**3.2 Fractal dimension through the box-counting method (box-dimension method)**

This is an application created in the beginning of the 1990s. It establishes that if a line is split in half, we will have two parts to recreate the original line. If it is split into four parts, we will use that to cover the original line. We may deduce that if we split a line's time, we will have such a number of segments covering the original line. Therefore:

It is also true that, if we take a square and split each side in half, we obtain four smaller parts of the original square. If we split each side into ¼, we will have 16 parts of the original. Therefore, if we repeat this for: *<sup>D</sup>* <sup>¼</sup> <sup>2</sup> : *Ns* <sup>¼</sup> <sup>1</sup> *s* <sup>2</sup> and for *<sup>D</sup>* <sup>¼</sup> <sup>3</sup> : *Ns* <sup>¼</sup> <sup>1</sup> *s* <sup>3</sup> *:*

1, 2, and 3 exponents in the above examples are fundamental for the concept of dimension. This may be generalized by Eq. (1):

$$\mathbf{N}\_s = \left(\frac{\mathbf{1}}{s}\right)^D \tag{1}$$


**Table 1.**

*Values of calculation of fractal dimension for each method used.*

where *D* is the dimension (whole or unit) as in the beginning, but if we take logarithms on both sides, we have Eq. (4):

$$\log\left(\mathbf{N}\_{\mathbf{s}}\right) = D \log\left(\frac{\mathbf{1}}{\mathbf{s}}\right) \tag{2}$$

In other words, we may estimate the dimension graphing log(*N*(*s*)) to *D* log(1/*s*). Its slope is the dimension. If the said dimension is not a whole, then it is a fractioned dimension.

The box-dimension method is the most popular one and counts boxes containing part of the fractal as a calculation procedure. Given an *A* fractal structure contained in a d-dimensional volume, the counting method implies distributing the space of the structure on a fixed d-dimensional mesh with square boxes of the same size *r*. Several algorithms [8–10] based on this method have been widely developed and used to estimate *Df*, and may be applied to systems with or without self-similarity. The fractal dimension obtained was 1.95322, **Table 1**.

#### **3.3 Fractal dimension through the perimeter-area dimension method**

The perimeter-area fractal dimension is attractive because it reflects the complexity of a shape through a range of spatial scales (size of fragments). However, as it's homologous to the fragment level (FRAC), the perimeter-area fractal dimension is only significant if the log-log ratio between perimeter and area is linear on the full range of fragment sizes. If it is not (and this should be determined separately), the fractal dimension is calculated separately for the range of patch sizes or fragments on which it is constant. Note that as this index uses regression analysis, it is subject to false results when there are small sample sizes. In mosaics with only some patches or fragments, it is common to obtain values above the theoretical limits of this index. PAFD (perimeter-area fractal dimension) is equal to 2, divided by the slope of the regression line obtained through regression of the mosaic piece mosaic area logarithm (in square meters) times the piece mosaic perimeter logarithm (in meters). That is, 2 divided by the coefficient *b*<sup>1</sup> derived from a regression of minimal squares for the equation ln(area) = *b*<sup>0</sup> + *b*<sup>1</sup> \* ln (perimeter). Note that PAFD excludes any background mosaic. It has no units and its range is 1 ≤ PAFD ≤2.

The formula for the calculation of the fractal dimension through this method is Eq. (5).

$$\begin{array}{l} \text{PAFD} = \frac{2}{\left[ \begin{smallmatrix} n\_i \sum\_{j=1}^n \left( \ln \left( p\_{\vec{y}} \right) \* \ln \left( a\_{\vec{\eta}} \right) \right) \end{smallmatrix} \right] - \left[ \left( \sum\_{j=1}^n \ln \left( a\_{\vec{\eta}} \right) \right) \left( \sum\_{j=1}^n \ln \left( a\_{\vec{\eta}} \right) \right) \right]}}{\left( \begin{smallmatrix} n\_i \sum\_{j=1}^n \ln \left( p\_{\vec{y}=1}^2 \right) \end{smallmatrix} \right) - \left( \sum\_{j=1}^n \ln \left( p\_{\vec{y}} \right) \right)^2} \end{array} \tag{3}$$

*aij* is the area (in square meters) of the fragment or piece *ij*, *pij* is the perimeter (in meters) of the fragment or piece *ij*, and *ni* is the number of pieces in the mosaic patch or fragment *i* (class). The fractal dimension thus obtained is 0.164771, **Table 1**.

#### **3.4 Information-dimension method**

The concept of fractal dimension may be seen as a basic vision of complex construction. Let us define the information function as Eq. (6), see [11]:

$$I = -\sum\_{i=1}^{N} P\_i(\varepsilon) \ln \left[ P\_i(\varepsilon) \right] \tag{4}$$

where *Pi*ð Þ*ϵ* is the natural measure or the probability that the element *i* is the index of the normalized population such that: P*<sup>N</sup> <sup>i</sup>*¼<sup>1</sup>*Pi*ð Þ¼ *<sup>ϵ</sup>* 1. The dimension of information *dinf* is Eq. (7):

$$d\_{\inf} \equiv -\lim\_{\epsilon \to 0^{+}} \frac{I}{\ln(\epsilon)} = \lim\_{\epsilon \to 0^{+}} \sum\_{i=1}^{N} \frac{P\_i(\epsilon) \ln\left[P\_i(\epsilon)\right]}{\ln\left(\epsilon\right)}\tag{5}$$

The fractal dimension, thus obtained is 0.0010445, **Table 1**.

#### **3.5 Mass-dimension method**

Unifractal shows something similar to a random Cantor dust. This is the version of the two dimensions of the Cantor set, attributed to the mathematician George Cantor [12]. The method was supported when Berger and Mandelbrot found data showing strong hierarchical grouping in a high number of superimposed levels. The model published is a discretized Lévy dust, but the authors here consider the first a model based on Cantor dust, see [13].

Entries *N* and *b* are not present in more realistic models, much less in reality, and should not be measured separately only through the similar dimension Eq. (8).

$$D = \frac{\log N}{\log b} \tag{6}$$

In the case of Lévy dust, no counter-physical grid and cascade are involved; hence, *N* and *b* are not needed. Here, the sole parameter *D* is an easily measurable concrete quantity. It is not a similarity dimension, but a mass dimension. That is, an interval of length *R* centered on the dust contains a mass of the order of *RD*. The fractal dimension obtained with this method is 0.0093884, **Table 1**.
