The Tammes problem for N=14

TL;DR

This paper presents a computer-assisted solution to the long-standing Tammes problem for N=14 points on a sphere, completing the known solutions for specific N values through enumeration of contact graphs.

Contribution

We provide the first proof of the optimal arrangement for N=14 points on a sphere, solving a 60-year-old open problem in geometric configuration.

Findings

Confirmed the optimal configuration for N=14 points on a sphere.

Developed a method using enumeration of irreducible contact graphs.

Extended the set of solved cases for the Tammes problem.

Abstract

The Tammes problem is to find the arrangement of N points on a unit sphere which maximizes the minimum distance between any two points. This problem is presently solved for several values of N, namely for N=3,4,6,12 by L. Fejes Toth (1943); for N=5,7,8,9 by Schutte and van der Waerden (1951); for N=10,11 by Danzer (1963) and for N=24 by Robinson (1961). Recently, we solved the Tammes problem for N=13. The optimal configuration of 14 points was conjectured more than 60 years ago. In the paper, we give a solution of this long-standing open problem in geometry. Our computer-assisted proof relies on an enumeration of the irreducible contact graphs.