Circle packing in a square

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Circle packing in a square is a packing problem in recreational mathematics, where the aim is to pack n unit circles into the smallest possible square; or, equivalently, to arrange n points in a unit square for the greatest minimal separation, dn, between points.[1] To convert between these two formulations of the problem, the square side for unit circles will be L=2+\frac{2}{d_n}.

Solutions (not necessarily optimal) have been computed for every N≤10,000.[2] Solutions up to N=20 are shown below.:[2]

Number of circles Square size (side length) dn[1] Number density Figure
1 2 0.25
2 2+\sqrt{2}
≈ 3.414...
\sqrt{2}
≈ 1.414...
0.172... Circles packed in square 2.svg
3 2+\frac{\sqrt{2}}{2}+\frac{\sqrt{6}}{2}
≈ 3.931...
\sqrt{6} - \sqrt{2}
≈ 1.035...
0.194... Circles packed in square 3.svg
4 4 1 0.25 Circles packed in square 4.svg
5 2+2\sqrt{2}
≈ 4.828...
\frac{1}{2} \sqrt{2}
≈ 0.707...
0.215... Circles packed in square 5.svg
6 2 + \frac{12}{\sqrt{13}}
≈ 5.328...
\frac{1}{6} \sqrt{13}
≈ 0.601...
0.211... Circles packed in square 6.svg
7 4+ \sqrt{3}
≈ 5.732...
4- 2\sqrt{3}
≈ 0.536...
0.213... Circles packed in square 7.svg
8 2 + \sqrt{2} + \sqrt{6}
≈ 5.863...
\frac{1}{2}(\sqrt{6} - \sqrt{2})
≈ 0.518...
0.233... Circles packed in square 8.svg
9 6 0.5 0.25 Circles packed in square 9.svg
10 6.747... 0.421... 0.220... Circles packed in square 10.svg
11 7.022... 0.398... 0.223... Circles packed in square 11.svg
12 2 + 15\sqrt{\frac{2}{17}}
≈ 7.144...
0.389... 0.235... Circles packed in square 12.svg
13 7.463... 0.366... 0.233... Circles packed in square 13.svg
14 6 + \sqrt{3}
≈ 7.732...
0.348... 0.226... Circles packed in square 14.svg
15 4 + \sqrt{2} + \sqrt{6}
≈ 7.863...
0.341... 0.243... Circles packed in square 15.svg
16 8 0.333... 0.25 Circles packed in square 16.svg
17 8.532... 0.306... 0.234... Circles packed in square 17.svg
18 2 + \frac{24}{\sqrt{13}}
≈ 8.656...
0.300... 0.240... Circles packed in square 18.svg
19 8.907... 0.290... 0.240... Circles packed in square 19.svg
20 \frac{130}{17} + \frac{16}{17} \sqrt{2}
≈ 8.978...
0.287... 0.248... Circles packed in square 20.svg

References[edit]

  1. ^ a b Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991). Unsolved Problems in Geometry. New York: Springer-Verlag. pp. 108–110. ISBN 0-387-97506-3. 
  2. ^ a b Eckard Specht (20 May 2010). "The best known packings of equal circles in a square". Retrieved 25 May 2010.