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Fractals
Published in Nicholas Stergiou, Nonlinear Analysis for Human Movement Variability, 2018
As well as naturally occurring fractal objects, exact geometric fractals can also be constructed using simple formula that are repeated over and over again. Well-known mathematical, or geometric fractals include the Sierpinski triangle (Figure 7.4). The basic rule for the construction of this triangle is as follows: connect the midpoints of each side of the equilateral triangle to form four separate triangles, and differentially color the triangle in the center (the downward pointing triangle). When this rule is iterated, that is, repeated, as illustrated in steps 1–4 of Figure 7.4, an interesting shape begins to emerge. If the process is continued (e.g., using a computer program), a rather intricate object appears. Also illustrated in Figure 7.4 is the fact that magnifying a portion of the Sierpinski triangle produces an exact copy of the original shape. This magnification process can conceptually go on to infinity in a geometric fractal. However, fractals found in nature are not so exact. They naturally include elements of randomness and the fractal scaling exists over a finite range of scales, as is the case with the Romanesco Broccoli.
Fractals and chaos
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
Among the well-known fractals are the Cantor set (after Georg Cantor, 1883), the Koch curve and snowflakes (after Helge von Koch, 1904), and the Sierpinski triangle (after Waclow Sierpinski, 1915) in the plane one- and two-dimensional spaces, and Mandelbrot (after Benoit Mandelbrot, 1960s; he coined the word ‘fractal’ to an object whose Hausdorff–Besicovitch dimension is greater than the topological dimension) and Julia (after Gaston Julia, 1918) sets in the complex plane. The Sierpinski triangle and Koch snowflake exhibit exact self-similarity. The Mandelbrot set is quasi self-similar, as the satellites are approximations of the entire set, but not exact copies. Fractals can be deterministic or stochastic. Trajectories of Brownian motion are generated by stochastic fractals. Random fractals can be used to describe many highly irregular objects. Other applications include fractal landscape, enzymology, generation of new music, signal and image compression, seismology, soil and rock mechanics, and video games, among many others.
Fractal Dimension of Biosignals
Published in Dinesh K. Kumar, Sridhar P. Arjunan, Behzad Aliahmad, Fractals, 2017
Dinesh K. Kumar, Sridhar P. Arjunan, Behzad Aliahmad
The concept of a fractal is most often associated with geometrical objects and the two most famous examples are the Sierpinski triangle and the Koch curve. Both satisfy two important properties: self-similarityfractional dimensions
Assessment of Scale Invariance Changes in Heart Rate Signal During Postural Shift
Published in IETE Journal of Research, 2022
Helen M. C. Mary, Dilbag Singh, K. K. Deepak
The structural characteristics of waveform are visually apparent but not captured by conventional measures like the average amplitude of the time series [9]. Fractals can be classified according to their self-similarity degree [8]. There are three types of self-similarity found in fractals [10] as follows: exact self-similarity – This is the strongest type of self-similarity; the fractal appears identical at different scales, e.g. Sierpinski triangle.quasi self-similarity – This is a loose form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales.statistical self-similarity – This is the weakest type of self-similarity; the fractal has numerical or statistical measures (mean, standard deviation, variance, etc.), which are preserved across scales, e.g. biomedical signals
Fractal analysis for the blast-induced damage in rock masses with one free boundary
Published in Mechanics of Advanced Materials and Structures, 2022
Changda Zheng, Renshu Yang, Qing Li, Chenxi Ding, Chenglong Xiao, Yong Zhao, Jie Zhao
The accuracy of the fractal dimension calculation program using MATLAB is verified before calculating the box-counting dimension of the specimens, as shown in Figure 9. The Sierpinski triangle image is imported into the fractal program, wherein the Sierpinski triangle image size is 256 × 256 pixel. The theoretical fractal dimension of the Sierpinski triangle image is 1.584962 and the fractal dimension calculated by the program is 1.585178. The relative error between the theoretical fractal dimension and the calculated fractal dimension is found to be 0.014% which proves that the fractal dimension calculated by the program is reliable.