{{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''' asks for 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. 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 | 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. 484–489, 2003.
Category:Discrete geometry Category:Covering problems
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