Sphere packing
|
|
In mathematics, sphere packing problems are problems concerning arrangements of non-overlapping identical spheres which fill a space. Usually the space involved is three-dimensional Euclidean space. However, sphere packing problems can be generalised to two dimensional space (where the "spheres" are circles), to n-dimensional space (where the "spheres" are hyperspheres) and to non-Euclidean spaces such as hyperbolic space.
A typical sphere packing problem is to find an arrangement in which the spheres fill as large a proportion of the space as possible. The proportion of space filled by the spheres is called the density of the arrangement. As the density of an arrangement can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume.
A regular arrangement (also called a periodic or lattice arrangement) is one in which the centres of the spheres form a very symmetric pattern called a lattice. Arrangements in which the spheres are not arranged in a lattice are called irregular or aperiodic arrangements. Regular arrangements are easier to handle than irregular ones — their high degree of symmetry makes it easier to classify them and to measure their densities.
| Contents |
Circle packing
In two dimensional Euclidean space, German mathematician Carl Friedrich Gauss proved that the regular arrangement of circles with the highest density is the hexagonal packing arrangement, in which the centres of the circles are arranged in a hexagonal lattice (like a honeycomb), and each circle is surrounded by 6 other circles. The density of this arrangement is
- <math>\frac{\pi}{\sqrt{12}} \simeq 0.9069<math>
In 1940, Hungarian mathematician László Fejes Tóth proved that the hexagonal lattice is the densest of all possible circle packings, both regular and irregular.
The branch of mathematics generally known as "circle packing", however, is not concerned with dense packing of equal-sized circles but with the geometry and combinatorics of packings of arbitrarily-sized circles; these give rise to discrete analogs of conformal mapping, Riemann surfaces and the like.
Sphere packing
In three dimensional Euclidean space, Gauss proved that the regular arrangements of spheres with the highest density are two very similar arrangements called cubic close packing (or face centred cubic) and hexagonal close packing. In both of these arrangements each sphere is surrounded by 12 other spheres, and both arrangements have an average density of
- <math>\frac{\pi}{\sqrt{18}} \simeq 0.74048<math>
In 1661 Johannes Kepler had conjectured that this is the maximum possible density for both regular and irregular arrangements — this became known as the Kepler conjecture. In 1998 Thomas Hales, 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 has almost certainly been proved.
Hypersphere packing
In dimensions higher than three, the densest regular packings of hyperspheres are known up to 8 dimensions. Very little is known about irregular hypersphere packings — it is possible that in some dimensions the densest packing may be irregular. Some support for this conjecture comes from the fact that in certain dimensions (e.g. 10) the densest known irregular packing is better than the densest known regular packing.
Dimension 24 is special due to the existence of the Leech lattice, which has the best kissing number and for a long time was suspected to be the densest lattice packing. In 2004, Cohn and Kumar 1 published a preprint proving this conjecture, and in addition showing that an irregular packing may improve over the Leech lattice packing, if at all, by no more than <math>2\times 10^{-30}<math>.
Another line of research in high dimensions is trying to find asymptotic bounds for the density of the densest packings. Currently the best known result is that there exists a lattice in dimension n with density bigger or equal to <math>cn2^{-n}<math> for some number c.
Hyperbolic space
Although the concept of circles and spheres can be extended to hyperbolic space, finding the densest packing becomes much more difficult. In a hyperbolic space there is no limit to the number of spheres that can surround another sphere (for example, Ford circles can be thought of as an arrangement of identical hyperbolic circles in which each circle is surrounded by an infinite number of other circles). The concept of average density also becomes much more difficult to define accurately.
Despite these difficulties, Charles Radin and Lewis Bowen of the University of Texas showed in May 2002 that the densest packings in any hyperbolic space are almost always irregular.
Other spaces
Sphere packing on the corners of a hypercube (with the spheres defined by Hamming distance) corresponds to designing error-correcting codes: if the spheres have radius d, then their centers are codewords of a d-error-correcting code. Lattice packings correspond to linear codes. There are other, subtler relationships between Euclidean sphere packing and error-correcting codes; thus, the binary Golay code is closely related to the 24-dimensional Leech lattice.
See also
References
- Conway, J.H. & Sloane, N.J.H. (1998) "Sphere Packings, Lattices and Groups" (Third Edition). ISBN 0387985859
- Lewis Bowen & Charles Radin (2003) "Densest Packings of Equal Spheres in Hyperbolic Space" (ftp://ftp.ma.utexas.edu/pub/papers/radin/hyper.pdf) (pre-print of article in Discrete & Computational Geometry)
- N. J. A. Sloane, The Sphere Packing Problem, [1] (http://arxiv.org/abs/math.CO/0207256) (A technical survey from 2002).
- C. A. Rogers, Existence Theorems in the Geometry of Numbers, The Annals of Mathematics, 2nd Ser., 48:4 (1947), 994-1002 (The <math>n2^{-n}<math> result mentioned above. Despite 60 years of research, only the constant was improved in this result).
- Henry Cohn and Abhinav Kumar, The densest lattice in twenty-four dimensions, [2] (http://front.math.ucdavis.edu/math.MG/0403263) (The solution for the 24 dimensional case).
External links
- Dana Mackenzie (May 2002) "A fine mess" (http://www.ma.utexas.edu/users/radin/reviews/newscientist2.html) (New Scientist)
- A non-technical overview of packing in hyperbolic space.de:Dichteste Kugelpackung