# Disk covering problem

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Disk_covering_problem
> Markdown URL: https://mediated.wiki/source/Disk_covering_problem.md
> Source: https://en.wikipedia.org/wiki/Disk_covering_problem
> Source revision: 1270913042
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{unsolved|mathematics|What is the smallest real number <math>r(n)</math> such that <math>n</math> disks of radius <math>r(n)</math> can be arranged in such a way as to cover the unit disk?}}

The '''disk [covering problem](/source/covering_problem)''' asks for the smallest [real number](/source/real_number)  <math>r(n)</math> such that <math>n</math> [disks](/source/disk_(mathematics)) of radius <math>r(n)</math> can be arranged in such a way as to cover the [unit disk](/source/unit_disk). Dually, for a given radius ''&epsilon;'', one wishes to find the smallest integer ''n'' such that ''n'' disks of radius ''&epsilon;'' 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
 | issue = 3
 | doi=10.2307/2371320| jstor = 2371320
 }}.</ref>

The best solutions known to date are as follows.<ref name=CirclesCoveringCircles>{{cite web |url=https://erich-friedman.github.io/packing/circovcir/|title=Circles Covering Circles|last=Friedman|first=Erich|access-date=4 October 2021}}</ref>

{| class="wikitable" border="1"
|-
! n
! r(n)
! Symmetry
|-
| 1
| 1
| All
|-
| 2
| 1
| All (2 stacked disks)
|-
| 3
| <math>\sqrt{3}/2</math> = 0.866025...
| 120°, 3 reflections
|-
| 4
| <math>\sqrt{2}/2</math> = 0.707107...
| 90°, 4 reflections
|-
| 5
| 0.609382... {{OEIS2C|A133077}}
| 1 reflection
|-
| 6
| 0.555905... {{OEIS2C|A299695}}
| 1 reflection
|-
| 7
| <math>1/2</math> = 0.5
| 60°, 6 reflections
|-
| 8
| 0.445041...
| ~51.4°, 7 reflections
|-
| 9
| 0.414213...
| 45°, 8 reflections
|-
| 10
| 0.394930... 
| 36°, 9 reflections
|-
| 11
| 0.380083...
| 1 reflection
|-
| 12
| 0.361141...
| 120°, 3 reflections
|}

==Method==
The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6. One of the covering disks is placed central and the remaining five in a symmetrical way around it.

File:DiscCoveringExample.svg

While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively.<ref name=CirclesCoveringCircles/> The corresponding angles θ are written in the "Symmetry" column in the above table.

==References==
{{reflist}}

==External links==
*{{MathWorld |title=Disk Covering Problem |id=DiskCoveringProblem}}
* Finch, S. R. "Circular Coverage Constants." §2.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp.&nbsp;484–489, 2003. 

Category:Discrete geometry
Category:Covering problems

{{geometry-stub}}

---
Adapted from the Wikipedia article [Disk covering problem](https://en.wikipedia.org/wiki/Disk_covering_problem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Disk_covering_problem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.