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The concept of fractal dimension described in this article is a basic view of a complicated construct.

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The examples discussed here were chosen for clarity, and the scaling unit and ratios were known ahead of time. 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. 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. For examples of how fractal patterns can be constructed, see Fractal, Sierpinski triangle, Mandelbrot set, Diffusion limited aggregation, L-System. Fractal dimensions are used to characterize a broad spectrum of objects ranging from the abstract [1] [3] to practical phenomena, including turbulence, [5] :97–104 river networks, :246–247 urban growth, [10] [11] human physiology, [12] [13] medicine, [9] and market trends.

The two fractal curves described above show a type of self-similarity that is exact with a repeating unit of detail that is readily visualized. It is found however that many physical surface phenomena cannot readily be interpreted with reference to such descriptors, thus fractal dimension is increasingly applied to establish correlations between surface structure in terms of scaling behavior and performance.

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No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles. Of note, images shown in this page are not true fractals because the scaling described by D {\displaystyle D} cannot continue past the point of their smallest component, a pixel. 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). 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. Every smaller piece is composed of an infinite number of scaled segments that look exactly like the first iteration.If the theoretical fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry. Various historical authorities credit him with also synthesizing centuries of complicated theoretical mathematics and engineering work and applying them in a new way to study complex geometries that defied description in usual linear terms. The earliest roots of what Mandelbrot synthesized as the fractal dimension have been traced clearly back to writings about nondifferentiable, infinitely self-similar functions, which are important in the mathematical definition of fractals, around the time that calculus was discovered in the mid-1600s.

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The length between any two points on these curves is infinite, no matter how close together the two points are, which means that it is impossible to approximate the length of such a curve by partitioning the curve into many small segments. Overall, fractals show several types and degrees of self-similarity and detail that may not be easily visualized. displaystyle D_{0}=\lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log {\frac {1}{\varepsilon }}}}. However, such neatly countable complexity is only one example of the self-similarity and detail that are present in fractals. 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.

A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale.

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displaystyle C_{H} These include, as examples, strange attractors for which the detail has been described as in essence, smooth portions piling up, [17] :49 the Julia set, which can be seen to be complex swirls upon swirls, and heart rates, which are patterns of rough spikes repeated and scaled in time. Fractal complexity may not always be resolvable into easily grasped units of detail and scale without complex analytic methods but it is still quantifiable through fractal dimensions. But we cannot do either of these things, because fractal curves have complexity in the form of self-similarity and detail that ordinary curves lack.Mean surface roughness, usually denoted R A, is the most commonly applied surface descriptor, however numerous other descriptors including mean slope, root mean square roughness (R RMS) and others are regularly applied. The relationship of an increasing fractal dimension with space-filling might be taken to mean fractal dimensions measure density, but that is not so; the two are not strictly correlated.