en>Jesse V.: various fixes using AWB

2012-04-12T04:55:12Z

various fixes using AWB

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The '''disk [[covering problem]]''' asks for the smallest [[real number]] <math>r(n)</math> such that <math>n</math> [[disk (mathematics)|disks]] of radius <math>r(n)</math> can be arranged in such a way as to cover the [[unit disk]]. Dually, for a given radius ''ε'', one wishes to find the smallest integer ''n'' such that ''n'' disks of radius ''ε'' can cover the unit disk.<ref>{{citation
| last = Kershner | first = Richard
| journal = American Journal of Mathematics
| mr = 0000043
| pages = 665–671
| title = The number of circles covering a set
| volume = 61
| year = 1939}}.</ref>

The best solutions to date are as follows:

{| class="wikitable" border="1"
|-
! n
! r(n)
|-
| 1
| 1
|-
| 2
| 1
|-
| 3
| <math>\sqrt{3}/2</math>
|-
| 4
| <math>\sqrt{2}/2</math>
|-
| 5
| 0.609382...
|-
| 6
| 0.555905...
|-
| 7
| <math>1/2</math>
|-
| 8
| 0.445041...
|-
| 9
| 0.414213...
|-
| 10
| 0.394930...
|}

==Method==
This is the best known layout strategy for r(9) and r(10):

[[File:DiscCoveringExample.svg]]

==References==
{{reflist}}

==External links==
* Weisstein, Eric W. "Disk Covering Problem." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/DiskCoveringProblem.html
* Finch, S. R. "Circular Coverage Constants." §2.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 484–489, 2003.
* Illustrations of [http://www2.stetson.edu/~efriedma/circovcir/ circles covering circles]

[[Category:Discrete geometry]]
[[Category:Mathematical problems]]

{{geometry-stub}}

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en>Jesse V.: various fixes using AWB