Editing Sphere packing

{{Short description|Geometrical structure}}
{{Use dmy dates|date=November 2017}}
[[File:Rye Castle, Rye, East Sussex, England-6April2011 (1) (cropped).jpg|thumb|Sphere packing finds practical application in the stacking of [[cannonball]]s.]]

In [[geometry]], a '''sphere packing''' is an arrangement of non-overlapping [[sphere]]s within a containing space. The spheres considered are usually all of identical size, and the space is usually three-[[dimension]]al [[Euclidean space]]. However, sphere [[packing problem]]s can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes [[circle packing]] in two dimensions, or [[hypersphere]] packing in higher dimensions) or to [[Non-Euclidean geometry|non-Euclidean]] spaces such as [[hyperbolic space]].

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 ''[[packing 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 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%.<ref>{{Cite journal |last1=Wu |first1=Yugong |last2=Fan |first2=Zhigang |last3=Lu |first3=Yuzhu |date=2003-05-01 |title=Bulk and interior packing densities of random close packing of hard spheres |url=https://doi.org/10.1023/A:1023597707363 |journal=Journal of Materials Science |language=en |volume=38 |issue=9 |pages=2019–2025 |doi=10.1023/A:1023597707363 |s2cid=137583828 |issn=1573-4803}}</ref>

== Classification and terminology ==
A '''[[lattice (group)|lattice]]''' arrangement (commonly called a '''regular''' arrangement) is one in which the centers of the spheres form a very symmetric pattern which needs only ''n'' vectors to be uniquely defined (in ''n''-[[dimension]]al [[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 '''non-periodic''') or '''[[randomness|random]]'''. Because of their high degree of [[symmetry]], lattice packings are easier to classify than non-lattice ones. Periodic lattices always have well-defined densities.

== Regular packing ==
[[File:Order and Chaos.tif|thumb|Regular arrangement of equal spheres in a plane changing to an irregular arrangement of unequal spheres (bubbles).]]
[[Image:close packing box.svg|thumb|right|160px| HCP lattice (left) and the FCC lattice (right) are the two most common highest density arrangements.]]

[[Image:Closepacking.svg|thumb|right|160px|Two ways to stack three planes made of spheres]]

===Dense packing===
{{main|Close-packing of equal spheres}}
In three-dimensional Euclidean space, the densest packing of equal spheres is achieved by a family of structures called [[Close-packing of spheres|close-packed]] structures. One method for generating such a structure is as follows. Consider a plane with a compact arrangement of spheres on it. Call it A. 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 for half of the holes in a second plane above the first, we create a new compact layer. There are two possible choices for doing this, call them B and C. Suppose that we chose B. Then one half of the hollows of B lies above the centers of the balls in A and one half lies above the hollows of A which were not used for B. Thus the balls of a third layer can be placed either directly above the balls of the first one, yielding a layer of type A, or above the holes of the first layer which were not occupied by the second layer, yielding a layer of type C. Combining layers of types A, B, and C produces various close-packed structures.

Two simple arrangements within the close-packed family correspond to regular lattices. One is called cubic close packing (or [[Cubic crystal system|face-centred cubic]], "FCC")—where the layers are alternated in the ABCABC... sequence. The other is called [[Hexagonal crystal family |hexagonal close packing]] ("HCP"), where the layers are alternated in the ABAB... sequence.{{Dubious|FCC and HCP are not both lattices?|date=April 2022}} 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,<ref name="grancrys">{{Cite journal |last1=Dai |first1=Weijing |last2=Reimann |first2=Joerg |last3=Hanaor |first3=Dorian |last4=Ferrero |first4=Claudio |last5=Gan |first5=Yixiang |date=2019-03-13 |title=Modes of wall induced granular crystallisation in vibrational packing |url=https://hal.science/hal-02355176/document/#page=6 |journal=Granular Matter |language=en |volume=21 |issue=2 |pages=26 |doi=10.1007/s10035-019-0876-8 |arxiv=1805.07865 |s2cid=254106945 |issn=1434-7636}}</ref> and the average density is
:<math>\frac{\pi}{3\sqrt{2}} \approx 0.74048.</math>

