Question
Is there a formula to calculate the number of circles that can be enclosed in a given rectangle? It's the staggering aspect of the cans that is of interest, as pack density is the main factor, I guess I will have to calculate the interlock distance to see how many more I can get in?
Answer
You dont say whether the circles can overlap. If so, the number is INFINITE!!!<BR><BR><BR>If not, then the number is <BR><BR>Interger [Length/ Width]
Questions about packing density can be both interesting and difficult. Lining up circles (or cans) in a staggered fashion, just as you suggest, gives the greatest packing density of all possible arrangements, but this was not <I>fully</I> proved until about 1940 or so, though Gauss came close to proving it much earlier (circa 1800)! Surprisingly, however, the greatest packing density doesn't always give you the maximum number of circles for a given rectangle. <BR><BR>Searching NSDL for "packing circles" returns several interesting results, some of which are quite advanced. My favorites for your question are "Circle Packing -- From MathWorld" (which has a nice diagram but gets a little complex) and "Erich's Packing Center" from the Math Forum, which includes some fun animated images. <BR><BR><A href=http://mathworld.wolfram.com/CirclePacking.html">http://mathworld.wolfram.com/CirclePacking.html</A><BR>and<BR><A href=http://www.stetson.edu/~efriedma/packing.html">http://www.stetson.edu/~efriedma/packing.html</A><BR><BR><BR>I think you can answer your own question by studying these, but it may be helpful to notice that the <I>centers</I> of circles (arranged in a staggered, honeycomb fashion) form a lattice of <I>equilateral triangles</I>, and the length of each side is exactly the diameter of the circles. Also, don't try to find a single formula that works for all cases--this problem is harder than that. <BR><BR>Of course your question doesn't really have a good answer at all unless you specify the size of the circles and state clearly that they cannot overlap, as with cans in a crate. Even with these assumptions, you might make the problem easier by limiting consideration to cases where the circle diameter bears certain relationships (which perhaps you can discover!) to the height and the width of the rectangle. <BR><BR>Finally (though we don't have it in the NSDL yet), there is another interesting resource, called "Packing Pennies in the Plane," at: <BR><A href=http://www.math.sunysb.edu/~tony/whatsnew/column/pennies-1200/cass3.html">http://www.math.sunysb.edu/~tony/whatsnew/column/pennies-1200/cass3.html</A> <BR><BR>Thanks for posing such a challenging question. <BR><BR>---- <BR><BR>Dave Fulker<BR></X-HTML>Principal investigator for NSDL Core Integration project; degrees in math;<BR>experience with computer use in science.
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