Disk covering problem

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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.[1]

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


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. 


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

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

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