Circle packing in a square#Circle packing in a rectangle

Circle packing in a square is a packing problem in recreational mathematics where the aim is to pack {{mvar|n}} unit circles into the smallest possible square. Equivalently, the problem is to arrange {{mvar|n}} points in a unit square in order to maximize the minimal separation, {{mvar|d{{sub|n}}}}, between points.{{cite book |title=Unsolved Problems in Geometry |last=Croft |first=Hallard T. |author2=Falconer, Kenneth J. |author3=Guy, Richard K. |year=1991 |publisher=Springer-Verlag |location=New York |isbn=0-387-97506-3 |pages=[https://archive.org/details/unsolvedproblems0000crof/page/108 108–110] |url=https://archive.org/details/unsolvedproblems0000crof/page/108 }} To convert between these two formulations of the problem, the square side for unit circles will be {{math|L {{=}} 2 + {{sfrac|2|d_n}}}}.

Solutions

Solutions (proven optimal for {{math|N ≤ 30}}) have been computed for every {{math|N ≤ 10,000}}.{{cite web |url=http://hydra.nat.uni-magdeburg.de/packing/csq/csq.html|title=The best known packings of equal circles in a square |author=Eckard Specht |date=22 Aug 2022 |access-date=4 Mar 2025}} Solutions up to {{math|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.

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Number of circles ({{mvar\|n}}) ! Square side length ({{mvar\|L}}) ! {{mvar\|d_n}} ! Number density ({{math\|{{sfrac\|n\|L²}}}}) ! Figure
1 \| 2 \| ∞ \| 0.25 \|
2 \| $2+\sqrt{2}$ ≈ 3.414... \| $\sqrt{2}$ ≈ 1.414... \| 0.172... \| 85x85px
3 \| $2+\frac{\sqrt{2}}{2}+\frac{\sqrt{6}}{2}$ ≈ 3.931... \| $\sqrt{6} - \sqrt{2}$ ≈ 1.035... \| 0.194... \| 85x85px
4 \| 4 \| 1 \| 0.25 \| 85x85px
5 \| $2+2\sqrt{2}$ ≈ 4.828... \| $\frac{\sqrt{2}}{2}$ ≈ 0.707... \| 0.215... \| 85x85px
6 \| $2 + \frac{12}{\sqrt{13}}$ ≈ 5.328... \| $\frac{\sqrt{13}}{6}$ ≈ 0.601... \| 0.211... \| 85x85px
7 \| $4+ \sqrt{3}$ ≈ 5.732... \| $4- 2\sqrt{3}$ ≈ 0.536... \| 0.213... \| 85x85px
8 \| $2 + \sqrt{2} + \sqrt{6}$ ≈ 5.863... \| $\frac{\sqrt{6}}{2} - \frac{\sqrt{2}}{2}$ ≈ 0.518... \| 0.233... \| 85x85px
9 \| 6 \| 0.5 \| 0.25 \| 85x85px
10 \| 6.747... \| 0.421... {{OEIS2C\|A281065}} \| 0.220... \| 85x85px
11 \| $3 + \sqrt{2} + \frac{\sqrt{6}}{2} + \frac{\sqrt{2+4\sqrt{2}}}{2}$ ≈ 7.022... \| 0.398... \| 0.223... \| 85x85px
12 \| $2 + 15\sqrt{\frac{2}{17}}$ ≈ 7.144... \| $\frac{\sqrt{34}}{15}$ ≈ 0.389... \| 0.235... \| 85x85px
13 \| 7.463... \| 0.366... \| 0.233... \| 85x85px
14 \| $6 + \sqrt{3}$ ≈ 7.732... \| $\frac{8}{13} - \frac{2\sqrt{3}}{13}$ ≈ 0.349... \| 0.226... \| 85x85px
15 \| $4 + \sqrt{2} + \sqrt{6}$ ≈ 7.863... \| $\frac{1}{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2}$ ≈ 0.341... \| 0.243... \| 85x85px
16 \| 8 \| 0.333... \| 0.25 \| 85x85px
17 \| 8.532... \| 0.306... \| 0.234... \| 85x85px
18 \| $2 + \frac{24}{\sqrt{13}}$ ≈ 8.656... \| $\frac{\sqrt{13}}{12}$ ≈ 0.300... \| 0.240... \| 85x85px
19 \| 8.907... \| 0.290... \| 0.240... \| 85x85px
20 \| $\frac{130}{17} + \frac{16}{17} \sqrt{2}$ ≈ 8.978... \| $\frac{3}{8} - \frac{\sqrt{2}}{16}$ ≈ 0.287... \| 0.248... \| 85x85px

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Number of circles ({{mvar|n}})

! Square side length ({{mvar|L}})

! {{mvar|d_n}}

! Number density ({{math|{{sfrac|n|L²}}}})

! Figure

| 2

| ∞

| 0.25

| $2+\sqrt{2}$
≈ 3.414...

| $\sqrt{2}$
≈ 1.414...

| 0.172...

Circle packing in a square#Circle packing in a rectangle

Solutions

Circle packing in a rectangle

See also

References