The Kissing Number Problem

Problem statement. When rigid balls touch each other, in technical terms, they “kiss”. In mathematical terms, the kissing number in D dimensions is the maximum number of D-spheres of radius R that can be arranged around a central D-sphere of radius R so that each of the surrounding spheres touches the central one without overlapping. The kissing number problem(KNP), well known in Combinatorial Geometry, is the problem of finding such number for a given dimension D. We indicate the Kissing Number Problem inD dimensions by KNP(D). In R 2 , the maximum kissing number is 6 (Fig. 1a). The situation is far from trivial in R 3 . The problem earned its fame because, according to Newton, the maximum kissing number in 3D is 12, whereas according to his contemporary fellow mathematician David Gregory, the maximum kissing number in 3D is 13 (this conjecture was stated without proof). This question was settled, at long last, mor e than 250 years after having been stated, when J. Leech finally proved that the solution in 3D is 12 [2]. The question for the 4-dimensional case was very recently settled by O. Musin [3], proving that the solution of KNP(4) is 24 spheres. Nonlinear models of the KNP for its solution using Global Optimization techniques were proposed by Kucherenko et al. [1].