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First of all I am not a mathematician, forgive me if this is a stupid question.

A circular area is given. How to place n plants within the area so that the minimal distance between any of two plants is maximized?

I would truly appreciate some visual answers for cases n up to 50.

Aleksandar M
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  • Can't understand the phrase '... so that minimal distance between any of two plants is maximised ?' – Anik Bhowmick Aug 24 '18 at 06:19
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    @AnikBhowmick That is, the points are "as evenly spaced as possible". The phrase is unambiguous: $\max(\min_{x,y} |x-y|)$. – Patrick Stevens Aug 24 '18 at 06:20
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    Okay... Understandable in mathematical terms. Thanks !! – Anik Bhowmick Aug 24 '18 at 06:22
  • @Patrick You are correct. – Aleksandar M Aug 24 '18 at 06:22
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    Please, write a more informative title. – Taroccoesbrocco Aug 24 '18 at 06:23
  • You know the answer for 7 points. Suggest you try to go from there. – Moti Aug 24 '18 at 06:27
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    I think the question is equivalent to putting 50 identical small circles inside a big circle and find the largest radius of the small circle. – mastrok Aug 24 '18 at 07:00
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    You could also phrase it as: so that the distance between the two closest plants is maximized. – gen-ℤ ready to perish Aug 24 '18 at 07:17
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    This site gives you an answer and draws you a picture. https://www.engineeringtoolbox.com/smaller-circles-in-larger-circle-d_1849.html – user121049 Aug 24 '18 at 08:07
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    Place the plants at the centers of the black disks shown here: Circle packing in a circle. For up to 2600 plants, see Packomania (click the number in the left column for a picture, for example 50). –  Aug 24 '18 at 09:25
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    Many of the comments above are helpful (especially Rahul's). But it seems to me that everyone has forgotten that the question does not constrain the packed smaller circles to be totally within the large circle --- only their centers need to be. So don't forget when using the above tools to increase the diameter of the large circle by your minimal distance (i.e., the diameter of the smaller circles). – Ron Kaminsky Aug 25 '18 at 19:08
  • @RonKaminsky Good point. Actually excellent point! I can increase the diameter of the large circle - but with not yet known small radius?? What guaranties that the optimal solution for circle packing is also the optimal solution for plant packing? I am not convinced. – Aleksandar M Aug 25 '18 at 19:39
  • @Ron, Aleksandar: The solution to the packing problem gives you the largest radius $r$ such that $n$ circles will fit into a unit circle. Then the centers of the circles are at least a distance $2r$ from each other and at least $r$ from the boundary of the unit circle. Scale all the center coordinates by $1-r$, and you have $n$ points in a unit circle as far apart as possible and with no gap from the unit circle. –  Aug 28 '18 at 04:38
  • @Rahul Consider two circle packings into unit circle yielding r_1 and r_2 smaller circle radiuses, and r_1>r_2. You have to scale them by factors 1/(1-r_1) and 1/(1-r_2) to transform them to plant packings. Resulting plant packing smaller circle radiuses will be r_1/(1-r_1) and r_2/(1-r_2). Since r_1/(1-r_1) > r_2/(1-r_2), packing 1 is better. Is this a correct proof of equivalence of circle and plant packing problems? – Aleksandar M Aug 28 '18 at 05:28

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