The Thirteen Spheres: A New Proof

The “thirteen spheres problem,” also known as the “Gregory-Newton problem” is to determine the maximum number of three-dimensional spheres that can simultaneously touch a given sphere, where all the spheres have the same radius. The history of the problem goes back to a disagreement between Isaac Newton and David Gregory in 1694. Using a combination of harmonic analysis and linear programming it can be shown that the maximum cannot exceed 13, but in fact 13 is impossible. The standard proof that the maximum is 12 uses an ad-hoc construction that does not appear to extend to higher dimensions. In this paper we describe a new proof that uses linear programming bounds and properties of spherical Delaunay triangulations.