Editing Packing problems (section)

=== Packing of circles ===
{{main article|Circle packing}}

People are given {{mvar|n}} [[unit circle]]s, and have to pack them in the smallest possible container. Several kinds of containers have been studied:

* [[Circle packing in a circle|Packing circles in a '''circle''']] - closely related to spreading points in a unit circle with the objective of finding the greatest minimal separation, {{mvar|d{{sub|n}}}}, between points. Optimal solutions have been proven for {{math|''n'' ≤ 13}}, and {{math|1=''n'' = 19}}.
* [[Circle packing in a square|Packing circles in a '''square''']] - closely related to spreading points in a unit square with the objective of finding the greatest minimal separation, {{mvar|d{{sub|n}}}}, between points. To convert between these two formulations of the problem, the square side for unit circles will be <math>L = 2 + 2/d_n</math>. [[File:15 circles in a square.svg|thumb|120px|right|The optimal packing of 15 circles in a square]]Optimal solutions have been proven for {{math|''n'' ≤ 30}}.
* [[Circle packing in a rectangle|Packing circles in a '''rectangle''']]
* [[Circle packing in an isosceles right triangle|Packing circles in an '''isosceles right triangle''']] - good estimates are known for {{math|''n'' < 300}}.
* [[Circle packing in an equilateral triangle|Packing circles in an '''equilateral triangle''']] - Optimal solutions are known for {{math|''n'' < 13}}, and [[conjecture]]s are available for {{math|''n'' < 28}}.<ref>{{Cite journal | last1 = Melissen | first1 = J. | title = Packing 16, 17 or 18 circles in an equilateral triangle | journal = Discrete Mathematics | volume = 145 | issue = 1–3 | pages = 333–342 | year = 1995 | doi = 10.1016/0012-365X(95)90139-C| url = https://research.utwente.nl/en/publications/packing-16-17-of-18-circles-in-an-equilateral-triangle(b2172f19-9654-4ff1-9af4-59da1b6bef3d).html | doi-access = free }}</ref>

{{Anchor|Packing squares}}