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.
n r(n) Symmetry 1 1 All 2 1 All (2 stacked disks) 3 = 0.866025... 120°, 3 reflections 4 = 0.707107... 90°, 4 reflections 5 0.609382... 1 reflection 6 0.555905... 1 reflection 7 = 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.
References
External links
Finch, S. R. "Circular Coverage Constants." §2.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 484–489, 2003.