Box-Counting Dimension

Robert P. Munafo, 1997 Jan 17.

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Box-counting dimension is a simple way of estimating the Hausdorff dimension for fractals.

You compute the box-counting dimension from a grid that is superimposed on a fractal image and counting how many boxes in the grid contain part of the fractal. Then you increase the number of boxes in the grid (but covering the same area: the boxes get smaller) and count again. If the number of boxes in the first and second grids are G₁ and G₂, and the counts are C₁ and C₂, then you compute a dimension by the formula:

D = log(C2/C1) / log(sqrt(G2/G1))

See delta Hausdorff dimension for a discussion of how the Mandelbrot Set behaves.

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From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo.     Mu-ency index

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