Disk covering problem

The disk covering problem asks for the smallest real number $$r(n)$$ such that $$n$$ disks of radius $$r(n)$$ can be arranged in such a way as to cover the unit disk. Dually, for a given radius &epsilon;, one wishes to find the smallest integer n such that n disks of radius &epsilon; can cover the unit disk.

The best solutions known to date are as follows.

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.