Editing Kissing number (section)

{{Short description|Geometric concept}}
{{unsolved|mathematics|What is the maximum possible kissing number for [[n-sphere|''n''-dimensional spheres]] in (''n'' + 1)-dimensional [[Euclidean space]]?}}
In [[geometry]], the '''kissing number''' of a [[mathematical space]] is defined as the greatest number of non-overlapping unit [[sphere]]s that can be arranged in that space such that they each touch a common unit sphere.  For a given [[sphere packing]] (arrangement of spheres) in a given space, a kissing number can also be defined for each individual sphere as the number of spheres it touches.  For a [[Lattice (group)|lattice]] packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another.

Other names for kissing number that have been used are '''Newton number''' (after the originator of the problem), and '''contact number'''.

In general, the '''kissing number problem''' seeks the maximum possible kissing number for [[n-sphere|''n''-dimensional spheres]] in (''n'' + 1)-dimensional [[Euclidean space]]. Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space.

Finding the kissing number when centers of spheres are confined to a line (the one-dimensional case) or a plane (two-dimensional case) is trivial. Proving a solution to the three-dimensional case, despite being easy to conceptualise and model in the physical world, eluded mathematicians until the mid-20th century.<ref name=Conway/><ref name=Brass/> Solutions in higher dimensions are considerably more challenging, and only a handful of cases have been solved exactly. For others, investigations have determined upper and lower bounds, but not exact solutions.<ref name=Mittlemann>{{cite journal|last1=Mittelmann|first1=Hans D.|last2=Vallentin|first2=Frank|title=High accuracy semidefinite programming bounds for kissing numbers|year=2010|pages=174–178|volume=19|journal=[[Experimental Mathematics (journal)|Experimental Mathematics]]|issue=2|doi=10.1080/10586458.2010.10129070|arxiv=0902.1105|bibcode=2009arXiv0902.1105M|s2cid=218279}}</ref>