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N circles of maximum possible equal radius are contained in a square of side 1 with no overlap among the circles. What is the % of the area of the square outside the circle for N=3? (would appreciate a way of thinking about solving this for the general case; also, is there an easy way to extrapolate this into higher dimensions? i.e. spheres of maximum possible equal radius inside a cube of side 1).

This is similar to the question here, but I am looking for a generalized way of thinking about this, for circles and cubes: Maximum area of $2$ circles in a square Thank you.

gnokem
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    Does the Wikipedia article help somewhat at least? https://en.wikipedia.org/wiki/Circle_packing_in_a_square – PrincessEev Nov 22 '20 at 08:09
  • No you cannot generally extrapolate to higher dimension. You should search for packing spheres in a cylinder for example. – Math Lover Nov 22 '20 at 10:27

1 Answers1

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enter image description here

Hint: This figure is drawn with condition that all three circles are mutually tangent. As can be seen maximum r is when three circle are symmetric about DB. In this case DB bisects $\angle SPT$ which is $60^o$. BD makes $45^o$ angle with PQ which is parallel with AD. Therefore:

$\angle QPS=45-30=15^o$

You can use this figure to find the ratio.

$AD=2 r +2r(1+ \cos (\angle QPS)=2r(1+ \cos (\angle QPS)$;

With $\angle (QPS)=15^o$ we get $r=0.25435$

Can do the rest?

sirous
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  • I would propose some edits: 1. Your image only shows that the bounding box of circles is strictly contained in a square with parallel sides. But what if the circles do not touch each other pairwise and what happens when not two of them share the same $x$-value? I think there are some details missing here... 2. What does $4r=1=0.25$ mean? – Jonas Linssen Nov 22 '20 at 10:13
  • @PrudiiArca, This is not specified. I considered the conditions like in two circles as in OP reference. I edited my answer. – sirous Nov 22 '20 at 12:25
  • @DanielMathias, You are right , I edited my answer. – sirous Nov 22 '20 at 14:06
  • +1, though there seems to be a typo in your equation. Also, the result should be $r\approx 0.254333$ – Daniel Mathias Nov 22 '20 at 14:13