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Optimal Rectangles for Circles

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number of circles
6
radius of circles
1
6 circles of radius 1 in a 6. by 4. box
circle area: 18.8496, box area: 24., fill %: 0.785398
Some circles must be packed into a minimal rectangle. For 4, 6, 8, 9, 10, or 12 circles, the square lattice works best. Beyond that, the hexagonal lattice works best. Sometimes, as with 49 circles, a hexagonal layout seeming to miss a single circle is most efficient.
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