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# Box-Counting Dimension

Robert P. Munafo, 2023 Feb 16.

Box-counting dimension, also called Minkowski-Bouligand 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))

A simple approach is to just create one fractal image and use G2 = 4 G1, so 4 pixels on the grid make a "box" for the purposes of counting C1, and each pixel is a "box" for the C2 count.

For Mandelbrot images this approach can be used with images created using the distance estimator method. See the delta Hausdorff dimension discussion for examples.

revisions: 19970117 oldest on record; 20100905 add "a simple approach" paragraph

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Feb 17. s.27