Infractal geometry, theMinkowski–Bouligand dimension, also known asMinkowski dimension orbox-counting dimension, is a way of determining thefractal dimension of a boundedset$S$ in aEuclidean space${\mathbb{R}}^n$, or more generally in ametric space$(X,d)$. It is named after thePolishmathematicianHermann Minkowski and theFrench mathematicianGeorges 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 tocover the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid finer by applying abox-counting algorithm.

Suppose that$N(\epsilon )$ is the number of boxes of side length$\epsilon$ required to cover the set. Then the box-counting dimension is defined as

$${\displaystyle {\dim }_{\text{box}}(S):=\underset{\epsilon \to 0}{lim}\frac{\log N(\epsilon )}{\log (1/\epsilon )}=-\underset{\epsilon \to 0}{lim}\frac{\log N(\epsilon )}{\log (\epsilon )}.}$$

Roughly speaking, this means that the dimension is the exponent$d$ such that$N(\epsilon )\approx C{\epsilon }^{-d}$, which is what one would expect in the trivial case where$S$ is a smooth space (amanifold) of integer dimension$d$.

If the abovelimit does not exist, one may still take thelimit superior and limit inferior, which respectively define theupper box dimension andlower box dimension. The upper box dimension is sometimes called theentropy dimension,Kolmogorov dimension,Kolmogorov capacity,limit capacity orupper Minkowski dimension, while the lower box dimension is also called thelower Minkowski dimension.

The upper and lower box dimensions are strongly related to the more popularHausdorff dimension. Only in very special applications is it important to distinguish between the three (seebelow). Yet another measure of fractal dimension is thecorrelation dimension.

It is possible to define the box dimensions using balls, with either thecovering number or the packing number. The covering number$N_{\text{covering}}(\epsilon )$ is theminimal number ofopen balls of radius$\epsilon$ required tocover the fractal, or in other words, such that their union contains the fractal. We can also consider the intrinsic covering number$N_{\text{covering}}^{\prime }(\epsilon )$, which is defined the same way but with the additional requirement that the centers of the open balls lie in the setS. The packing number$N_{\text{packing}}(\epsilon )$ is themaximal number ofdisjoint open balls of radius$\epsilon$ one can situate such that their centers would be in the fractal. While$N$,$N_{\text{covering}}$,$N_{\text{covering}}^{\prime }$ and$N_{\text{packing}}$ 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:

$${\displaystyle N_{\text{packing}}(\epsilon )\le N_{\text{covering}}^{\prime }(\epsilon )\le N_{\text{covering}}(\epsilon /2)\le N_{\text{covering}}^{\prime }(\epsilon /2)\le N_{\text{packing}}(\epsilon /4).}$$

These, in turn, follow either by definition or with little effort from thetriangle inequality.

The advantage of using balls rather than squares is that this definition generalizes to anymetric space. In other words, the box definition isextrinsic – one assumes the fractal spaceS is contained in aEuclidean space, and defines boxes according to the external geometry of the containing space. However, the dimension ofS should beintrinsic, independent of the environment into whichS is placed, and the ball definition can be formulated intrinsically. One defines an internal ball as all points ofS within a certain distance of a chosen center, and one counts such balls to get the dimension. (More precisely, theN_covering definition is extrinsic, but the other two are intrinsic.)

The advantage of using boxes is that in many casesN(ε) may be easily calculated explicitly, and that for boxes the covering and packing numbers (defined in an equivalent way) are equal.

Thelogarithm of the packing and covering numbers are sometimes referred to asentropy numbers and are somewhat analogous to the concepts ofthermodynamic entropy andinformation-theoretic entropy, in that they measure the amount of "disorder" in the metric space or fractal at scaleε and also measure how many bits or digits one would need to specify a point of the space to accuracyε.

Another equivalent (extrinsic) definition for the box-counting dimension is given by the formula

$${\displaystyle {\dim }_{\text{box}}(S)=n-\underset{r\to 0}{lim}\frac{\log \text{vol}(S_r)}{\log r},}$$

where for eachr > 0, the set$S_r$ is defined to be ther-neighborhood ofS, i.e. the set of all points in$R^n$ that are at distance less thanr fromS (or equivalently,$S_r$ is the union of all the open balls of radiusr which have a center that is a member of S).

The upper box dimension is finitely stable, i.e. if {A₁, ...,A_n} is a finite collection of sets, then

$${\displaystyle {\dim }_{\text{upper box}}(A_1\cup \cdots \cup A_n)=max\{{\dim }_{\text{upper box}}(A_1),\dots ,{\dim }_{\text{upper box}}(A_n)\}.}$$

However, it is notcountably stable, i.e. this equality does not hold for aninfinite sequence of sets. For example, the box dimension of a single point is 0, but the box dimension of the collection ofrational numbers in the interval [0, 1] has dimension 1. TheHausdorff dimension by comparison, is countably stable. The lower box dimension, on the other hand, is not even finitely stable.

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. IfA andB are two sets in a Euclidean space, thenA +B is formed by taking all the pairs of pointsa, b wherea is fromA andb is fromB and addinga + b. One has

$${\displaystyle {\dim }_{\text{upper box}}(A+B)\le {\dim }_{\text{upper box}}(A)+{\dim }_{\text{upper box}}(B).}$$

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 theopen set condition (OSC).^[1] For example, theHausdorff dimension, lower box dimension, and upper box dimension of theCantor set are all equal to log(2)/log(3). However, the definitions are not equivalent.

The box dimensions and the Hausdorff dimension are related by the inequality

$${\displaystyle {\dim }_{\text{Haus}}\le {\dim }_{\text{lower box}}\le {\dim }_{\text{upper box}}.}$$

In general, both inequalities may bestrict. 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 set of numbers in the interval [0, 1] satisfying the condition

The digits in the "odd place-intervals", i.e. between digits 2²ⁿ⁺¹ and 2²ⁿ⁺² − 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 calculatingN(ε) for${\displaystyle \epsilon ={10}^{-2^n}}$ and noting that their values behave differently forn even and odd.

Another example: the set of rational numbers$\mathbb{Q}$, a countable set with${\dim }_{\text{Haus}}=0$, has${\dim }_{\text{box}}=1$ because its closure,$\mathbb{R}$, has dimension 1. In fact,

$${\displaystyle {\dim }_{\text{box}}\left\{0,1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots \right\}=\frac{1}{2}.}$$

These examples show that adding a countable set can change box dimension, demonstrating a kind of instability of this dimension.

• Correlation dimension
• Packing dimension
• Uncertainty exponent
• Weyl–Berry conjecture
• Lacunarity

• Falconer, Kenneth (1990).Fractal geometry: mathematical foundations and applications. Chichester: John Wiley. pp. 38–47.ISBN 0-471-92287-0.Zbl 0689.28003.
• Weisstein, Eric W."Minkowski-Bouligand Dimension".MathWorld.

• 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).
• FracLac: online user guide and softwareImageJ and FracLac box counting plugin; free user-friendly open source software for digital image analysis in biology

• Fractals
• Dimension theory
• Hermann Minkowski

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