21
$\begingroup$

I've just came back from my Mathematics of Packing and Shipping lecture, and I've run into a problem I've been trying to figure out.

Let's say I have a rectangle of length $l$ and width $w$.

Is there a simple equation that can be used to show me how many circles of radius $r$ can be packed into the rectangle, in the optimal way? So that no circles overlap. ($r$ is less than both $l$ and $w$)

I'm rather in the dark as to what the optimum method of packing circles together in the least amount of space is, for a given shape.

An equation with a non-integer output is useful to me as long as the truncated (rounded down) value is the true answer.

(I'm not that interested in how the circles would be packed, as I am going to go into business and only want to know how much I can demand from the packers I hire to pack my product)

  • 3
    I had an answer before, but I looked into it a bit more and my answer was incorrect so I removed it. This link may be of help: http://en.wikipedia.org/wiki/Circle_packing_in_a_square2010-07-26
  • 1
    @Cam: Looks like there's no optimal solution yet. Maybe you could just put this comment as an answer.2010-07-26
  • 0
    Might be a good question to work out how to answer problems which actually aren't solved yet in advanced maths. (if there is not an optimal solution yet)2010-07-26
  • 0
    @KennyTM: Sure.2010-07-26
  • 0
    Why can't we just use $\left \lfloor {\frac{\text{Ar. of Rectangle}}{\text{Area of Circle}} }\right \rfloor$ ?2015-10-22
  • 2
    @NeilRoy That is an upper bound but it assumes that the circles pack basically perfectly in the rectangle, which is obviously asymptotically false as there must be space between the circles.2016-01-05
  • 0
    I know this thread is very old, but I have put together a working solution for this problem which I use at work which I can share? (Optimal packing of circular glass bottles on to pallets).2017-08-31

2 Answers 2