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Is there a reference about determining the minimum radius of a circle that would cover n circles of radius 1 that are in a square packing configuration ( see Wolfram's MathWorld packing packing description)?

This is a different problem than the "best known packing of equal circles in a circle", though for n=1, 2, and 4 it would have the same result.

For a hexagonal packing configuration, n=1, 2, 3, 7, would have the same result as "best know packing of equal circles in a circle". Believe 6 also does, but would actually cover 7.

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For circles packing in square, it is solved problem. The formulas are here en.wikipedia.org/wiki/Circle_packing_in_a_square and here hydra.nat.uni-magdeburg.de/packing/csq/csq.html (there is a huge PDF there also to download) –  Nasser Dec 21 '12 at 19:38
    
Nasser: Not interested in circles fitting in a square. Interested in covering by a circle of n circles that are square packed. See diagram for what constitutes "square packing of circles". Klett: You are right it -- question probably should have gone there. Thought I was on that site until after I hit send. My apologies. –  Refactor Dec 21 '12 at 21:19

1 Answer 1

If you don't have a square number of small circles, you need to define which ones are missing. It is easy to say "the corners" if you have at least $n^2-4$, but what about if you have $89$ small circles? Asking for the minimum radius is fair.

A partial answer is that if you have a diagonal of $n$ unit circles, the length from one end to the other is $1+(n-1)\sqrt 2$, so the radius of the enclosing circle would be half that. The next layer would be if the farthest center is $n$ units one direction and $m$ units the other from the center (where $n,m$ can both be half integral or integral) the radius is $\frac 12(1+\sqrt{n^2+m^2})$ The minimum is just a constant increase over the size of the circle to contain a given number of lattice points. This is the Gauss circle problem, still unsolved.

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While my problem is similar to the Gauss's circle problem, I think it is slightly different. My understanding is Gauss's circle problem has the restriction of the covering circle be centered at the origin. That is, it has requires one of the lattice points to serve as the center of the covering circle. –  Refactor Dec 23 '12 at 0:14
    
The solution for the next layer seems usable but I think it would have the solution for <br> –  Refactor Dec 23 '12 at 0:28
    
The solution for the next layer seems usable but I think it would have the solution for 5 would be the same solution as for 6. However 5 solution for the problem should for a + formation aka 1 3 1 find a radius 1.5 circle which covers 5 circles of radius 1. –  Refactor Dec 23 '12 at 0:33

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