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Kepler's conjecture confirmed

Nature volume 395, pages 435–436 (1998)Cite this article

One of the oldest unsolved problems in mathematics appears to have been settled. On 9 August, Thomas C. Hales announced that he had proved Kepler's assertion of 1611 that no packing of spheres can be denser than a face-centred-cubic lattice.

There are three reasons. First, technical difficulties come from the fact that the density of a packing is defined as the limit of the fraction of space occupied by the balls as the number of balls goes to infinity. This means that (say) a million balls can be thrown away without changing the density.

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Figure 1: Cannonballs stacked in a face-centred-cubic lattice (Arlington, Virginia, about 1863).

MATHEW B. BRADY/LIBRARY OF CONGRESS

Figure 2: Stacking schemes.

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  1. the AT&T Shannon Laboratory, 180 Park Avenue, Florham Park, 07932-0971, New Jersey, USA

    Neil J. A. Sloane

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  1. Neil J. A. Sloane
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Sloane, N. Kepler's conjecture confirmed. Nature 395, 435–436 (1998). https://doi.org/10.1038/26609

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