Editing Sphere packing (section)

==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>