Circle packing in a square

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

Circle packing in a square is a packing problem in applied 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 aiming to get 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 .

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 (n) Square size (side length (L)) dn[1] Number density (n/L^2) Figure
1 2 0.25
2
≈ 3.414...

≈ 1.414...
0.172... 2 circles in a square.svg
3
≈ 3.931...

≈ 1.035...
0.194... 3 circles in a square.svg
4 4 1 0.25 4 circles in a square.svg
5
≈ 4.828...

≈ 0.707...
0.215... 5 circles in a square.svg
6
≈ 5.328...

≈ 0.601...
0.211... 6 circles in a square.svg
7
≈ 5.732...

≈ 0.536...
0.213... 7 circles in a square.svg
8
≈ 5.863...

≈ 0.518...
0.233... 8 circles in a square.svg
9 6 0.5 0.25 9 circles in a square.svg
10 6.747... 0.421... OEISA281065 0.220... 10 circles in a square.svg
11 7.022... 0.398... 0.223... 11 circles in a square.svg
12
≈ 7.144...
0.389... 0.235... 12 circles in a square.svg
13 7.463... 0.366... 0.233... 13 circles in a square.svg
14
≈ 7.732...
0.348... 0.226... 14 circles in a square.svg
15
≈ 7.863...
0.341... 0.243... 15 circles in a square.svg
16 8 0.333... 0.25 16 circles in a square.svg
17 8.532... 0.306... 0.234... 17 circles in a square.svg
18
≈ 8.656...
0.300... 0.240... 18 circles in a square.svg
19 8.907... 0.290... 0.240... 19 circles in a square.svg
20
≈ 8.978...
0.287... 0.248... 20 circles in a square.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.