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| [[image:Great Britain Box.svg|thumb|450px|Estimating the box-counting dimension of the coast of Great Britain]]
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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 (mathematics)|set]] ''S'' in a [[Euclidean space]] '''R'''<sup>''n''</sup>, or more generally in a [[metric space]] (''X'', ''d'').
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| 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 (topology)|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|box-counting]] algorithm.
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| 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:
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| :<math>\dim_{\rm box}(S) := \lim_{\varepsilon \to 0} \frac {\log N(\varepsilon)}{\log (1/\varepsilon)}.</math>
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| If the [[limit of a function|limit]] does not exist then one must talk about the '''upper box dimension''' and the '''lower box dimension''' which correspond to the [[Limit superior and limit inferior|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'''.
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| 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 [[#Relations to the Hausdorff dimension|below]] for more details. Also, another measure of fractal dimension is the [[correlation dimension]].
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| == Alternative definitions ==
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| [[image:Great Britain coverings.svg|thumb|350px|3 types of coverings or packings]]
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| It is possible to define the box dimensions using balls, with either the [[covering number]] or the packing number. The covering number <math>N_{\rm covering}(\varepsilon)</math> is the ''minimal'' number of [[open ball]]s of radius ε required to [[cover (topology)|cover]] the fractal, or in other words, such that their union contains the fractal. We can also
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| consider the intrinsic covering number <math>N'_{\rm covering}(\varepsilon)</math>, 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 <math>N_{\rm packing}(\varepsilon)</math> is the ''maximal'' number of [[Disjoint sets|disjoint]] open balls of radius ε one can situate such that their centers would be inside the fractal. While ''N'', ''N''<sub>covering</sub>, ''N'''<sub>covering</sub> and ''N''<sub>packing</sub> 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:
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| :<math> N'_\text{covering}(2\varepsilon) \leq N_\text{covering}(\varepsilon), \quad N_\text{packing}(\varepsilon) \leq N'_\text{covering}(\varepsilon). \, </math>
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| These, in turn follow with a little effort from the [[triangle inequality]].
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| The advantage of using balls rather than squares is that this definition generalizes to any [[metric space]]. In other words, the box definition is "external" — one needs to assume the fractal is contained in a [[Euclidean space]], and define boxes according to the external structure "imposed" by the containing space. The ball definition is "internal". One can imagine the fractal disconnected from its environment, define balls using the distance between points on the fractal and calculate the dimension (to be more precise, the ''N''<sub>covering</sub> definition is also external, but the other two are internal).
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| 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.
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| The [[logarithm]] of the packing and covering numbers are sometimes referred to as ''entropy numbers'', and are somewhat analogous (though not identical) to the concepts of [[entropy|thermodynamic entropy]] and [[entropy (information theory)|information-theoretic entropy]], in that they measure the amount of "disorder" in the metric space or fractal at scale <math>\varepsilon</math>, and also measure how many "bits" one would need to describe an element of the metric space or
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| fractal to accuracy <math>\varepsilon</math>.
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| Another equivalent definition for the box counting dimension, which is again "external", is given by the formula
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| :<math>\dim_\text{box}(S) = n - \lim_{r \to 0} \frac{\log \text{vol}(S_r)}{\log r},</math>
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| where for each ''r'' > 0, the set <math>S_r</math> is defined to be the ''r''-neighborhood of ''S'', i.e. the set of all points in <math>R^n</math> which are at distance less than ''r'' from ''S'' (or equivalently, <math>S_r</math> is the union of all the open balls of radius ''r'' which are centered at a point in ''S'').
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| == Properties ==
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| Both box dimensions are finitely additive, i.e. if { ''A''<sub>1</sub>, .... ''A''<sub>''n''</sub> } is a finite collection of sets then
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| :<math>\dim (A_1 \cup \dotsb \cup A_n) = \max \{ \dim A_1 ,\dots, \dim A_n \}. \, </math> | |
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| However, they are not [[countable set|countably]] additive, i.e. this equality does not hold for an ''infinite'' sequence of sets. For example, the box dimension of a single point is 0, but the box dimension of the collection of [[rational number]]s in the interval [0, 1] has dimension 1. The [[Hausdorff measure]] by comparison, is countably additive.
