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 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 to date are as follows:

n	r(n)	Symmetry
1	1	All
2	1	All (2 stacked disks)
3	$\sqrt{3}/2$ = 0.866025...	120°, 3 reflections
4	$\sqrt{2}/2$ = 0.707107...	90°, 4 reflections
5	0.609382...	1 reflection
6	0.555905...	1 reflection
7	$1/2$ = 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

This is the best known layout strategy for r(9) and r(10):

References

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

External links

Weisstein, Eric W., "Disk Covering Problem", MathWorld.
Finch, S. R. "Circular Coverage Constants." §2.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 484–489, 2003.
Illustrations of circles covering circles