Circle packing in a square
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... | ![]() |
3 |
≈ 3.931... |
≈ 1.035... |
0.194... | ![]() |
4 | 4 | 1 | 0.25 | ![]() |
5 |
≈ 4.828... |
≈ 0.707... |
0.215... | ![]() |
6 |
≈ 5.328... |
≈ 0.601... |
0.211... | ![]() |
7 |
≈ 5.732... |
≈ 0.536... |
0.213... | ![]() |
8 |
≈ 5.863... |
≈ 0.518... |
0.233... | ![]() |
9 | 6 | 0.5 | 0.25 | ![]() |
10 | 6.747... | 0.421... ![]() |
0.220... | ![]() |
11 | 7.022... | 0.398... | 0.223... | ![]() |
12 |
≈ 7.144... |
0.389... | 0.235... | ![]() |
13 | 7.463... | 0.366... | 0.233... | ![]() |
14 |
≈ 7.732... |
0.348... | 0.226... | ![]() |
15 |
≈ 7.863... |
0.341... | 0.243... | ![]() |
16 | 8 | 0.333... | 0.25 | ![]() |
17 | 8.532... | 0.306... | 0.234... | ![]() |
18 |
≈ 8.656... |
0.300... | 0.240... | ![]() |
19 | 8.907... | 0.290... | 0.240... | ![]() |
20 |
≈ 8.978... |
0.287... | 0.248... | ![]() |
References
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