In 1998 Thomas Hales, presently Andrew Mellon Professor at the University of Pittsburgh, announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving checking of many individual cases using complex computer calculations. Referees have said that they are "99% certain" of the correctness of Hales' proof. So the Kepler conjecture is now very close to becoming a theorem.

Table of contents |

2 Origins 3 Nineteenth century 4 Twentieth century 5 Hales' proof 6 A formal proof 7 References 8 External links |

Experiment shows that dropping the spheres in randomly will achieve a density of around 65%. However, a higher density can be achieved by carefully arranging the spheres as follows. Start with a layer of spheres in a hexagonal lattice, then put the next layer of spheres in the lowest points you can find above the first layer, and so on - this is just the way you see oranges stacked in a shop. This natural method of stacking the spheres creates one of two similar patterns called cubic close packing and hexagonal close packing. Each of these two arrangements has an average density of

This meant that any packing arrangement that disproved the Kepler conjecture would have to be an irregular one. But eliminating all possible irregular arrangements is very difficult, and this is what made the Kepler conjecture so hard to prove. In fact, there are irregular arrangements that are denser than the cubic close packing arrangement over a small enough volume, but any attempt to extend these arrangements to fill a larger volume always reduces their density.

After Gauss, no further progress was made towards proving the Kepler conjecture in the nineteenth century. In 1900 David Hilbert included it in his list of twenty three unsolved problems of mathematics - it forms part of Hilbert's eighteenth problem.

Meanwhile, attempts were made to find an upper bound for the maximum density of any possible arrangement of spheres. English mathematician Claude Ambrose Rogers established an upper bound value of about 78% in 1958, and subsequent efforts by other mathematicians reduced this value slightly, but this was still a long way above the cubic close packing density of 74%.

There were also some failed proofs. American architect and geometer Buckminster Fuller claimed to have a proof in 1975, but this was soon found to be incorrect. In 1993 Wu-Yi-Hsang at the University of California, Berkeley published a paper in which he claimed to prove the Kepler conjecture using geometric methods. This was also found to be incorrect.

When presenting the progress of his project in 1996, Hales said that the end was in sight, but it might take "a year or two" to complete. In August 1998 Hales announced that the proof was complete. At that stage it consisted of 250 pages of notes and 3 gigabytes of computer programs, data and results.

Despite the unusual nature of the proof, the editors of the *Annals of Mathematics* agreed to publish it, provided it was accepted by a panel of twelve referees. In 2003, after four years of work, the head of the referee's panel Gábor Fejes Tóth (son of László Fejes Tóth) reported that the panel were "99% certain" of the correctness of the proof, but they could not certify the correctness of all of the computer calculations.

In February 2003 Hales published a 100-page paper describing the non-computer part of his proof in detail.

- L.G. Szpiro (2003)
*Kepler's Conjecture*Wiley, John & Sons Inc. ISBN 0471086010 - Thomas C. Hales (2003)
*A Proof of the Kepler Conjecture*