Optimal size of n circles to fit an area

Let us say that I have a rectangular area that has to always look "filled" with circles. (the void spaces with the given number of circles should be minimal)(Goal)

Let us assume that, I am also told that there will be n circles that I want to fit into the container and I am also given relative sizes of different circles (i.e say that n/2 circles have to be twice the size of the remaining n/2 circles). Now the question is whether there is a way to synthesize the size of the circles such that te container looks "filled".

• Appolonian gaskets (en.wikipedia.org/wiki/Apollonian_gasket) spring to mind... – Johannes Jan 20 '15 at 13:25
• I think this is more a general math problem for mathsSE than a puzzle, unless you specify a very distinct problem to which a nice solution exists. I have voted for closure as off-topic asking to relocate the question. This is not meant as a turn-down, though. – BmyGuest Jan 22 '15 at 10:37
• The optimal size and packing will depend on r, the aspect ratio of the rectangle. Might as well pick, for illustration here, r = 2.0 – smci Feb 13 '18 at 9:42

Your question falls into the area of "circle packing" problems. Even some highly restricted special cases of circle packing are very hard and messy, and there exists no general theory for attacking them.

A good survey on the current knowledge of the area is given at http://www.packomania.com/

• Thanks. After going through the link, I realise it is a difficult problem. But are there any safe approximations that might be possible? – suzee Jan 20 '15 at 10:19
• Another key difference is that we are also free to choose the size of the circles (satisfying certain constrainsts, ofcourse), doesnt it make the problem simpler? – suzee Jan 20 '15 at 12:22
• Stupid but probably good approximation: $n$ small circles fall into the container. Shake the container. Simulate physics. Grow the circles until they won't grow any more. – Lopsy Jan 21 '15 at 13:51
• @Lopsy hexagonal packing is, I believe, the most dense packing. Nature does it. (Think bee hives with circles inscribed into hexagons) - ah, I misread your comment. But your solution will create the condition after the solution. (Robin Hood firing anywhere and then proclaiming that was exactly where the arrow should have gone...) – BmyGuest Jan 22 '15 at 10:40