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Say you have a shape you want to fill up with circles, where by the circles overlap just enough to cover the whole surface area of the shape. The circles will remain as a fixed size however the shape they fill may change. Is there some sort of mathematical formula that could achieve this to efficiently fill a shape with circles?

for example you have a rectangle thats 2500cm2 and circles that are 500cm2 what would you do to work it out so that they fill the whole surface area of the shape using the minimum number of circles.

I don't have much of a background in maths so I have no idea if this is a simple or complex problem.

Images below not to scale just to demonstrate the concept.

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  • $\begingroup$ "a rectangle that is 2500 cm²" doesn't tell us enough. $\endgroup$ – Yves Daoust Dec 13 '19 at 9:48
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In the first place, don't use a square grid but an hexagonal one.

enter image description here

In the general case, the problem is quite arduous (in fact even intractable) if you want a truly optimal solution. Optimal solutions are known for a small number of particular cases. For instance https://en.wikipedia.org/wiki/Disk_covering_problem, https://erich-friedman.github.io/packing/circovsqu/.

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This is a somewhat complex problem. You want to minimize the number of circles while still having the overlap. If you settle for good, but not perfect solutions, you can write algorithms to find good solutions. For instance genetic algorithms will "evolve" solutions as your shape evolves. However, if you want to do any of this, you need at least some understanding of the equations that define distance, circles, and the shapes you desire to cover.

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  • $\begingroup$ Ohh sounds a little out side my scope, I haven't done much in the way of maths since GCSE. $\endgroup$ – Nphot Apr 10 '15 at 21:08

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