disk covering problem
{{unsolved|mathematics|What is the smallest real number such that disks of radius can be arranged in such a way as to cover the unit disk?}}
The disk covering problem asks for the smallest real number such that disks of radius 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.{{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
}}.
The best solutions known to date are as follows.{{cite web |url=https://erich-friedman.github.io/packing/circovcir/|title=Circles Covering Circles|last=Friedman|first=Erich|access-date=4 October 2021}}
class="wikitable" border="1" |
n
! r(n) ! Symmetry |
---|
1
| 1 | All |
2
| 1 | All (2 stacked disks) |
3
| = 0.866025... | 120°, 3 reflections |
4
| = 0.707107... | 90°, 4 reflections |
5
| 0.609382... {{OEIS2C|A133077}} | 1 reflection |
6
| 0.555905... {{OEIS2C|A299695}} | 1 reflection |
7
| = 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.
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. 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. 484–489, 2003.
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