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.[1]
The best solutions known to date are as follows.[2]
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... OEIS: A133077
1 reflection
6
0.555905... OEIS: 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
^Kershner, Richard (1939), "The number of circles covering a set", American Journal of Mathematics, 61 (3): 665–671, doi:10.2307/2371320, JSTOR 2371320, MR 0000043.
^Friedman, Erich. "Circles Covering Circles". Retrieved 4 October 2021.
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