# 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.

A. Mughal, H. K. Chan, D. Weaire, and S. Hutzler. Dense packings of spheres in cylinders: Simulations. Phys. Rev. E 85, 051305 – Published 11 May 2012

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Welcome to MSE! Recommend adding the name of the article and authors in case the web site is gone, does not work or otherwise. This way, future searches gives that information to search on. Regards –  Amzoti Feb 4 '13 at 18:06
the above mentioned paper is not accessible. Is it possible to get a free version of the material –  La Rias Aug 15 '14 at 11:41