Disk covering problem

The disk covering problem asks for the smallest real number [math]\displaystyle{ r(n) }[/math] such that [math]\displaystyle{ n }[/math] disks of radius [math]\displaystyle{ r(n) }[/math] 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.

n	r(n)	Symmetry
1	1	All
2	1	All (2 stacked disks)
3	[math]\displaystyle{ \sqrt{3}/2 }[/math] = 0.866025...	120°, 3 reflections
4	[math]\displaystyle{ \sqrt{2}/2 }[/math] = 0.707107...	90°, 4 reflections
5	0.609382... OEIS: A133077	1 reflection
6	0.555905... OEIS: A299695	1 reflection
7	[math]\displaystyle{ 1/2 }[/math] = 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.

External links

• Weisstein, Eric W.. "Disk Covering Problem". http://mathworld.wolfram.com/DiskCoveringProblem.html. 
• Finch, S. R. "Circular Coverage Constants." §2.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 484–489, 2003.