In 1611, [[Johannes Kepler]] conjectured that this is the maximum possible density amongst both regular and irregular arrangements—this became known as the [[Kepler conjecture]]. [[Carl Friedrich Gauss]] proved in 1831 that these packings have the highest density amongst all possible lattice packings.<ref>{{cite journal|first=C. F.|last=Gauß|author-link=Carl Friedrich Gauss|title=Besprechung des Buchs von L. A. Seeber: ''Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen'' usw|trans-title=Discussion of L. A. Seeber's book: ''Studies on the characteristics of positive ternary quadratic forms'' etc|journal=Göttingsche Gelehrte Anzeigen|year=1831}}</ref> 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.<ref>{{cite web|url=https://code.google.com/p/flyspeck/wiki/AnnouncingCompletion|website=Google Code Archive|title=Long-term storage for Google Code Project Hosting}}</ref>

=== Other common lattice packings ===

Some other lattice packings are often found in physical systems. These include the cubic lattice with a density of <math> \frac{\pi}{6} \approx 0.5236</math>, the hexagonal lattice with a density of <math>\frac{\pi}{3\sqrt{3}}\approx 0.6046</math> and the tetrahedral lattice with a density of <math>\frac{\pi\sqrt{3}}{16}\approx 0.3401</math>.<ref>{{cite web | title=Wolfram Math World, Sphere packing | url=http://mathworld.wolfram.com/SpherePacking.html}}</ref>

===Jammed packings with a low density===
Packings where all spheres are constrained by their neighbours to stay in one location are called rigid or [[Jamming (physics)|jammed]]. The strictly jammed (mechanically stable even as a finite system) regular sphere packing with the lowest known density is a diluted ("tunneled") fcc crystal with a density of only {{math|''π''{{sqrt|2}}/9 ≈ 0.49365}}.<ref>{{Cite journal | last1 = Torquato | first1 = S. | author-link1 = Salvatore Torquato | last2 = Stillinger | first2 = F. H. | title = Toward the jamming threshold of sphere packings: Tunneled crystals | journal = Journal of Applied Physics | volume = 102 | year = 2007 | issue = 9 | pages = 093511–093511–8 | doi = 10.1063/1.2802184 | arxiv = 0707.4263 | bibcode = 2007JAP...102i3511T | s2cid = 5704550 }}
</ref> The loosest known regular jammed packing has a density of approximately 0.0555.<ref>{{cite web | title=Wolfram Math World, Sphere packing | url=http://mathworld.wolfram.com/SpherePacking.html}}</ref>

== Irregular packing ==
{{main|Random close pack}}
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.<ref>{{cite journal|title=Random thoughts|first=Paul|last=Chaikin|date=June 2007|journal=Physics Today|page=8|issn=0031-9228|publisher=American Institute of Physics|volume=60|issue=6|doi=10.1063/1.2754580|bibcode = 2007PhT....60f...8C }}</ref> Vibration of a random loose packing can result in the arrangement of spherical particles into regular packings, a process known as [[Granular material#Pattern formation|granular crystallisation]]. Such processes depend on the geometry of the container holding the spherical grains.<ref name=grancrys/>

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%<ref name="nature">{{cite journal |last1=Song |first1=C. |last2=Wang |first2=P. |last3=Makse |first3=H. A. |date=29 May 2008 |title=A phase diagram for jammed matter |journal=[[Nature (journal)|Nature]] |volume=453 |pages=629–632 |doi=10.1038/nature06981 |pmid=18509438 |issue=7195 |bibcode = 2008Natur.453..629S |arxiv = 0808.2196 |s2cid=4420652 }}</ref> 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 [[circle packing|packing circles]] on a plane. In one dimension it is packing line segments into a linear universe.<ref>{{cite journal | last1 = Griffith | first1 = J.S. | year = 1962 | title = Packing of equal 0-spheres | journal = Nature | volume = 196 | issue = 4856| pages = 764–765 | doi = 10.1038/196764a0 | bibcode = 1962Natur.196..764G | s2cid = 4262056 }}</ref>

