The strong thirteen spheres problem

TL;DR

This paper solves the long-standing strong thirteen spheres problem by providing a computer-assisted proof that determines the optimal arrangement and maximum radius of 13 equal spheres touching a unit sphere.

Contribution

The paper presents the first definitive solution to the strong thirteen spheres problem using a computer-assisted approach based on irreducible graph enumeration.

Findings

Established the optimal configuration for 13 spheres touching a unit sphere.

Determined the maximum radius of the spheres in this configuration.

Provided a rigorous proof confirming the uniqueness of the solution.

Abstract

The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can 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 spheres problem (or the Tammes problem for 13 points) which asks to find an arrangement and the maximum radius of 13 equal size nonoverlapping spheres touching the unit sphere. In the paper we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on a enumeration of the so-called irreducible graphs.