Sphere packing
In geometry, a sphere packing is an arrangement of nonoverlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually threedimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, ndimensional Euclidean space (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to nonEuclidean 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 local density of a packing in an infinite space 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.
For equal spheres the densest packing uses approximately 74% of the volume. Random packing of equal spheres generally have a density around 64%.
Classification and terminology
A lattice arrangement (commonly called a regular arrangement) is one in which the centers of the spheres form a very symmetric pattern which only needs n vectors to be uniquely defined (in ndimensional Euclidean space). Lattice arrangements are periodic. Arrangements in which the spheres do not form a lattice (often referred to as irregular) can still be periodic, but also aperiodic (properly speaking nonperiodic) or random. Lattice arrangements are easier to handle than irregular ones—their high degree of symmetry makes it easier to classify them and to measure their densities.
Regular packing
Dense packing
In threedimensional Euclidean space, the densest packing of equal spheres is achieved by a family of structures called closepacked structures. One method for generating such a structure is as follows. Consider a plane with a compact arrangement of spheres on it. For any three neighbouring spheres, a fourth sphere can be placed on top in the hollow between the three bottom spheres. If we do this "everywhere" in a second plane above the first, we create a new compact layer. A third layer can be placed directly above the first one, or the spheres can be offset, vertically above another set of hollows of the first layer. There are thus three types of planes, called A, B and C.
Two simple arrangements within the closepacked family correspond to regular lattices. One is called cubic close packing (or face centred cubic, "FCC") — where the layers are alternated in the ABCABC… sequence. The other is called hexagonal close packing ("HCP") — where the layers are alternated in the ABAB… sequence. But many layer stacking sequences are possible (ABAC, ABCBA, ABCBAC, etc.), and still generate a closepacked structure. In all of these arrangements each sphere is surrounded by 12 other spheres, and the average density is
Gauss proved in 1831 that these packings have the highest density amongst all possible lattice packings.^{}
In 1611 Johannes Kepler had conjectured that this is the maximum possible density amongst both regular and irregular arrangements — this became known as the Kepler conjecture. In 1998, Thomas Callister Hales, following the approach suggested by László Fejes Tóth in 1953, announced a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof. On 10 August 2014 Hales announced the completion of a formal proof using automated proof checking, removing any doubt.^{}
Other common lattice packings
Some other lattice packings are often found in physical systems. These include the cubic lattice with a density of , the hexagonal lattice with a density of , and the tetrahedral lattice with a density of and loosest possible at a density of 0.0555.^{}
Jammed packings with a low density
Packings where all spheres are constrained by their neighbours to stay in one location are called rigid or jammed. The strictly jammed sphere packing with the lowest density is a diluted ("tunneled") fcc crystal with a density of only 0.49365.^{}
Irregular packing
If we attempt to build a densely packed collection of spheres we will be tempted to always place the next sphere in a hollow between three packed spheres. If five spheres are assembled in this way, they will be consistent with one of the regularly packed arrangements described above. However, the sixth sphere placed in this way will render the structure inconsistent with any regular arrangement. This results in the possibility of a random close packing of spheres which is stable against compression.^{}
When spheres are randomly added to a container and then compressed, they will generally form what is known as an "irregular" or "jammed" packing configuration when they can be compressed no more. This irregular packing will generally have a density of about 64%. Recent research predicts analytically that it cannot exceed a density limit of 63.4%^{} This situation is unlike the case of one or two dimensions, where compressing a collection of 1dimensional or 2dimensional spheres (i.e. line segments or disks) will yield a regular packing.
Hypersphere packing
The sphere packing problem is the threedimensional version of a class of ballpacking problems in arbitrary dimensions. In two dimensions, the equivalent problem is packing circles on a plane.
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 denser 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 is the densest lattice packing. No better irregular packing is known, and an irregular packing could, at best, improve over the Leech lattice packing by a factor of less than 1+2×10^{−30}.^{}
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 for some number c.^{}
Unequal sphere packing
Many problems in the chemical and physical sciences can be related to packing problems where more than one size of sphere is available. Here there is a choice between separating the spheres into regions of closepacked equal spheres, or combining the multiple sizes of spheres into a compound or interstitial packing. When many sizes of spheres (or a distribution) are available, the problem quickly becomes intractable, but some studies of binary hard spheres (two sizes) are available.
