The Problem of the Thirteen Spheres
Professor Kurt Anstreicher
The "Thirteen Spheres Problem," also known as the "Gregory-Newton Problem,"is to determine the maximum number of 3-dimensional spheres that can simultaneously touch another sphere, where all the spheres have the same diameter. The history of the problem goes back to a disagreement between Isaac Newton and David Gregory in 1694. A combination of harmonic analysis and linear programming can be used to show that the maximum cannot exceed 13, but in fact only 12 is possible. The proof that the maximum is 12 uses an ad-hoc construction that does not appear to extend to higher dimensions. We describe a new proof that utilizes linear programming bounds together with properties of spherical Delaunay triangulations.