Editing Kissing number (section)

==Known greatest kissing numbers==
===One dimension===
In one dimension,<ref>Note that in one dimension, "spheres" are just pairs of points separated by the unit distance.  (The vertical dimension of one-dimensional illustration is merely evocative.) Unlike in higher dimensions, it is necessary to specify that the interior of the spheres (the unit-length open intervals) do not overlap in order for there to be a finite packing density.</ref> the kissing number is 2:

[[File:Kissing-1d.svg|center]]

===Two dimensions===
In two dimensions, the kissing number is 6:

[[File:Kissing-2d.svg|center]]

'''Proof''': Consider a circle with center ''C'' that is touched by circles with centers ''C''<sub>1</sub>, ''C''<sub>2</sub>, .... Consider the rays ''C'' ''C''<sub>''i''</sub>. These rays all emanate from the same center ''C'', so the sum of angles between adjacent rays is 360°.

Assume by contradiction that there are more than six touching circles. Then at least two adjacent rays, say ''C'' ''C''<sub>1</sub> and ''C'' ''C''<sub>2</sub>, are separated by an angle of less than 60°. The segments ''C C<sub>i</sub>'' have the same length – 2''r'' – for all ''i''. Therefore, the triangle ''C'' ''C''<sub>1</sub> ''C''<sub>2</sub> is [[isosceles]], and its third side – ''C''<sub>1</sub> ''C''<sub>2</sub> – has a side length of less than 2''r''. Therefore, the circles 1 and 2 intersect – a contradiction.<ref>See also Lemma 3.1 in {{Cite journal | last1 = Marathe | first1 = M. V. | last2 = Breu | first2 = H. | last3 = Hunt | first3 = H. B. | last4 = Ravi | first4 = S. S. | last5 = Rosenkrantz | first5 = D. J. | title = Simple heuristics for unit disk graphs | doi = 10.1002/net.3230250205 | journal = Networks | volume = 25 | issue = 2 | pages = 59 | year = 1995 | arxiv = math/9409226 }}</ref>

[[File:Kissing-3d.png|thumb|A highly symmetrical realization of the kissing number 12 in three dimensions is by aligning the centers of outer spheres with vertices of a [[regular icosahedron]]. This leaves slightly more than 0.1 of the radius between two nearby spheres.]]

===Three dimensions===
In three dimensions, the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two. It is easy to arrange 12 spheres so that each touches a central sphere, with a lot of space left over, and it is not obvious that there is no way to pack in a 13th sphere. (In fact, there is so much extra space that any two of the 12 outer spheres can exchange places through a continuous movement without any of the outer spheres losing contact with the center one.) This was the subject of a famous disagreement between mathematicians [[Isaac Newton]] and [[David Gregory (mathematician)|David Gregory]]. Newton correctly thought that the limit was 12; Gregory thought that a 13th could fit. Some incomplete proofs that Newton was correct were offered in the nineteenth century, most notably one by [[Reinhold Hoppe]], but the first correct proof (according to Brass, Moser, and Pach) did not appear until 1953.<ref name=Conway>{{cite book |first=John H. |last=Conway |author-link=John Horton Conway |author2=Neil J.A. Sloane |author-link2=Neil Sloane  |year=1999 |title=Sphere Packings, Lattices and Groups |edition=3rd |publisher=Springer-Verlag |location=New York |isbn=0-387-98585-9|page=[https://books.google.com/books?id=upYwZ6cQumoC&pg=PA21 21]}}</ref><ref  name=Brass>{{cite book |first1=Peter |last1=Brass |first2=W. O. J. |last2=Moser |first3=János |last3=Pach |author-link3=János Pach |title=Research problems in discrete geometry |publisher=Springer |year=2005 |isbn=978-0-387-23815-9 |page=[https://books.google.com/books?hl=en&id=cT7TB20y3A8C&pg=PA93 93]}}</ref><ref>{{cite book
 | last = Zong | first = Chuanming
 | editor1-last = Goodman | editor1-first = Jacob E.
 | editor2-last = Pach | editor2-first = J├ínos
 | editor3-last = Pollack | editor3-first = Richard
 | contribution = The kissing number, blocking number and covering number of a convex body
 | doi = 10.1090/conm/453/08812
 | location = Providence, RI
 | mr = 2405694
 | pages = 529–548
 | publisher = American Mathematical Society
 | series = Contemporary Mathematics
 | title = Surveys on Discrete and Computational Geometry: Twenty Years Later (AMS-IMS-SIAM Joint Summer Research Conference, June 18ÔÇô22, 2006, Snowbird, Utah)
 | volume = 453
 | year = 2008| isbn = 9780821842393
 }}.</ref>

