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.^[1] To convert between these two formulations of the problem, the square side for unit circles will be $L = 2 + .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠2/dn⁠$ .

Solutions

Solutions (proven optimal for $N \leq 30$ ) have been computed for every $N \leq 10,000$ .^[2] Solutions up to $N = 20$ are shown below.^[2] 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.^[2]

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

Circle packing in a rectangle

Dense packings of circles in non-square rectangles have also been the subject of investigations.^[3]^[4]

References

^ ^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.
^ ^a ^b ^c Eckard Specht (22 Aug 2022). "The best known packings of equal circles in a square". Retrieved 4 Mar 2025.
^ Lubachevsky, Boris D.; Graham, Ronald L. (2009). "Minimum perimeter rectangles that enclose congruent non-overlapping circles". Discrete Mathematics. 309 (8). Elsevier BV: 1947–1962. arXiv:math/0412443. doi:10.1016/j.disc.2008.03.017. ISSN 0012-365X. S2CID 783236.
^ Specht, E. (2013). "High density packings of equal circles in rectangles with variable aspect ratio". Computers & Operations Research. 40 (1). Elsevier BV: 58–69. doi:10.1016/j.cor.2012.05.011. ISSN 0305-0548.

[upg-1] 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.

[bestcirclepacking-2] Eckard Specht (22 Aug 2022). "The best known packings of equal circles in a square". Retrieved 4 Mar 2025.

[Lubachevsky2009-3] Lubachevsky, Boris D.; Graham, Ronald L. (2009). "Minimum perimeter rectangles that enclose congruent non-overlapping circles". Discrete Mathematics. 309 (8). Elsevier BV: 1947–1962. arXiv:math/0412443. doi:10.1016/j.disc.2008.03.017. ISSN 0012-365X. S2CID 783236.

[Specht2013-4] Specht, E. (2013). "High density packings of equal circles in rectangles with variable aspect ratio". Computers & Operations Research. 40 (1). Elsevier BV: 58–69. doi:10.1016/j.cor.2012.05.011. ISSN 0305-0548.

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[2]

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v t e Packing problems
Abstract packing	Bin Set
Circle packing	In a circle / equilateral triangle / isosceles right triangle / square Apollonian gasket Circle packing theorem Tammes problem (on sphere)
Sphere packing	Apollonian Finite In a sphere In a cube In a cylinder Close-packing Kissing number Sphere-packing (Hamming) bound
Other 2-D packing	Square packing
Other 3-D packing	Tetrahedron Ellipsoid
Puzzles	Conway Slothouber–Graatsma

Circle packing in a square

Solutions

Circle packing in a rectangle

See also

References