The Kepler conjecture, named after the 17th-century German mathematician and astronomer Johannes Kepler, is a 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 Hales, following an approach suggested by Fejes Tóth (1953), announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the 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.
Imagine filling a large container with small equal-sized spheres. The density of the arrangement is equal to the collective volume of the spheres divided by the volume of the container. To maximize the number of spheres in the container means to create 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
The Kepler conjecture says that this is the best that can be done—no other arrangement of spheres has a higher average density.
The conjecture was first stated by Johannes Kepler (1611) in his paper 'On the six-cornered snowflake'. He 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.
Kepler did not have a proof of the conjecture, and the next step was taken by Carl Friedrich Gauss (1831), who 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.
The next step toward a solution was taken by Hungarian mathematician László Fejes Tóth. Fejes Tóth (1953) 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 (1958) established an upper bound value of about 78%, and subsequent efforts by other mathematicians reduced this value slightly, but this was still much larger than the cubic close packing density of about 74%.
Hsiang (1993, 2001) claimed to prove the Kepler conjecture using geometric methods. However Gábor Fejes Tóth (the son of László Fejes 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." Hales (1994) gave a detailed criticism of Hsiang's work, to which Hsiang (1995) responded. The current consensus is that Hsiang's proof is incomplete.
Following the approach suggested by Fejes Tóth (1953), 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.
Hales (2005) published a 100-page paper describing the non-computer part of his proof in detail. Hales & Ferguson (2006) and several subsequent papers described the computational portions. Hales and Ferguson received the 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. This project is called 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.
Christian Marchal's simplified proof
Many attempts of proofs have used cells associated to each sphere and as small as possible. These cells are disjoint from each other, but their union fills the entire space, and each of them is in the immediate vicinity of the corresponding sphere (no use of the paradoxes of infinity). The ratios of the volume of each sphere to that of its cell give some information about the coefficient of occupation of space; if all these ratios are equal they define this coefficient.
The most famous cells are the Voronoi cells, that is the portion of space closer to the sphere of interest than to any other sphere of the considered packing.
In 1998 Sean MacLaughin has proved that the smallest Voronoi cell is the regular dodecahedron circumscribing the sphere, see MacLaughin and Hales (2010). The ratio of the volumes of these two figures is 0.754697..., and thus this proof has given the certainty that the coefficient of occupation of space by equal spheres cannot be above this ratio, very close to the density 0.74048... of Kepler's limit. Kepler's conjecture was then almost proven.
The same year Thomas C. Hales presented a 250 pages proof of Kepler's conjecture to the Annals of Mathematics, using sophisticated mathematics and long computations. This proof has been reduced to 120 pages, Hales (2005), and the panel of referees in charge of checking it claims that it is almost sure of the validity of the proof.
A much simpler proof was presented by C. Marchal (2011), published online in 2009. Nevertheless, also this proof must use a definition of cells using two successive steps, beginning with the definition of four types of associated domains. The proof leads from the analysis of general packings (with an infinite number of parameters) to the analysis of the thirteen nearest spheres surrounding a given sphere placed at the origin. This latter analysis requires a final lengthy but simple numerical verification that has been successfully accomplished by Georges Hoynant in March 2011. The proof is thus complete and Kepler is right.
- 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. Henk and Ziegler attribute this result to Lagrange, in 1773 (see references, pag. 770).
- 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.
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- Hales, Thomas C. (2000), "Cannonballs and honeycombs", Notices of the American Mathematical Society 47 (4): 440–449, ISSN 0002-9920, MR 1745624 An elementary exposition of the proof of the Kepler conjecture.
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- Kepler, Johannes (1611), Strena seu de nive sexangula (The six-cornered snowflake), ISBN 978-1-58988-053-5, MR 0927925, lay summary
- MacLaughin, Sean; Hales, Thomas (2010), The dodecahedral conjecture, J. Amer. Math. Soc. 23 (2), pp. 299–344
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- Weisstein, Eric W., "Kepler Conjecture", MathWorld.
- Front page of 'On the six-cornered snowflake'
- Thomas Hales' home page
- Flyspeck project 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