Pack identical rectangles in a circle

Asked Apr 24 '20 at 16:14

Active Apr 24 '20 at 16:22

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I have identical rectangles of size (w, h). I would like to calculate the maximum number of such rectangles that I can pack (no overlap) in a circle of radius r. All the rectangles should also have the same orientation.

If $h = w$, I found $N=\pi(r-w)^2/w^2$.

Thanks.

edited Apr 24 '20 at 16:22

asked Apr 24 '20 at 16:14

user130104

I am looking for an upper bound. But I couldn't find a formula even for the square case, so I was putting an approximation. – user130104 Apr 24 '20 at 16:23
Related : Is there an algorithm out there that will show you how to fit the most rectangles into a circle? – sammy gerbil Apr 24 '20 at 16:30
There is no formula even for squares. The best you can expect is a table such as provided by Erich Friedman. A separate table would be required for each value of aspect ratio $a=w/h$. – sammy gerbil Apr 24 '20 at 17:50
More related questions : Number of Squares in a circle, Mathematica SE : packing squares into a circle, How many squares fit in a circle?, Fitting a grid inside a circle, is the solution always symmetric?, – sammy gerbil Apr 24 '20 at 19:38
For the unit square case $(h=w=1)$ you can use curve fitting to find an approximate formula from the 35 data points in Erich Friedman's website. The power law $r=an^b$ is a very good fit with $a=0.7682, b=0.44189$ and $R^2=0.996683$. Here $n$ is the number of unit squares and $r$ is the radius of the smallest circle they can be fitted into. – sammy gerbil Apr 24 '20 at 23:09
Did you try an approach that scales it by setting a procedure of packing n element and based on that calculate the radius? – Moti Apr 25 '20 at 01:52
Is orientation limited? – Moti Apr 25 '20 at 01:52

Pack identical rectangles in a circle

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