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.
The best solutions known to date are as follows:
nrSymmetry
11All
21All
3= 0.866025...120°, 3 reflections
4= 0.707107...90°, 4 reflections
50.609382... 1 reflection
60.555905... 1 reflection
7= 0.560°, 6 reflections
80.445041...~51.4°, 7 reflections
90.414213...45°, 8 reflections
100.394930...36°, 9 reflections
110.380083...1 reflection
120.361141...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, 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, r, r, and r, respectively. The corresponding angles θ are written in the "Symmetry" column in the above table. Pictures showing these arrangements can be found at