In dimensions higher than three, the densest lattice packings of hyperspheres are known for 8 and 24 dimensions.<ref>{{MathWorld |title=Hypersphere Packing|urlname=HyperspherePacking}}</ref> 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.<ref>{{cite journal | last=Sloane |first=N. J. A. | title=The Sphere-Packing Problem | year=1998 | pages=387–396 | journal=Documenta Mathematica|volume=3 | arxiv=math/0207256|bibcode = 2002math......7256S }}</ref>

In 2016, [[Maryna Viazovska]] announced a proof that the [[E8 lattice|E<sub>8</sub> lattice]] provides the optimal packing (regardless of regularity) in eight-dimensional space,<ref>{{Cite journal|last=Viazovska|first=Maryna|date=1 January 2017|title=The sphere packing problem in dimension 8|url=http://annals.math.princeton.edu/2017/185-3/p07|journal=Annals of Mathematics|language=en-US|volume=185|issue=3|pages=991–1015|arxiv=1603.04246|doi=10.4007/annals.2017.185.3.7|s2cid=119286185|issn=0003-486X}}</ref> and soon afterwards she and a group of collaborators announced a similar proof that the [[Leech lattice]] is optimal in 24 dimensions.<ref>{{Cite journal|last1=Cohn|first1=Henry|last2=Kumar|first2=Abhinav|last3=Miller|first3=Stephen|last4=Radchenko|first4=Danylo|last5=Viazovska|first5=Maryna|date=1 January 2017|title=The sphere packing problem in dimension 24|url=http://annals.math.princeton.edu/2017/185-3/p08|journal=Annals of Mathematics|language=en-US|volume=185|issue=3|pages=1017–1033|arxiv=1603.06518|doi=10.4007/annals.2017.185.3.8|s2cid=119281758|issn=0003-486X}}</ref> This result built on and improved previous methods which showed that these two lattices are very close to optimal.<ref>{{Citation | last1=Cohn | first1=Henry | last2=Kumar | first2=Abhinav | title=Optimality and uniqueness of the Leech lattice among lattices | doi=10.4007/annals.2009.170.1003 | mr=2600869 | zbl=1213.11144 | year=2009 | journal=Annals of Mathematics | issn=1939-8980 | volume=170 | issue=3 | pages=1003–1050 | arxiv=math.MG/0403263 | s2cid=10696627 }} {{Citation | last1=Cohn | first1=Henry | last2=Kumar | first2=Abhinav | title=The densest lattice in twenty-four dimensions | doi=10.1090/S1079-6762-04-00130-1 | mr=2075897 | year=2004 | journal=Electronic Research Announcements of the American Mathematical Society | issn=1079-6762 | volume=10 | issue=7 | pages=58–67 |arxiv=math.MG/0408174 | bibcode=2004math......8174C | s2cid=15874595 }}</ref>
The new proofs involve using the [[Laplace transform]] of a carefully chosen [[modular function]] to construct a [[Rotational symmetry|radially symmetric]] function {{mvar|f}} such that {{mvar|f}} and its [[Fourier transform]] {{mvar|f̂}}&nbsp; both equal 1 at the [[origin (mathematics)|origin]], and both vanish at all other points of the optimal lattice, with {{mvar|f}} negative outside the central sphere of the packing and {{mvar|f̂}} positive. Then, the [[Poisson summation formula]] for {{mvar|f}}&nbsp; is used to compare the density of the optimal lattice with that of any other packing.<ref>{{citation|url=https://www.youtube.com/watch?v=8qlZjarkS_g |archive-url=https://ghostarchive.org/varchive/youtube/20211221/8qlZjarkS_g |archive-date=2021-12-21 |url-status=live|first=Stephen D.|last=Miller|date=4 April 2016|title=The solution to the sphere packing problem in 24 dimensions via modular forms|publisher=[[Institute for Advanced Study]]}}{{cbignore}}. Video of an hour-long talk by one of Viazovska's co-authors explaining the new proofs.</ref> Before the proof had been [[Scholarly peer review|formally refereed]] and published, mathematician [[Peter Sarnak]] called the proof "stunningly simple" and wrote that "You just start reading the paper and you know this is correct."<ref>{{citation|last1=Klarreich|first1=Erica|author-link1=Erica Klarreich|title=Sphere Packing Solved in Higher Dimensions|url=https://www.quantamagazine.org/sphere-packing-solved-in-higher-dimensions-20160330/|magazine=Quanta Magazine|date=30 March 2016}}</ref>

