Circle packing in a square

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 in order to maximize the minimal separation, d_n, between points. To convert between these two formulations of the problem, the square side for unit circles will be L = 2 + ⁠2/d_n⁠.

Solutions

Solutions (proven optimal for N ≤ 30) have been computed for every N ≤ 10,000. Solutions up to N = 20 are shown below. The obvious square packing is optimal for 1, 4, 9, 16, 25, and 36 circles (the six smallest square numbers), but ceases to be optimal for larger squares from 49 onwards.

Number of circles (n)	Square side length (L)	dn	Number density (⁠n/L2⁠)	Figure
1	2	∞	0.25
2	$2+\sqrt{2}$ ≈ 3.414...	$\sqrt{2}$ ≈ 1.414...	0.172...
3	$2+\frac{\sqrt{2}}{2}+\frac{\sqrt{6}}{2}$ ≈ 3.931...	$\sqrt{6}-\sqrt{2}$ ≈ 1.035...	0.194...	Error creating thumbnail:
4	4	1	0.25	File:4 circles in a square.svg
5	$2+2\sqrt{2}$ ≈ 4.828...	$\frac{\sqrt{2}}{2}$ ≈ 0.707...	0.215...	File:5 circles in a square.svg
6	$2+\frac{12}{\sqrt{13}}$ ≈ 5.328...	$\frac{\sqrt{13}}{6}$ ≈ 0.601...	0.211...	File:6 circles in a square.svg
7	$4+\sqrt{3}$ ≈ 5.732...	$4-2\sqrt{3}$ ≈ 0.536...	0.213...	File:7 circles in a square.svg
8	$2+\sqrt{2}+\sqrt{6}$ ≈ 5.863...	$\frac{\sqrt{6}}{2}-\frac{\sqrt{2}}{2}$ ≈ 0.518...	0.233...	File:8 circles in a square.svg
9	6	0.5	0.25	File:9 circles in a square.svg
10	6.747...	0.421... OEIS: A281065	0.220...	File:10 circles in a square.svg
11	$3+\sqrt{2}+\frac{\sqrt{6}}{2}+\frac{\sqrt{2+4\sqrt{2}}}{2}$ ≈ 7.022...	0.398...	0.223...	File:11 circles in a square.svg
12	$2+15\sqrt{\frac{2}{17}}$ ≈ 7.144...	$\frac{\sqrt{34}}{15}$ ≈ 0.389...	0.235...	File:12 circles in a square.svg
13	7.463...	0.366...	0.233...	File:13 circles in a square.svg
14	$6+\sqrt{3}$ ≈ 7.732...	$\frac{8}{13}-\frac{2\sqrt{3}}{13}$ ≈ 0.349...	0.226...	File:14 circles in a square.svg
15	$4+\sqrt{2}+\sqrt{6}$ ≈ 7.863...	$\frac{1}{2}+\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2}$ ≈ 0.341...	0.243...	File:15 circles in a square.svg
16	8	0.333...	0.25	File:16 circles in a square.svg
17	8.532...	0.306...	0.234...	File:17 circles in a square.svg
18	$2+\frac{24}{\sqrt{13}}$ ≈ 8.656...	$\frac{\sqrt{13}}{12}$ ≈ 0.300...	0.240...	File:18 circles in a square.svg
19	8.907...	0.290...	0.240...	File:19 circles in a square.svg
20	$\frac{130}{17}+\frac{16}{17}\sqrt{2}$ ≈ 8.978...	$\frac{3}{8}-\frac{\sqrt{2}}{16}$ ≈ 0.287...	0.248...	File:20 circles in a square.svg

Circle packing in a rectangle

Dense packings of circles in non-square rectangles have also been the subject of investigations.

See also

• Square packing in a circle
• Circle packing in a circle
• Sphere packing in a cube