volume 31, issue 4, P613-625 2004
DOI: 10.1007/s00454-003-0819-2
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Abstract: The "thirteen spheres problem," also known as the "Gregory-Newton problem," is to determine the maximum number of three-dimensional spheres that can simultaneously touch a given sphere, where all the spheres have the same radius. The history of the problem goes back to a disagreement between Isaac Newton and David Gregory in 1694. Using a combination of harmonic analysis and linear programming it can be shown that the maximum cannot exceed 13, but in fact 13 is impossible. The standard proof that the maximum …

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