Another line of research in high dimensions is trying to find [[asymptotic]] bounds for the density of the densest packings. It is known that for large {{mvar|n}}, the densest lattice in dimension {{mvar|n}} has density <math>\theta(n)</math> between {{math|''cn'' &sdot; 2<sup>−''n''</sup>}} (for some constant {{mvar|c}}) and {{math|2<sup>−(0.599+o(1))''n''</sup>}}.<!--<ref>{{cite journal|last=Rogers|first=C. A.|title=Existence Theorems in the Geometry of Numbers|journal=Annals of Mathematics |series=Second Series|volume=48|issue=4|year=1947|pages=994–1002|jstor=1969390|doi=10.2307/1969390}}</ref>--><ref>{{Citation | last1=Cohn | first1=Henry | title=A conceptual breakthrough in sphere packing | doi=10.1090/noti1474 | mr=3587715 | year=2017 | journal=Notices of the American Mathematical Society| url=https://www.ams.org/journals/notices/201702/rnoti-p102.pdf| issn=0002-9920| volume=64 | issue=2 | pages=102–115 |arxiv=1611.01685| s2cid=16124591 }}</ref>  Conjectural bounds lie in between.<ref>{{Citation| last1=Torquato| first1=S.| last2=Stillinger |first2=F. H.| title=New conjectural lower bounds on the optimal density of sphere packings| journal=Experimental Mathematics| year=2006| volume=15| issue=3| pages=307&ndash;331 | url=https://www.tandfonline.com/doi/pdf/10.1080/10586458.2006.10128964|doi=10.1080/10586458.2006.10128964| arxiv=math/0508381| mr=2264469| s2cid=9921359}}</ref> In a 2023 preprint, Marcelo Campos, Matthew Jenssen, Marcus Michelen and [[Julian Sahasrabudhe]] announced an improvement to the lower bound of the maximal density to <math>\theta(n)\geq (1-o(1))\frac{n\ln n}{2^{n+1}}</math>,<ref>{{Cite arXiv |last1=Campos |first1=Marcelo |last2=Jenssen |first2=Matthew |last3=Michelen |first3=Marcus |last4=Sahasrabudhe |first4=Julian |date=2023 |title=A new lower bound for sphere packing |class=math.MG |eprint=2312.10026}}</ref><ref>{{Cite web |last=Houston-Edwards |first=Kelsey |date=2024-04-30 |title=To Pack Spheres Tightly, Mathematicians Throw Them at Random |url=https://www.quantamagazine.org/to-pack-spheres-tightly-mathematicians-throw-them-at-random-20240430/ |access-date=2024-04-30 |website=Quanta Magazine |language=en}}</ref> among their techniques they make use of the [[Rödl nibble]].

==Unequal sphere packing{{anchor|Unequal}}==
{{see also|Unequal circle packing}}
[[File:Binary sphere packing LS3.png|thumb|A dense packing of spheres with a radius ratio of 0.64799 and a density of 0.74786<ref name="doi10.1021/jp206115p">{{Cite journal | last1 = O'Toole | first1 = P. I. | last2 = Hudson | first2 = T. S. | doi = 10.1021/jp206115p | title = New High-Density Packings of Similarly Sized Binary Spheres | journal = The Journal of Physical Chemistry C | volume = 115 | issue = 39 | pages = 19037 | year = 2011 }}</ref>]]
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 [[interstitial compound|interstitial]] packing. When many sizes of spheres (or a [[particle size distribution|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 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.<ref>{{Cite journal | last1 = Hudson | first1 = D. R. | title = Density and Packing in an Aggregate of Mixed Spheres | doi = 10.1063/1.1698327 | journal = Journal of Applied Physics | volume = 20 | issue = 2 | pages = 154–162| year = 1949 |bibcode = 1949JAP....20..154H }}</ref> 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.<ref>{{Cite journal | last1 = Zong | first1 = C. | title = From deep holes to free planes | doi = 10.1090/S0273-0979-02-00950-3 | journal = Bulletin of the American Mathematical Society | volume = 39 | issue = 4 | pages = 533–555 | year = 2002 | doi-access = free }}</ref>

