2010/02/28 by Oleg R. Musin, Oleg Musin, Alexey S. Tarasov +1 · 2 citations
Computer Science · Mathematics · #Combinatorics #Computational Geometry and Mesh Generation #Computer science #Digital Image Processing Techniques #Enumeration #Geometry #Mathematical Approximation and Integration #Mathematics #Physics #RADIUS #SPHERES #Unit (ring theory) #Unit sphere #Van der Waerden's theorem #math.CO #math.MG #msc:05B40 #msc:52C15
paper · pdf · doi:10.1007/s00454-011-9392-2
published as Discrete & Computational Geometry, 48:1 (2012), 128-141 · Modified lemma 2, 16 pages, 12 figures. Uploaded program package
arxiv created 2012/01/16 · openalex publication_date 2012/02/28 · arxiv updated 2015/03/12 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
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.