Please wait while we load your article...

Home > Kepler Conjecture

Learn more about "Kepler Conjecture"

Kepler conjecture

The '''Kepler conjecture''', named after Johannes Kepler, is a mathematics|mathematical 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 slightly greater than 74%. In 1998 Thomas Callister Hales|Thomas Hales, following an approach suggested by , 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.

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. At each step there are two choices of where to put the next layer, so this natural method of stacking the spheres creates an uncountably infinite number of equally dense packings, the best known of which are called cubic close packing and hexagonal close packing. Each of these arrangements has an average density of
-

\frac{\pi}{\sqrt{18}} \simeq 0.74048.

The Kepler conjecture says that this is the best that can be done—no other arrangement of spheres has a higher average density.

Origins

The conjecture was first stated by , who had started to study arrangements of spheres as a result of his correspondence with the England|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.

Nineteenth century

Kepler did not have a proof of the conjecture, and the next step was taken by , who proved that the Kepler conjecture is true if the spheres have to be arranged in a regular lattice (group)|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 Hilbert's problems|twenty three unsolved problems of mathematics—it forms part of Hilbert's eighteenth problem.

Twentieth century

The next step towards a solution was taken by Magyars|Hungarian mathematician László Fejes Tóth. showed that the problem of determining the maximum density of all arrangements (regular and irregular) could be reduced to a finite set|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 established an upper bound value of about 78%, and subsequent efforts by other mathematicians reduced this value slightly, but this was still larger than the cubic close packing density of about 74%. claimed to prove the Kepler conjecture using geometric methods. However Tóth stated in his review of the paper "As far as details are concerned, my opinion is that many of the key statements have no acceptable proofs." gave a detailed criticism of Hsiang's work, to which responded. The current consensus is that Hsiang's proof is incomplete.

Hales' proof

Following the approach suggested by , 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 program 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. published a 100-page paper describing the non-computer part of his proof in detail. and several subsequent papers described the computational portions. Hales and Ferguson received the Fulkerson Prize|Fulkerson Prize for outstanding papers in the area of discrete mathematics for 2009.

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 theorem prover|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.

References

* *
-
- An elementary exposition of the proof of the Kepler conjecture. * * * * * *
- * * *

External links

*
- Thomas Hales' home page
- Overview of Hales' proof
- Article in American Scientist by Dana Mackenzie
- Flyspeck I: Tame Graphs, verified enumeration of tame plane graphs as defined by Thomas C. Hales in his proof of the Kepler Conjecture Category:Discrete geometry Category:Conjectures Category:Johannes Kepler Category:Hilbert's problems

Related Images

- Diagrams of cubic close packing (left) and hexagonal close packing (right).
- One of the diagrams from ''Strena Seu de Nive Sexangula'', illustrating the Kepler conjecture

Sources: StartLearningNow, Wikipedia | Usage license: GNU FDL

Welcome to Start Learning Now. Explore to your heart's content, and we hope you enjoy reading the material we have assembled for you here!

Related News

Further Resources


	home lucky pick

Related Resources