The Kissing Number Problem: A New Result from Global Optimization

Determining the maximum number of D-dimensional spheres of radius r that can be adjacent to a central sphere of radius r is known as the Kissing Number Problem (KNP). The problem has been solved for 2 and 3 dimensions. The smallest open case is 4 dimensions: a solution with 24 spheres is known, and an upper bound of 25 has been found. We present a new nonlinear mathematical programming model for the solution of the KNP. This problem is solved using a quasi Monte Carlo variant of a multi level single linkage algorithm for global optimization. The numerical results indicate that the solution of the KNP is 24 spheres, and not 25.