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 bounded set S {\textstyle S} in a Euclidean space R n {\textstyle \mathbb {R} ^{n}} , or more generally in a metric space ( X , d ) {\textstyle (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 {\textstyle 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 ) {\textstyle N(\varepsilon )} is the number of boxes of side length ε {\textstyle \varepsilon } required to cover the set. Then the box-counting dimension is defined as
dim box ( S ) := lim ε → 0 log N ( ε ) log ( 1 / ε ) = − lim ε → 0 log N ( ε ) log ( ε ) . {\displaystyle \dim _{\text{box}}(S):=\lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log(1/\varepsilon )}}=-\lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log(\varepsilon )}}.}
Roughly speaking, this means that the dimension is the exponent d {\textstyle d} such that N ( ε ) ≈ C ε − d {\textstyle N(\varepsilon )\approx C\varepsilon ^{-d}} , which is what one would expect in the trivial case where S {\textstyle S} is a smooth space (a manifold) of integer dimension d {\textstyle d} . If the above limit does not exist, one may still take the limit superior and limit inferior, which respectively define the upper box dimension and lower box dimension. The upper box dimension is sometimes called the entropy dimension, Kolmogorov dimension, Kolmogorov capacity, limit capacity or upper Minkowski dimension, while the lower box dimension is also called the lower Minkowski dimension. The upper and lower box dimensions are strongly related to the more popular Hausdorff dimension. Only in very special applications is it important to distinguish between the three (see below). Yet another measure of fractal dimension is the correlation dimension.
Alternative definitions
It is possible to define the box dimensions using balls, with either the covering number or the packing number. The covering number N covering ( ε ) {\textstyle N_{\text{covering}}(\varepsilon )} is the minimal number of open balls of radius ε {\textstyle \varepsilon } required to cover the fractal, or in other words, such that their union contains the fractal. We can also consider the intrinsic covering number N covering ′ ( ε ) {\textstyle N'_{\text{covering}}(\varepsilon )} , which is defined the same way but with the additional requirement that the centers of the open balls lie in the set S. The packing number N packing ( ε ) {\textstyle N_{\text{packing}}(\varepsilon )} is the maximal number of disjoint open balls of radius ε {\textstyle \varepsilon } one can situate such that their centers would be in the fractal. While N {\textstyle N} , N covering {\textstyle N_{\text{covering}}} , N covering ′ {\textstyle N'_{\text{covering}}} and N packing {\textstyle N_{\text{packing}}} are not exactly identical, they are closely related to each other and give rise to identical definitions of the upper and lower box dimensions. This is easy to show once the following inequalities are proven:
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