Sphere packing
In
A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the
For equal spheres in three dimensions, the densest packing uses approximately 74% of the volume. A random packing of equal spheres generally has a density around 63.5%.[1]
Classification and terminology
A , lattice packings are easier to classify than non-lattice ones. Periodic lattices always have well-defined densities.
Regular packing
Dense packing
In three-dimensional Euclidean space, the densest packing of equal spheres is achieved by a family of structures called
Two simple arrangements within the close-packed 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.[dubious ] But many layer stacking sequences are possible (ABAC, ABCBA, ABCBAC, etc.), and still generate a close-packed structure. In all of these arrangements each sphere touches 12 neighboring spheres,[2] and the average density is
In 1611,
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 .[5]
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 (mechanically stable even as a finite system) sphere packing with the lowest known density is a diluted ("tunneled") fcc crystal with a density of only π√2/9 ≈ 0.49365.[6] The loosest known jammed packing has a density of approximately 0.0555.[7]
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.[8] Vibration of a random loose packing can result in the arrangement of spherical particles into regular packings, a process known as granular crystallisation. Such processes depend on the geometry of the container holding the spherical grains.[2]
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%[9] This situation is unlike the case of one or two dimensions, where compressing a collection of 1-dimensional or 2-dimensional spheres (that is, line segments or circles) will yield a regular packing.
Hypersphere packing
The sphere packing problem is the three-dimensional version of a class of ball-packing problems in arbitrary dimensions. In two dimensions, the equivalent problem is packing circles on a plane. In one dimension it is packing line segments into a linear universe.[10]
In dimensions higher than three, the densest lattice packings of hyperspheres are known up to 8 dimensions.[11] 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.[12]
In 2016, Maryna Viazovska announced a proof that the E8 lattice provides the optimal packing (regardless of regularity) in eight-dimensional space,[13] and soon afterwards she and a group of collaborators announced a similar proof that the Leech lattice is optimal in 24 dimensions.[14] This result built on and improved previous methods which showed that these two lattices are very close to optimal.[15] The new proofs involve using the
Another line of research in high dimensions is trying to find
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 close-packed equal spheres, or combining the multiple sizes of spheres into a compound or
When the second sphere is much smaller than the first, it is possible to arrange the large spheres in a close-packed 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.[22] Even if the large spheres are not in a close-packed arrangement, it is always possible to insert some smaller spheres of up to 0.29099 of the radius of the larger sphere.[23]
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 close-packed 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.[21][24]
Upper bounds for the density that can be obtained in such binary packings have also been obtained.[25]
In many chemical situations such as
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.[26]
Despite this difficulty, K. Böröczky gives a universal upper bound for the density of sphere packings of hyperbolic n-space where n ≥ 2.[27] In three dimensions the Böröczky bound is approximately 85.327613%, and is realized by the horosphere packing of the order-6 tetrahedral honeycomb with Schläfli symbol {3,3,6}.[28] In addition to this configuration at least three other horosphere packings are known to exist in hyperbolic 3-space that realize the density upper bound.[29]
Touching pairs, triplets, and quadruples
The
The problem of finding the arrangement of n identical spheres that maximizes the number of contact points between the spheres is known as the "sticky-sphere problem". The maximum is known for n ≤ 11, and only conjectural values are known for larger n.[31]
Other spaces
Sphere packing on the corners of a hypercube (with the spheres defined by
For further details on these connections, see the book Sphere Packings, Lattices and Groups by Conway and Sloane.[32]
See also
- Close-packing of equal spheres
- Apollonian sphere packing
- Finite sphere packing
- Hermite constant
- Inscribed sphere
- Kissing number
- Sphere-packing bound
- Random close pack
- Cylinder sphere packing
References
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- ^ Gauß, C. F. (1831). "Besprechung des Buchs von L. A. Seeber: Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen usw" [Discussion of L. A. Seeber's book: Studies on the characteristics of positive ternary quadratic forms etc]. Göttingsche Gelehrte Anzeigen.
- ^ "Long-term storage for Google Code Project Hosting". Google Code Archive.
- ^ "Wolfram Math World, Sphere packing".
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- ^ Weisstein, Eric W. "Hypersphere Packing". MathWorld.
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- ^ Miller, Stephen D. (4 April 2016), The solution to the sphere packing problem in 24 dimensions via modular forms, Institute for Advanced Study, archived from the original on 21 December 2021. Video of an hour-long talk by one of Viazovska's co-authors explaining the new proofs.
- ^ Klarreich, Erica (30 March 2016), "Sphere Packing Solved in Higher Dimensions", Quanta Magazine
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- arXiv:2312.10026 [math.MG].
- ^ .
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- ^ Marshall, G. W.; Hudson, T. S. (2010). "Dense binary sphere packings". Contributions to Algebra and Geometry. 51 (2): 337–344.
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- ^ "The Science of Sticky Spheres". American Scientist. 6 February 2017. Retrieved 14 July 2020.
- ISBN 0-387-98585-9.
Bibliography
- Aste, T.; Weaire, D. (2000). ISBN 0-7503-0648-3.
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External links
- Dana Mackenzie (May 2002) "A fine mess" (New Scientist)
- A non-technical overview of packing in hyperbolic space.
- Weisstein, Eric W. "Circle Packing". MathWorld.
- "Kugelpackungen (Sphere Packing)" (T. E. Dorozinski)
- "3D Sphere Packing Applet" Archived 26 April 2009 at the Wayback Machine Sphere Packing java applet
- "Densest Packing of spheres into a sphere" java applet
- "Database of sphere packings" (Erik Agrell)