The Strong Thirteen Spheres Problem

The thirteen spheres problem asks if 13 equal-size non-overlapping spheres in three dimensions can simultaneously touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schutte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen-sphere problem (or the Tammes problem for 13 points), which calls for finding the maximum radius of and an arrangement for 13 equal-size non-overlapping spheres touching the unit sphere. In this paper, we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on an enumeration of irreducible graphs.