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.<ref name="doi10.1021/jp206115p"/><ref>{{cite journal|first1=G. W.|last1=Marshall|first2=T. S.|last2=Hudson|journal=Contributions to Algebra and Geometry|title=Dense binary sphere packings|volume=51|issue=2|pages=337–344|year=2010|url=http://www.emis.de/journals/BAG/vol.51/no.2/3.html}}</ref>

Upper bounds for the density that can be obtained in such binary packings have also been obtained.<ref>{{cite journal |last1=de Laat |first1=David |last2=de Oliveira Filho |first2=Fernando Mário |last3=Vallentin |first3=Frank |title=Upper bounds for packings of spheres of several radii|arxiv=1206.2608|date=12 June 2012 |doi=10.1017/fms.2014.24 |volume=2 |journal=Forum of Mathematics, Sigma|s2cid=11082628 }}</ref>

In many chemical situations such as [[ionic crystal]]s, 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.

The upper bound for the density of a strictly jammed sphere packing with any set of radii is 1{{snd}}an example of such a packing of spheres is the Apollonian sphere packing. The lower bound for such a sphere packing is 0{{snd}}an example is the Dionysian sphere packing.<ref>{{cite journal |last1=Dennis |first1=Robert |last2=Corwin |first2=Eric |title=Dionysian Hard Sphere Packings Are Mechanically Stable at Vanishingly Low Densities |arxiv=2006.11415|date=2 September 2021 | journal=[[Physical Review]] |volume=128 | issue = 1 | pages = 018002 | doi=10.1103/PhysRevLett.128.018002 }}</ref>

==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 circle]]s can be thought of as an arrangement of identical hyperbolic circles in which each circle is surrounded by an [[Infinity|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.<ref>{{Cite journal | last1 = Bowen | first1 = L. | last2 = Radin | first2 = C. | doi = 10.1007/s00454-002-2791-7 | title = Densest Packing of Equal Spheres in Hyperbolic Space | journal = Discrete and Computational Geometry | volume = 29 | pages = 23–39 | year = 2002 | doi-access = free }}</ref>

Despite this difficulty, K. Böröczky gives a universal upper bound for the density of sphere packings of hyperbolic ''n''-space where ''n''&nbsp;≥&nbsp;2.<ref>{{Cite journal | last1 = Böröczky | first1 = K. | title = Packing of spheres in spaces of constant curvature | doi = 10.1007/BF01902361 | doi-access= | journal = [[Acta Mathematica Hungarica|Acta Mathematica Academiae Scientiarum Hungaricae]] | volume = 32 | issue = 3–4 | pages = 243–261 | year = 1978 | s2cid = 122561092 }}</ref> 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}.<ref>{{Cite journal | last1 = Böröczky | first1 = K. | last2 = Florian | first2 = A. | doi = 10.1007/BF01897041 | doi-access= | title = Über die dichteste Kugelpackung im hyperbolischen Raum | journal = [[Acta Mathematica Hungarica|Acta Mathematica Academiae Scientiarum Hungaricae]] | volume = 15 | pages = 237–245 | year = 1964 | issue = 1–2 | s2cid = 122081239 }}</ref> 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.<ref>{{Cite journal | last1 = Kozma | first1 = R. T. | last2 = Szirmai | first2 = J. | doi = 10.1007/s00605-012-0393-x | title = Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types | journal = Monatshefte für Mathematik | volume = 168 | pages = 27–47 | year = 2012 | arxiv = 1007.0722 | s2cid = 119713174 }}</ref>

==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 3-cycles 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 ''n'' dimensions that the cardinality of the set of ''n''-simplices in the contact graph gives the number of touching (''n''&nbsp;+&nbsp;1)-tuples in the sphere packing). In the case of 3-dimensional Euclidean space, non-trivial upper bounds on the number of touching pairs, triplets, and quadruples<ref>{{cite journal| last1 = Bezdek | first1 = Karoly | last2 = Reid | first2 = Samuel | title=Contact Graphs of Sphere Packings Revisited | year = 2013 |arxiv=1210.5756 |journal=Journal of Geometry |volume = 104 |issue = 1 | pages = 57–83 |doi = 10.1007/s00022-013-0156-4| s2cid = 14428585 }}</ref> were proved by [[Karoly Bezdek]] and Samuel Reid at the University of Calgary.

