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There are 86 pages on this site alone under a search for: area, square, circle, convergent, approximation. I found one that arguably asks the same question here, but I am not sure.

The question is, if we are trying to approximate the area of a square with identical circles, does the sequence of approximations converge (and does it converge to the area of the square)?

Because for 1,4,9,16, and 25 circles, the ratio r of the area of the approximating circles to that of the square is the same ($\frac{\pi}{4}$), I thought that this pattern surely continued. So my original question was whether, if we remove the square-numbered approximations, we get a convergent sequence.

I see that this is related to the problem of packing circles into a square, and having looked at the sequence of ratios (given here) I see that after 25 the "best" packing is not a nice military arrangement but something else, something that allows the circles to swell a bit. So the "best" packing improves for square numbers as well.

And so it does appear that we have a non-monotone sequence of approximations that might or might not converge to a ratio r = 1. But appearances can be deceiving.

I did a quick review of the literature (at the link above) but it is addressed primarily at the question of optimal packing for given numbers of circles, and not convergence, which I hope is much simpler.

Thanks for any insight.

Community
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daniel
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  • This question is similar to: The minimum number of circles in order to obtain a COVER of a specific square. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. – Clemens Bartholdy Sep 03 '24 at 09:24
  • @CantorDustDrachen: In this question the circles cannot overlap and the coverage reaches the ratio given below, 0.9069. In the COVERING question, the circles are allowed to overlap. I don't think they answer the same question. If you think otherwise, please let me know why. – daniel Sep 03 '24 at 10:56
  • perhaps you should add that in into the question @daniel – Clemens Bartholdy Sep 03 '24 at 11:08
  • @CantorDustDrachen: I will do so, but your comment induced 3 "close" votes (now down to 1). If you read the 2 questions carefully you would have seen the difference. If I had been away from my computer, my question could have been closed because you did not read carefully. Am I mischaracterizing this? – daniel Sep 03 '24 at 11:35
  • You are corret, but then again, where your question to be use of other people, then eventually it'd be re-opened, or it'd also be reoopened if the community thought it was incorrectly be closed. – Clemens Bartholdy Sep 03 '24 at 15:32

1 Answers1

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If there were no walls and you were just trying to squeze circles together as tightly as possible, then the best way to do it is the hexagonal packing. Even in this packing the circles only cover 90.69% of the area, the other 9.31% lies in the gaps between the circles. So the approximation is always going to be less than 90.69% of the total area.

Now consider putting really small circles into your square. You can use a hexagonal packing in the middle, and continue it out toward the edges. Since the circles are very small, you'll be able to get very close to the edges without them messing up your pattern. So almost all the square will be covered with the hexagonal packing. So the limit does in fact converge to the ratio 0.9069...

Oscar Cunningham
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    I'm sure that is true, though it is slightly discouraging that the table daniel links to shows a better packing for 39 circles than the "best found so far" for 9996 circles. – Henry Feb 12 '12 at 23:32