Kepler conjecture
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In mathematics, the Kepler conjecture is a conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is a little over 74%.
In 1998 Thomas Hales, currently 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 being accepted as a theorem.
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[edit] Background
Imagine filling a large container with small equal-sized spheres. The density of the arrangement is the proportion of the volume of the container that is taken up by the spheres. In order to maximize the number of spheres in the container, you need to find an arrangement with the highest possible density, so that the spheres are packed together as closely as possible.
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
The Kepler conjecture says that this is the best that can be done—no other arrangement of spheres has a higher average density.
[edit] Origins
The conjecture is named after Johannes Kepler, who stated the conjecture in 1611 in Strena sue de nive sexangula (On the Six-Cornered Snowflake). Kepler had started to study arrangements of spheres as a result of his correspondence with the English mathematician and astronomer Thomas Harriot in 1606. Harriot was a friend and assistant of Sir Walter Raleigh, who had set Harriot the problem of determining how best to stack cannon balls on the decks of his ships. Harriot published a study of various stacking patterns in 1591, and went on to develop an early version of atomic theory.
[edit] Nineteenth century
Kepler did not have a proof of the conjecture, and the next step was taken by German mathematician Carl Friedrich Gauss, who published a partial solution in 1831. Gauss proved that the Kepler conjecture is true if the spheres have to be arranged in a regular lattice.
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.
[edit] Twentieth century
The next step towards a solution was taken by Hungarian mathematician László Fejes Tóth. In 1953 Fejes Tóth showed that the problem of determining the maximum density of all arrangements (regular and irregular) could be reduced to a finite (but very large) number of calculations. This meant that a proof by exhaustion was, in principle, possible. As Fejes Tóth realised, a fast enough computer could turn this theoretical result into a practical approach to the 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 Hsiang at the University of California, Berkeley published a paper in which he claimed to prove the Kepler conjecture using geometric methods. Some experts countered, claiming he gave insufficient support for some of his claims. Although nothing incorrect per se was found in Hsiang's work, general consensus has been reached, concluding that Hsiang's proof is incomplete. One of the most vocal critics was Thomas Hales, who at the time was working on his own proof.
[edit] Hales' proof
Following the approach suggested by Fejes Tóth, Thomas Hales, then at the University of Michigan, determined that the maximum density of all arrangements could be found by minimizing a function with 150 variables. In 1992, assisted by his graduate student Samuel Ferguson, he embarked on a research programme to systematically apply linear programming methods to find a lower bound on the value of this function for each one of a set of over 5,000 different configurations of spheres. If a lower bound (for the function value) could be found for every one of these configurations that was greater than the value of the function for the cubic close packing arrangement, then the Kepler conjecture would be proved. To find lower bounds for all cases involved solving around 100,000 linear programming problems.
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.
The Annals of Mathematics is going ahead with publishing the theoretical portions of Hales' proof. The computational portions will be published in a separate journal, Discrete and Computational Geometry.
[edit] A formal proof
In January 2003 Hales announced the start of a collaborative project to produce a complete formal proof of the Kepler conjecture. The aim is to remove any remaining uncertainty about the validity of the proof by creating a formal proof that can be verified by automated proof-checking software such as HOL. This project is called Project FlysPecK - the F, P and K standing for Formal Proof of Kepler. Hales estimates that producing a complete formal proof will take around 20 years of work.
[edit] Related problems
- Thue's theorem
- The regular hexagonal packing is the densest sphere packing in the plane. (1890)
- The 2-dimensional analog of the Kepler conjecture; the proof is elementary.
- The hexagonal honeycomb conjecture
- The most efficient partition of the plane into equal areas is the regular hexagonal tiling. Hales' proof (1999).
- Related to Thue's theorem.
- The dodecahedron conjecture
- The volume of the Voronoi polyhedron of a sphere in a packing of equal spheres is at least the volume of a regular dodecahedron with inradius 1. McLaughlin's proof, for which he received the 1999 Morgan Prize.
- A related problem, whose proof uses similar techniques to Hales' proof of the Kepler conjecture. Conjecture by L. Fejes Tóth in the 1950s.
- The Kelvin problem
- What is the most efficient foam in 3 dimensions? This was conjectured to be solved by the Kelvin structure, and this was widely believed for over 100 years, until disproved by the discovery of the Weaire-Phelan structure. The surprising discovery of the Weaire-Phelan structure and disproof of the Kelvin conjecture is one reason for the caution in accepting Hales' proof of the Kepler conjecture.
- Sphere packing in higher dimensions
- The optimal sphere packing question in dimensions bigger than 3 is still open.
[edit] References
- G.G. Szpiro (2003) Kepler's Conjecture Wiley, John & Sons Inc. ISBN 0-471-08601-0
- Thomas C. Hales (2003) A Proof of the Kepler ConjecturePDF (686 KiB)
- Thomas C. Hales (1999) Cannonballs and Honeycombs. An elementary exposition of the proof of the Kepler conjecture.
- T. Aste and D. Weaire "The Pursuit of Perfect Packing" (Institute Of Physics Publishing London 2000) ISBN 0-7503-0648-3