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''.<ref>{{Cite web|date=2017-02-06|title=The Science of Sticky Spheres|url=https://www.americanscientist.org/article/the-science-of-sticky-spheres|access-date=2020-07-14|website=American Scientist|language=en}}</ref>

==Other spaces==
Sphere packing on the corners of a hypercube (with [[Hamming ball]]s, spheres defined by [[Hamming distance]]) corresponds to designing [[error-correcting codes]]: if the spheres have radius ''t'', then their centers are codewords of a (2''t''&nbsp;+&nbsp;1)-error-correcting code. Lattice packings correspond to linear codes. There are other, subtler relationships between Euclidean sphere packing and error-correcting codes. For example, the [[binary Golay code]] is closely related to the 24-dimensional Leech lattice.

For further details on these connections, see the book ''Sphere Packings, Lattices and Groups'' by [[John Horton Conway|Conway]] and [[Neil Sloane|Sloane]].<ref>{{Cite book|url=https://books.google.com/books?id=ITDvBwAAQBAJ|title=Sphere Packings, Lattices and Groups|last1=Conway|first1=John H.|author-link=John Horton Conway|last2=Sloane|first2=Neil J. A.|author-link2=Neil Sloane|publisher=Springer Science & Business Media|year=1998|isbn=0-387-98585-9|edition=3rd}}</ref>

==See also==
{{div col|colwidth=20em}}
* [[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]]
* [[Sphere packing in a sphere]]
{{div col end}}

==References==
{{Reflist|2}}

==Bibliography==
* {{cite book |last1=Aste |first1=T. |last2=Weaire |first2=D. |title=The Pursuit of Perfect Packing|title-link=The Pursuit of Perfect Packing |publisher=Institute of Physics Publishing |location=London |year=2000 |isbn=0-7503-0648-3}}
* {{cite book |last1=Conway |first1=J. H. |author-link=John Horton Conway |last2=Sloane |first2=N. J. H. |author-link2=Neil Sloane |year=1998 |title=Sphere Packings, Lattices and Groups |publisher=Springer |url=https://archive.org/details/spherepackingsla0000conw_b8u0 |url-access=registration |edition=3rd |isbn=0-387-98585-9}}
* {{cite journal | last1 = Sloane | first1 = N. J. A. | author-link1=Neil Sloane | year = 1984 | title = The Packing of Spheres | journal = Scientific American | volume = 250 | pages = 116–125 |bibcode = 1984SciAm.250e.116G |doi = 10.1038/scientificamerican0584-116 }}

==External links==
* Dana Mackenzie (May 2002) [http://www.ma.utexas.edu/users/radin/reviews/newscientist2.html "''A fine mess''"] (New Scientist)
:A non-technical overview of packing in hyperbolic space.
* {{MathWorld|urlname=CirclePacking |title=Circle Packing }}
* [http://www.3doro.de/e-kp.htm "Kugelpackungen (Sphere Packing)"] (T. E. Dorozinski)
*[http://alecjacobson.com/graphics/hw10b/ "3D Sphere Packing Applet"] {{Webarchive|url=https://web.archive.org/web/20090426114733/http://alecjacobson.com/graphics/hw10b/ |date=26 April 2009 }} Sphere Packing java applet
*[http://www.randomwalk.de/sphere/insphr/spheresinsphr.html "Densest Packing of spheres into a sphere"] java applet
*[http://codes.se/packings/ "Database of sphere packings"] (Erik Agrell)

{{Packing problem}}

[[Category:Discrete geometry]]
[[Category:Crystallography]]
[[Category:Packing problems]]
[[Category:Spheres]]