Disk covering problem

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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... OEISA133077 1 reflection
6 0.555905... OEISA299695 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.


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. 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, 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

Retrieved from "https://en.wikipedia.org/w/index.php?title=Disk_covering_problem&oldid=827201412"
This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/Disk_covering_problem
This page is based on the copyrighted Wikipedia article "Disk covering problem"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA