Editing Kissing number (section)

===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.