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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 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 }}}}} 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]

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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.Unlike topological dimensions, the fractal index can take non- integer values, [18] indicating that a set fills its space qualitatively and quantitatively differently from how an ordinary geometrical set does. [1] [2] [3] For instance, a curve with a fractal dimension very near to 1, say 1.10, behaves quite like an ordinary line, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface. Similarly, a surface with fractal dimension of 2.1 fills space very much like an ordinary surface, but one with a fractal dimension of 2.9 folds and flows to fill space rather nearly like a volume. [17] :48 [notes 1] This general relationship can be seen in the two images of fractal curves in Fig.2 and Fig. 3 – the 32-segment contour in Fig. 2, convoluted and space filling, has a fractal dimension of 1.67, compared to the perceptibly less complex Koch curve in Fig. 3, which has a fractal dimension of approximately 1.2619. 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 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. This scaling rule typifies conventional rules about geometry and dimension – referring to the examples above, it quantifies that D = 1 {\displaystyle D=1} for lines because N = 3 {\displaystyle N=3} when ε = 1 3 {\displaystyle \varepsilon ={\tfrac {1}{3}}} , and that D = 2 {\displaystyle D=2} for squares because N = 9 {\displaystyle N=9} when ε = 1 3 . {\displaystyle \varepsilon ={\tfrac {1}{3}}.}

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One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension of 1, but it is by no means rectifiable: the length of the curve between any two points on the Koch snowflake is infinite. No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively by thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional. [7] Therefore its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, which is often a number between one and two; in the case of the Koch snowflake, it is approximately 1.2619. 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: 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] Information dimension: D considers how the average information needed to identify an occupied box scales with box size; p {\displaystyle p} is a probability. Figure 5. The first four iterations of the Koch snowflake, which has a Hausdorff dimension of approximately 1.2619.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] 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 }}}}.} 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]

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Figure 3. The Koch curve is a classic iterated fractal curve. It is a theoretical construct that is made by iteratively scaling a starting segment. As shown, each new segment is scaled by 1/3 into 4 new pieces laid end to end with 2 middle pieces leaning toward each other between the other two pieces, so that if they were a triangle its base would be the length of the middle piece, so that the whole new segment fits across the traditionally measured length between the endpoints of the previous segment. Whereas the animation only shows a few iterations, the theoretical curve is scaled in this way infinitely. Beyond about 6 iterations on an image this small, the detail is lost. 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] For examples of how fractal patterns can be constructed, see Fractal, Sierpinski triangle, Mandelbrot set, Diffusion limited aggregation, L-System.Subject of Fractalic_anna Chaturbate room: welcome to my forest // GOAL: slap my ass (30 times) [99 tokens left] #hairy #hairyarmpits #natural #bush #bigass