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So my motivation here is actually because I want to evenly distribute plants in a green house and ideally I would like to maximize the distance between the plants and the walls. It seems like there is probably a way to do this. I suppose the only thing of interest here is the nearest neighbor distance, ie, this is what I would like to maximize.

This seemed like an simple enough question, so I did some googling. It seems like it might fall under the umbrella of circle packing: http://en.wikipedia.org/wiki/Circle_packing. However, in my case, I don't have circles with a fixed radius so I'm unsure if this is applicable?

Does anybody have any suggestions? I suppose in the end it will have little affect on the efficiency or output of the greenhouse, but now I am curious. Thanks!

Fractal20
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  • Possible duplicate? This may shed some light: http://math.stackexchange.com/questions/15624/distribute-a-fixed-number-of-points-uniformly-inside-a-polygon/ – dls May 18 '14 at 17:53
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    You found the right link; circle packing is indeed applicable. Packing circles of radius $r$ in a rectangle of size $w\times h$ is equivalent to packing points with minimum distance $2r$ in a rectangle of size $(r+w+r)\times(r+h+r)$. –  May 18 '14 at 17:55
  • Certainly a lattice arrangement would be most efficient in this particular case? – Elle Najt May 18 '14 at 17:56
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    @user54092 If we take the number of plants $n$ to be a fixed parameter, then many values of $n$ don't correspond to a lattice and thus adjustments must be made at the boundary. I expect this effect becomes less significant for very large $n$, but see the variety of configurations at http://en.wikipedia.org/wiki/Circle_packing_in_a_square – Erick Wong May 18 '14 at 18:02
  • @ErickWong Interesting, thanks. I guess an underlying assumption in my mind was that the original poster was going to buy however many plants where necessary to be most efficient, and my guess was that could be achieved by buying enough and arranging them in a lattice. I have no idea how to prove that guess though. – Elle Najt May 18 '14 at 18:09
  • @ErickWong Based on the table you sent me it does seem like lattices are most efficient overall. – Elle Najt May 18 '14 at 18:10
  • To the first response, that seems oriented towards uniform distributions, I don't want a sampling from a uniform distribution, I want to maximize the distance between points. To the second comment, I'm still unsure of how to deal with the radius. I don't have a specified radius...? As for the lattice, I was looking at that page as well. It seems like for an even number of circles it is often a lattice. I'm thinking of doing either 10 or 12 plants (to be clear, these are circular planting pots, not individual plants). – Fractal20 May 18 '14 at 18:10
  • @Fractal20 The idea is to use the centers of the circles as the locations of the pots. The radius of the circles doesn't correspond to your pots in any way, it just represents a minimum (half-)distance between the centers. You just need to take the optimal point configuration and rescale, as Rahul points out. (Admittedly this takes a bit of iteration in the rectangular case, as the aspect ratio of a rectangle with margins removed depends on the size of the margin...) – Erick Wong May 18 '14 at 18:21

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