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Estimating the box-counting dimension of the coast of Great Britain

In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set S in a Euclidean space Rⁿ, or more generally in a metric space (X, d). It is named after the Polish mathematician Hermann Minkowski and the French mathematician Georges Bouligand.

To calculate this dimension for a fractal S, imagine this fractal lying on an evenly spaced grid and count how many boxes are required to cover the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid finer by applying a box-counting algorithm.

Suppose that N(ε) is the number of boxes of side length ε required to cover the set. Then the box-counting dimension is defined as

$\dim _{\text{box}}(S):=\lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log(1/\varepsilon )}}.$

Roughly speaking, this means that the dimension is the exponent d such that N(1/n) ≈ C n^d, which is what one would expect in the trivial case where S is a smooth space (a manifold) of integer dimension d.