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| An interesting property of the upper box dimension not shared with either the lower box dimension or the Hausdorff dimension is the connection to set addition. If ''A'' and ''B'' are two sets in a Euclidean space then ''A'' + ''B'' is formed by taking all the pairs of points ''a,b'' where ''a'' is from ''A'' and ''b'' is from ''B'' and adding ''a+b''. One has
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| :<math>\dim_\text{upper box}(A+B)\leq \dim_\text{upper box}(A)+\dim_\text{upper box}(B).</math> | |
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| == Relations to the Hausdorff dimension ==
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| 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 [[Hausdorff dimension#The open set condition|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). However, the definitions are not equivalent.
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| The box dimensions and the Hausdorff dimension are related by the inequality
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| :<math>\dim_{\operatorname{Haus}} \leq \dim_{\operatorname{lower box}} \leq \dim_{\operatorname{upper box}}.</math>
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| In general both inequalities may be [[strict inequality|strict]]. The upper box dimension may be bigger than the lower box dimension if the fractal has different behaviour in different scales. For example, examine the [[interval (mathematics)|interval]] [0, 1], and examine the set of numbers satisfying the condition
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| :for any ''n'', all the digits between the 2<sup>2''n''</sup>-th digit and the (2<sup>2''n''+1</sup> − 1)th digit are zero | |
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| The digits in the "odd places", i.e. between 2<sup>2''n''+1</sup> and 2<sup>2''n''+2</sup> − 1 are not restricted and may take any value. This fractal has upper box dimension 2/3 and lower box dimension 1/3, a fact which may be easily verified by calculating ''N''(''ε'') for <math>\varepsilon=10^{-2^n}</math> and noting that their values behaves differently for ''n'' even and odd. To see that the Hausdorff dimension may be smaller than the lower box dimension, return to the example of the rational numbers in [0, 1] discussed above. The Hausdorff dimension of this set is 0.
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| Another example: The set of rational numbers <math>\mathbb{Q}</math>, a countable set with <math>\dim_{\operatorname{Haus}} = 0</math>, has <math>\dim_{\operatorname{box}} = 1</math> because its closure, <math>\mathbb{R}</math>, has dimension 1.
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| Box counting dimension also lacks certain stability properties one would expect of a dimension. For instance, one might expect that adding a countable set would have no effect on the dimension of a set. This property fails for box dimension. In fact
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| :<math> \dim_{\operatorname{box}} \left\{0,1,\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\right\} = \frac{1}{2}. </math> | |
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| == See also ==
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| *[[Correlation dimension]]
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| *[[Packing dimension]]
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| *[[Uncertainty exponent]]
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| *[[Weyl–Berry conjecture]]
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| *[[Lacunarity]]
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| ==References==
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| * {{cite book | zbl=0689.28003 | last=Falconer | first=Kenneth | authorlink=Kenneth Falconer (mathematician) | title=Fractal geometry: mathematical foundations and applications | location=Chichester | publisher=John Wiley | year=1990 | isbn=0-471-92287-0 | pages=38–47 }}
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| * {{mathworld| urlname=Minkowski-BouligandDimension|title=Minkowski-Bouligand Dimension}}
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| == External links ==
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| * [http://code.google.com/p/frakout/ FrakOut!: an OSS application for calculating the fractal dimension of a shape using the box counting method] (Does not automatically place the boxes for you).
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| * FracLac: online user guide and software [http://rsb.info.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm ImageJ and FracLac box counting plugin; free user-friendly open source software for digital image analysis in biology]
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| {{DEFAULTSORT:Minkowski-Bouligand dimension}}
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| [[Category:Fractals]]
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| [[Category:Dimension theory]]
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