Let's say that I have a circle, which is the "middle circle" (red in the pictures below). I also have a number (n) of identical circles, that should appear around the middle one, without touching. For the first few Ns, they can basically hug the middle circle (without touching), and they will fit. At some point, there will be too many, so they will have to be further and further away from the middle circle, to maintain that round formation without touching each other.

I am wondering if there is a formula, that given n the number of outer circles, d the starting angle (there will always be a circle at this angle if n > 0), and k the (constant) minimum distance between two circles (always >= 0, so that they won't touch), and i the circle number, I could get its exact location (assuming the middle circle is at (0, 0)), or just its distance from the middle circle?

A picture is worth a thousand words, so here is what the first few Ns would look like.

Please forgive the horrible paint drawings.

A few of the first Ns

So what formula could I use in that situation?

  • $\begingroup$ for two I don't think minimum exists it must be infimum or something like that i think the problem continues for larger number of circles (you can always get the circles slightly closer and so on..) $\endgroup$ – Yanko Aug 7 '17 at 22:22
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    $\begingroup$ Do all the circles have to go in one layer? You can essentially pretend the circles are touching and keep them a distance $\varepsilon > 0$ apart (this will propogate a term through to the final result, but it will be negligible), so it makes sense to just deal with the problem where the circles are tangent. $\endgroup$ – platty Aug 7 '17 at 22:24
  • $\begingroup$ I think you want to ask this question: Find the smallest (infimum) radius $r$ such that the ball of radius $r$ such that it will contain $n$ balls of a given radius. $\endgroup$ – Yanko Aug 7 '17 at 22:24
  • $\begingroup$ @yanko that's probably a better wording than what I wrote. I'm not a mathematician, so I wasn't sure how to describe this problem correctly. Feel free to suggest an edit. $\endgroup$ – Kaito Kid Aug 7 '17 at 22:37
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    $\begingroup$ Hint: for $n \ge 3$ the centers of the black circles are the vertices of a regular polygon with $n$ sides. For $n \gt 6$ the side length of the regular polygon is $2r+k$ where $r$ is the common radius of the circles. From there, you can determine the distance from the center of the red circle to the center of any black circle and, once you have that, the rest is just working out some straightforward calculations. $\endgroup$ – dxiv Aug 7 '17 at 23:05

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