Editing Kissing number (section)

===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.]]