Discrete & Computational Geometry

, Volume 48, Issue 1, pp 128–141

The Strong Thirteen Spheres Problem

Authors

  • Oleg R. Musin
    • Department of MathematicsUniversity of Texas at Brownsville
    • Institute for System AnalysisRussian Academy of Science
Article

DOI: 10.1007/s00454-011-9392-2

Cite this article as:
Musin, O.R. & Tarasov, A.S. Discrete Comput Geom (2012) 48: 128. doi:10.1007/s00454-011-9392-2

Abstract

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 Schütte 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.

Copyright information

© Springer Science+Business Media, LLC 2012