The twelve neighbors of the central sphere correspond to the maximum bulk [[coordination number]] of an atom in a [[crystal lattice]] in which all atoms have the same size (as in a chemical element). A coordination number of 12 is found in a [[cubic close-packed]] or a [[hexagonal close-packed]] structure.

===Larger dimensions===
In four dimensions, the kissing number is 24. This was proven in 2003 by Oleg Musin.<ref name="Musin">{{cite journal |author=O. R. Musin |title=The problem of the twenty-five spheres |year=2003 |journal=Russ. Math. Surv. |volume=58 |pages=794–795 |doi=10.1070/RM2003v058n04ABEH000651 |issue=4|bibcode=2003RuMaS..58..794M |s2cid=250839515 }}</ref><ref>{{Cite journal|last1=Pfender|first1=Florian|last2=Ziegler|first2=Günter M.|author-link2=Günter M. Ziegler|title=Kissing numbers, sphere packings, and some unexpected proofs|journal=Notices of the American Mathematical Society|date=September 2004|pages=873–883|url=https://www.ams.org/notices/200408/fea-pfender.pdf}}.</ref> Previously, the answer was thought to be either 24 or 25: it is straightforward to produce a packing of 24 spheres around a central sphere (one can place the spheres at the vertices of a suitably scaled [[24-cell]] centered at the origin), but, as in the three-dimensional case, there is a lot of space left over — even more, in fact, than for ''n'' = 3 — so the situation was even less clear.

The existence of the highly symmetrical [[E8 lattice|''E''<sub>8</sub> lattice]] and [[Leech lattice]] has allowed known results for ''n'' = 8 (where the kissing number is 240), and  ''n'' = 24 (where it is 196,560).<ref>{{cite journal| last=Levenshtein | first=Vladimir I. | author-link=Vladimir Levenshtein | year=1979 | title=О границах для упаковок в n-мерном евклидовом пространстве |trans-title=On bounds for packings in ''n''-dimensional Euclidean space | journal=[[Doklady Akademii Nauk SSSR]] | volume=245 | issue=6 | language=ru | pages=1299–1303}}</ref><ref>{{cite journal
| last1=Odlyzko | first1=A. M. | authorlink1=Andrew Odlyzko
| last2=Sloane | first2=N. J. A. | authorlink2=N.J.A. Sloane
| title=New bounds on the number of unit spheres that can touch a unit sphere in n dimensions
| journal=[[Journal of Combinatorial Theory]] | series=Series A
| volume=26
| issue=2
| date=1979
| pages=210–214
| doi=10.1016/0097-3165(79)90074-8 | doi-access=free}}</ref>
The kissing number in ''n'' [[dimension]]s is unknown for other dimensions.

If arrangements are restricted to ''lattice'' arrangements, in which the centres of the spheres all lie on points in a [[Lattice (group)|lattice]], then this restricted kissing number is known for ''n'' = 1 to 9 and ''n'' = 24 dimensions.<ref>{{MathWorld | urlname=KissingNumber |title=Kissing Number}}</ref> For 5, 6, and 7 dimensions the arrangement with the highest known kissing number found so far is the optimal lattice arrangement, but the existence of a non-lattice arrangement with a higher kissing number has not been excluded.