Filling a Square with Random Disks

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A unit square is filled randomly with non-overlapping circles of decreasing radius according to a power law; the resulting set has fractal properties.

Contributed by: Enrique Zeleny (July 2013)
Open content licensed under CC BY-NC-SA



The area of each circle decreases as , where is an integer and is an exponent between 1.1 and 1.5, which ensures convergence and interesting pictures. In fact, summing all areas gives the Riemann zeta function, with a starting area for the first circle satisfying the relation


For a high number of steps and values of near 1.5, the number of iterations required grows; in that case the number of iterations is limited to one million.

For a variety of designs, extensions to 3D, and estimates of fractal dimensions, visit Paul Bourke's web page.

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