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Fractal Theory
Published in Fredrick Madaraka Mwema, Esther Titilayo Akinlabi, Oluseyi Philip Oladijo, Sputtered Thin Films, 2021
Fredrick Madaraka Mwema, Esther Titilayo Akinlabi, Oluseyi Philip Oladijo
There are several examples of fractal features (either existing in nature or artificial/man made) and they have various applications and uses. Most objects in nature exhibit fractal behavior; and some of the common examples of natural fractals include the clouds, mountains, river connections, biochemical structures, and blood vessels in animals and humans. Trees and crops such as cauliflower, broccoli, ferns, and leaves are more examples of fractals in nature. These natural systems can be modeled via recursive computer algorithms due to the self-similar characteristics. For instance, a branch from a tree is a minute replica of the whole tree although it is not the same but identical. Other natural fractals include snowflake, nautilus shell, gecko’s foot, flowerheads of most plants, frost crystals, and lightning bolt (some of these fractals are as shown in Figure 2.2).
Elements of View
Published in Lisa Heschong, Visual Delight in Architecture, 2021
There is another mathematical area that promises to help us better understand the appeal of views: fractal analysis. Fractals are patterns of self-similar shapes that repeat at many scales. The term was invented by the mathematician Benoit Mandelbrot in 1975, to describe shapes that were generated from a formula that he devised, that generated geometric shapes of great complexity, but with infinitely repeating themes at ever smaller and ever larger scales. The original Mandelbrot Set he devised still amazes with its repeating, ever-folding, and flowering complexity. (It is best viewed as a video to observe the continuous transformations.25) Artists soon discovered the visual fascination of fractal patterns, and a whole new discipline of fractal art was popularized, facilitated by ever more powerful computer graphics and data analysis.
Multiscale Modeling of Porous Media
Published in Peng Xu, Agus P. Sasmito, Arun S. Mujumdar, Heat and Mass Transfer in Drying of Porous Media, 2020
Peng Xu, Arun S. Mujumdar, Agus P. Sasmito, Boming Yu
A fractal can be defined as a geometrical shape with self-similarity independent of scale. It should be noted that most natural objects that are self-similar or self-affine have that property only in a statistical sense. That is, certain property (measured quantity) of a fractal object follows power-law relation or long-term dependence on another property of the same object. Mandelbrot (1982) argued that the unique property of fractal objects M is that they are independent of the unit of measurement (scale L) and follow the scaling law in the form of: M(L)∼LDf
Frequency Reconfigurable Fractal Patch Circularly Polarized Antennas for GSM/Wi-Fi/Wi-MAX Applications
Published in IETE Journal of Research, 2022
The idea of fractal was first presented by Benoit Mandelbrot in 1975. He has used the term fractal while describing the complex shapes. A fractal is naturally a mathematical set that repeats same pattern at different levels. Fractal concepts have been applied to many applications from, compression, natural modeling, computer graphics, statistical analyses, and microstrip antennas. The dimensions of the fractals are of non integers. The space-filling property of fractals can result to the compactness of radiating elements. The execution of a frequency agile modified Sierpinski fractal patch with the utilization of a moving feed is portrayed by M. M. Ajith Kumar et al. [17]. An adaptable coaxial probe is connected to the primary microstrip line feed of a perturbed Sierpinski monopole gasket, the movement of probe is controlled with computer-controlled motor mechanism. A U-Koch slot structure with partial ground plane is examined for frequency reconfiguration by Ramadan et al.. [18]. Reconfigurable antennas using Fractal-based tree concept are proposed [19]. However, most of these antennas [17–19] reports linear polarization characteristics. In this paper, asymmetrical fractal boundary antennas are utilized for CP emission, such fractal antennas connected with microwave diodes generates frequency reconfigurable CP radiation. Here, Minkowski, Half-circled, and Koch fractal curves are employed for generating CP radiation.
Study on the optimal compaction effort of asphalt mixture based on the distribution of contact points of coarse aggregates
Published in Road Materials and Pavement Design, 2021
Xu Cai, Kuanghuai Wu, Wenke Huang
Table 3 also provides the fractal dimensions of the gradations of asphalt mixture. The fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. Song (2010) suggested that the gradations of asphalt mixture can be characterised by fractal dimensions, that the values of the fractal dimensions are related to the complexity of the structural system of asphalt mixture, and that the gradation with a greater fractal dimension generally indicates a compact structure. To simplify a calculation, Song studied the correlation between the slope k of the logarithmic curve of a mineral aggregate gradation and the fractal dimension value D, and proposed a simplified fractal dimension calculation method, as shown in Equation (1). where k represents the slope of a mineral aggregate gradation in the logarithmic curve coordinate system and D represents the fractal dimension value of a gradation.
The fractal behaviour of gravity field elements: case study in Egypt
Published in Journal of Spatial Science, 2023
The concept of fractals was intensively highlighted by B.B. Mandelbrot (1983). Fractal geometry contributed to multidisciplinary applications on a variety of scientific fields. Most of such applications were oriented towards natural phenomena. Examples are the analyses of the shapes of clouds, trees, coasts, rivers and stream networks. Also, geophysical features; such as seismic signals, geomagnetic fields, faults and earthquakes; have been analyzed as fractals (Turcotte 1989). Moreover, fractal analysis has been widely applied for the characterization and modeling of terrain surfaces (McClean 1990, Jiang 1998). Furthermore, sea bathymetry and sea surface topography were investigated regarding their fractalness (Mareschal 1989, Shih et al. 1999).