# Problem of packing spheres of radius $\rho$ into a cylinder

Given a cylinder of radius $R$ and length $L$, I need to find the number of spheres which is possible to pack into the cylinder as a function of the radius $\rho$ of the spheres. I found something about the problem of packing into cubes or other solids, but nothing in a cylinder. Does someone know the solution? Thanks.

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I typed $$\rm packing\ spheres\ in\ a\ cylinder$$ into Google and got heaps of results. Why not have a look at what's there and report back to us? – Gerry Myerson Mar 20 '12 at 11:46
@Gerry Myerson: I typed the same into Google but I didn't find any formula linking $R$,$\rho$ and $L$ – Riccardo.Alestra Mar 20 '12 at 12:30
@Riccardo.Alestra: It is a research-level problem: ingentaconnect.com/content/bpl/itor/2010/00000017/00000001/… in optimization theory. – Jon Mar 20 '12 at 14:39
As long as $\rho \geq 2R/(2+\sqrt{3})$, all optimal packings follow the same strategy: you can view it as the 2-dimensional problem of packing circles into a tall rectangular strip, with the circles touching the left and right walls alternately. In this case, you can get a formula. For smaller $\rho$, the problem will be very complicated, like most packing problems, and is certainly unsolved in general; the best you could hope for are upper and lower bounds. A simple upper bound is given by the result that spheres can fill no more than $74.048\ldots$ percent of space under any packing. – Jonas Kibelbek Mar 20 '12 at 14:42

This is what you need. It includes numerical results up to $\rho$/R of about 2.9.