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The strong thirteen spheres problem

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arxiv 1002.1439 v3 pith:JBOLZ4CO submitted 2010-02-07 math.MG math.CO

The strong thirteen spheres problem

classification math.MG math.CO
keywords problemspheressizethirteenequalnonoverlappingspherestrong
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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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.

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