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 known 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

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.

Similar arrangements of six, seven, eight respectively nine disks around a central disk all having same radius result in the best known layout strategies for r(7), r(8), r(9), respectively r(10). 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.

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