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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''n'', 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: : Roughly speaking, this means the dimension is the exponent ''d'' such that ''N''(1/''n'') ≈ ''C nd'', which is what one would expect in the trivial case where ''S'' is a smooth space (a manifold) of integer dimension 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 is the ''minimal'' number of open balls of radius ε required to cover the fractal, or in other words, such that their union contains the fractal. We can also consider the intrinsic covering number , which is defined the same way but with the additional requirement that the centers of the open balls lie inside the set ''S''. The packing number is the ''maximal'' number of disjoint open balls of radius ε one can situate such that their centers would be inside the fractal. While ''N'', ''N''covering, ''Ncovering and ''N''packing are not exactly identical, they are closely related, and give rise to identical definitions of the upper and lower box dimensions. This is easy to prove once the following inequalities are proven: : These, in turn, follow with a little effort from the triangle inequality. The advantage of using balls rather than squares is that this definition generalizes to any metric space. In other words, the box definition is extrinsic — one assumes the fractal space ''S'' is contained in a Euclidean space, and defines boxes according to the external geometry of the containing space. However, the dimension of ''S'' should be intrinsic, independent of the environment into which ''S'' is placed, and the ball definition can be formulated intrinsically. One defines an internal ball as all points of ''S'' within a certain distance of a chosen center, and one counts such balls to get the dimension. (More precisely, the ''N''covering definition is extrinsic, but the other two are intrinsic.) The advantage of using boxes is that in many cases ''N''(''ε'') may be easily calculated explicitly, and that for boxes the covering and packing numbers (defined in an equivalent way) are equal. The logarithm of the packing and covering numbers are sometimes referred to as ''entropy numbers'', and are somewhat analogous to the concepts of thermodynamic entropy and information-theoretic entropy, in that they measure the amount of "disorder" in the metric space or fractal at scale ''ε'', and also measure how many bits or digits one would need to specify a point of the space to accuracy ''ε''. Another equivalent (extrinsic) definition for the box-counting dimension, is given by the formula: : where for each ''r'' > 0, the set is defined to be the ''r''-neighborhood of ''S'', i.e. the set of all points in which are at distance less than ''r'' from ''S'' (or equivalently, is the union of all the open balls of radius ''r'' which are centered at a point in ''S''). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Minkowski–Bouligand dimension」の詳細全文を読む スポンサード リンク
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