When the second sphere is much smaller than the first, it is possible to arrange the large spheres in a closepacked arrangement, and then arrange the small spheres within the octahedral and tetrahedral gaps. The density of this interstitial packing depends sensitively on the radius ratio, but in the limit of extreme size ratios, the smaller spheres can fill the gaps with the same density as the larger spheres filled space.^{} Even if the large spheres are not in a closepacked arrangement, it is always possible to insert some smaller spheres of up to 0.29099 of the radius of the larger sphere.^{}
When the smaller sphere has a radius greater than 0.41421 of the radius of the larger sphere, it is no longer possible to fit into even the octahedral holes of the closepacked structure. Thus, beyond this point, either the host structure must expand to accommodate the interstitials (which compromises the overall density), or rearrange into a more complex crystalline compound structure. Structures are known which exceed the close packing density for radius ratios up to 0.659786.^{}^{}
Upper bounds for the density that can be obtained in such binary packings have also been obtained.^{}
In many chemical situations such as ionic crystals, the stoichiometry is constrained by the charges of the constituent ions. This additional constraint on the packing, together with the need to minimize the Coulomb energy of interacting charges leads to a diversity of optimal packing arrangements.
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. The densest packings in any hyperbolic space are almost always irregular.^{}
Despite this difficulty, K. Böröczky gives a universal upper bound for the density of sphere packings of hyperbolic nspace where .^{} In three dimensions the Böröczky bound is approximately 85.327613%, and is realized by the horosphere packing of the order6 tetrahedral honeycomb with Schläfli symbol {3,3,6}.^{} In addition to this configuration at least three other horosphere packings are known to exist in hyperbolic 3space that realize the density upper bound.^{}
Touching pairs, triplets, and quadruples
The contact graph of an arbitrary finite packing of unit balls is the graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. The cardinality of the edge set of the contact graph gives the number of touching pairs, the number of 3cycles in the contact graph gives the number of touching triplets, and the number of tetrahedrons in the contact graph gives the number of touching quadruples (in general for a contact graph associated with a sphere packing in ndimensions that the cardinality of the set of nsimplices in the contact graph gives the number of touching (n+1)tuples in the sphere packing). In the case of 3dimensional Euclidean space, nontrivial upper bounds on the number of touching pairs, triplets, and quadruples^{} were proved by Karoly Bezdek and Samuel Reid at the University of Calgary.
Other spaces
Sphere packing on the corners of a hypercube (with the spheres defined by Hamming distance) corresponds to designing errorcorrecting codes: if the spheres have radius t, then their centers are codewords of a 2t+1errorcorrecting code. Lattice packings correspond to linear codes. There are other, subtler relationships between Euclidean sphere packing and errorcorrecting codes. For example, the binary Golay code is closely related to the 24dimensional Leech lattice.
For further details on these connections, see the book of Conway and Sloane.
See also
 Closepacking of equal spheres
 Apollonian sphere packing
 Hermite constant
 Kissing number problem
 Spherepacking bound
 Random close pack
References
Bibliography
 Aste T, Weaire D (2000). The Pursuit of Perfect Packing. London: Institute of Physics Publishing. ISBN 0750306483.
 Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0387985859.
 Sloane, N. J. A. (1984). "The Packing of Spheres". Scientific American 250: 116–125. Bibcode:1984SciAm.250..116G. doi:10.1038/scientificamerican0584116.
External links
 Dana Mackenzie (May 2002) "A fine mess" (New Scientist)
 A nontechnical overview of packing in hyperbolic space.
 Weisstein, Eric W., "Circle Packing", MathWorld.
 "Kugelpackungen (Sphere Packing)" (T.E. Dorozinski)
 "3D Sphere Packing Applet" Sphere Packing java applet
 "Densest Packing of spheres into a sphere" java applet
 "Database of sphere packings" (Erik Agrell)
