Minkowski–Bouligand Dimension

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).

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:

If the limit does not exist then one must talk about the upper box dimension and the lower box dimension which correspond to the upper limit and lower limit respectively in the expression above. In other words, the box-counting dimension is well defined only if the upper and lower box dimensions are equal. The upper box dimension is sometimes called the entropy dimension, Kolmogorov dimension, Kolmogorov 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 specialized applications is it important to distinguish between the three. See below for more details. Also, another measure of fractal dimension is the correlation dimension.

Read more about Minkowski–Bouligand Dimension:  Alternative Definitions, Properties, Relations To The Hausdorff Dimension

Other articles related to "dimension, dimensions":

Minkowski–Bouligand Dimension - Relations To The Hausdorff Dimension
... The box-counting dimension is one of a number of definitions for dimension that can be applied to fractals ... For many well behaved fractals all these dimensions are equal in particular, these dimensions coincide whenever the fractal satisfies the open set condition (OSC) ... For example, the Hausdorff dimension, lower box dimension, and upper box dimension of the Cantor set are all equal to log(2)/log(3) ...

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