensions 5 and 9, and the uniform 5D packings containing the known kissing configurations are classified.