Disk covering problem

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, although updated bounds can be found [here|https://mathworld.wolfram.com/DiskCoveringProblem.html].

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

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. Pictures showing these arrangements can be found at Friedman, Erich. "circles covering circles". Retrieved 2016-05-04.

References
  1. Kershner, Richard (1939), "The number of circles covering a set", American Journal of Mathematics, 61: 665–671, doi:10.2307/2371320, MR 0000043.


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