# Fractal Dimension

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## Fractal Dimension

A number that quantitatively describes how an object fills its space. In Euclidean, or Plane geometry, objects are solid and continuous. That is, they have no holes or gaps. As such, they have integer dimensions. Fractals are rough and often discontinuous, like a wiffle ball, and so have fractional, or fractal dimensions.

## Fractal Dimension

A description of how a discontinuous object fills the space it occupies. For example, a basketball is not smooth; rather it has lines and grooves all over it. Because it is not a perfect sphere, it requires fractal dimensions to explain how it exists.
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After many experiments with different methods, we decided to apply the fractal dimension calculation method for the time series available in the well-known program Benoa (BENOIT, version 1.31), which proved to be very good in the calculations used earlier by different authors.
Mandelbrot, a French mathematician, proposed the fractal theory to study a complicated phenomenon with similar features and suggested making a quantitative description of fractal complexity with fractal dimension (Mandelbrot, 1982; Roslee, 2018).
Burrough, "Fractal dimensions of landscapes and other environmental data," Nature,1 vol.
This is one of the critical steps in using digital imaging for fractal analysis and different thresholding value can lead to different fractal dimensions. Thresholding is not straightforward and changing the threshold value can slightly change the fractal dimension.
(16) designed a new methodology based on fractal geometry that includes the creation of the concept of degrees of similarity, which allows the comparison of the fractal dimensions of the ventricular contours in systole, diastole and totality from images with a diagnosis of cardiac ventricular normality and severe abnormality.
In this paper, damage variables are redefined by AE parameters, the number of cycles, and the fractal dimension, and the expression of damage variable is corrected.
Moreover, the analysis is further extended to study the fractal dimensions, multifractal spectrum, and their dependence on the energy and mass of the projectile.
In addition, as shown in Figures 7(a) and 7(b), despite the different initial PSDs and particle sizes between all types of specimens, the PSD data of the finer region in which the values of fractal dimensions exceed 2.29 are shown to be strictly self-similar, and those of the coarser region in which the values of fractal dimensions exceed 1.85 are shown to be strictly self-similar.
On the other hand, pore channel is also analyzed using the fractal dimensions, as shown in Table 5.
The fractal values obtained from average of fractal dimensions calculated using 10 different grid positions are shown in Table 4.
The fractal dimensions (box dimension and information dimension) were extracted as RF fingerprints to identify 3 FM stations in .

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