Two-dimensional packing problem
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 . Equivalently, the problem is to arrange n points in a unit square aiming to get the greatest minimal separation, dn , between points.[ 1] L = 2 + 2 / dn
Solutions
Solutions (not necessarily optimal) have been computed for every N ≤ 10,000[ 2] N = 20[ 2] square numbers ), but ceases to be optimal for larger squares from 49 onwards.[ 2]
Number of circles (n )
Square side length (L )
dn [ 1] Number density (n / L 2
Figure
1
2
∞
0.25
2
2
+
2
{\displaystyle 2+{\sqrt {2}}}
2
{\displaystyle {\sqrt {2}}}
0.172...
3
2
+
2
2
+
6
2
{\displaystyle 2+{\frac {\sqrt {2}}{2}}+{\frac {\sqrt {6}}{2}}}
6
− − -->
2
{\displaystyle {\sqrt {6}}-{\sqrt {2}}}
0.194...
4
4
1
0.25
5
2
+
2
2
{\displaystyle 2+2{\sqrt {2}}}
2
2
{\displaystyle {\frac {\sqrt {2}}{2}}}
0.215...
6
2
+
12
13
{\displaystyle 2+{\frac {12}{\sqrt {13}}}}
13
6
{\displaystyle {\frac {\sqrt {13}}{6}}}
0.211...
7
4
+
3
{\displaystyle 4+{\sqrt {3}}}
4
− − -->
2
3
{\displaystyle 4-2{\sqrt {3}}}
0.213...
8
2
+
2
+
6
{\displaystyle 2+{\sqrt {2}}+{\sqrt {6}}}
6
2
− − -->
2
2
{\displaystyle {\frac {\sqrt {6}}{2}}-{\frac {\sqrt {2}}{2}}}
0.233...
9
6
0.5
0.25
10
6.747...
0.421... OEIS : A281065
0.220...
11
3
+
2
+
6
2
+
2
+
4
2
2
{\displaystyle 3+{\sqrt {2}}+{\frac {\sqrt {6}}{2}}+{\frac {\sqrt {2+4{\sqrt {2}}}}{2}}}
0.398...
0.223...
12
2
+
15
2
17
{\displaystyle 2+15{\sqrt {\frac {2}{17}}}}
34
15
{\displaystyle {\frac {\sqrt {34}}{15}}}
0.235...
13
7.463...
0.366...
0.233...
14
6
+
3
{\displaystyle 6+{\sqrt {3}}}
8
13
− − -->
2
3
13
{\displaystyle {\frac {8}{13}}-{\frac {2{\sqrt {3}}}{13}}}
0.226...
15
4
+
2
+
6
{\displaystyle 4+{\sqrt {2}}+{\sqrt {6}}}
1
2
+
2
2
− − -->
3
2
{\displaystyle {\frac {1}{2}}+{\frac {\sqrt {2}}{2}}-{\frac {\sqrt {3}}{2}}}
0.243...
16
8
0.333...
0.25
17
8.532...
0.306...
0.234...
18
2
+
24
13
{\displaystyle 2+{\frac {24}{\sqrt {13}}}}
13
12
{\displaystyle {\frac {\sqrt {13}}{12}}}
0.240...
19
8.907...
0.290...
0.240...
20
130
17
+
16
17
2
{\displaystyle {\frac {130}{17}}+{\frac {16}{17}}{\sqrt {2}}}
3
8
− − -->
2
16
{\displaystyle {\frac {3}{8}}-{\frac {\sqrt {2}}{16}}}
0.248...
Circle packing in a rectangle
Dense packings of circles in non-square rectangles have also been the subject of investigations.[ 3] [ 4]
See also
References
