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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 R, or more generally in a metric space. 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 is the number of boxes of side length ε required to cover the set. Then the box-counting dimension is defined as
Roughly speaking, this means that the dimension is the exponent d such that N ≈ C n, which is what one would expect in the trivial case where S is a smooth space of integer dimension d.