Fractalic Awakening - A Seeker's Guide

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D 1 = lim ε → 0 − ⟨ log ⁡ p ε ⟩ log ⁡ 1 ε {\displaystyle D_{1}=\lim _{\varepsilon \to 0}{\frac {-\langle \log p_{\varepsilon }\rangle }{\log {\frac {1}{\varepsilon }}}}} The concept of fractal dimension described in this article is a basic view of a complicated construct. The examples discussed here were chosen for clarity, and the scaling unit and ratios were known ahead of time. In practice, however, fractal dimensions can be determined using techniques that approximate scaling and detail from limits estimated from regression lines over log vs log plots of size vs scale. Several formal mathematical definitions of different types of fractal dimension are listed below. Although for compact sets with exact affine self-similarity all these dimensions coincide, in general they are not equivalent: Figure 6. Two L-systems branching fractals that are made by producing 4 new parts for every 1/3 scaling so have the same theoretical D {\displaystyle D} as the Koch curve and for which the empirical box counting D {\displaystyle D} has been demonstrated with 2% accuracy. [8] Fractal dimensions were first applied as an index characterizing complicated geometric forms for which the details seemed more important than the gross picture. [16] For sets describing ordinary geometric shapes, the theoretical fractal dimension equals the set's familiar Euclidean or topological dimension. Thus, it is 0 for sets describing points (0-dimensional sets); 1 for sets describing lines (1-dimensional sets having length only); 2 for sets describing surfaces (2-dimensional sets having length and width); and 3 for sets describing volumes (3-dimensional sets having length, width, and height). But this changes for fractal sets. If the theoretical fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry. [17]

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D 2 = lim M → ∞ lim ε → 0 log ⁡ ( g ε / M 2 ) log ⁡ ε {\displaystyle D_{2}=\lim _{M\to \infty }\lim _{\varepsilon \to 0}{\frac {\log(g_{\varepsilon }/M Correlation dimension: D is based on M {\displaystyle M} as the number of points used to generate a representation of a fractal and g ε, the number of pairs of points closer than ε to each other.D 0 = lim ε → 0 log ⁡ N ( ε ) log ⁡ 1 ε . {\displaystyle D_{0}=\lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log {\frac {1}{\varepsilon }}}}.} Ultimately, the term fractal dimension became the phrase with which Mandelbrot himself became most comfortable with respect to encapsulating the meaning of the word fractal, a term he created. After several iterations over years, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants." [6] It is also a measure of the space-filling capacity of a pattern, and it tells how a fractal scales differently, in a fractal (non-integer) dimension. [1] [2] [3]

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Information dimension: D considers how the average information needed to identify an occupied box scales with box size; p {\displaystyle p} is a probability. As is the case with dimensions determined for lines, squares, and cubes, fractal dimensions are general descriptors that do not uniquely define patterns. [24] [25] The value of D for the Koch fractal discussed above, for instance, quantifies the pattern's inherent scaling, but does not uniquely describe nor provide enough information to reconstruct it. Many fractal structures or patterns could be constructed that have the same scaling relationship but are dramatically different from the Koch curve, as is illustrated in Figure 6. In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured. For examples of how fractal patterns can be constructed, see Fractal, Sierpinski triangle, Mandelbrot set, Diffusion limited aggregation, L-System.That is, for a fractal described by N = 4 {\displaystyle N=4} when ε = 1 3 {\displaystyle \varepsilon ={\tfrac {1}{3}}} , such as the Koch snowflake, D = 1.26185 … {\displaystyle D=1.26185\ldots } , a non-integer value that suggests the fractal has a dimension not equal to the space it resides in. [3]