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# 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.

## 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

In 1940, Hungarian mathematician László Fejes Tóth proved that the hexagonal lattice is the densest of all possible circle packings, both regular and irregular.

## Sphere packing

In three dimensional Eucldiean 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

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.

## Hyperbolic space

Although the concept of circles and spheres can be extende 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 Rabin and Lewis Bowen of the University of Texas showed in May 2002 that the densest packings in any hyperbolic space are almost always irregular.