# 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 G_{1} and G_{2}, and the counts are C_{1} and C_{2}, 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 G_{2}
= 4 G_{1}, so 4 pixels on the grid make a "box" for the purposes of
counting C_{1}, and each pixel is a "box" for the C_{2} 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.

Mu-ency main page — index — recent changes — DEMZ

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