Fractal dimension

. The theoretical fractal dimension for this fractal is 5/3 ≈ 1.67; its empirical fractal dimension from box counting analysis is ±1% using fractal analysis software. 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. human physiology, medicine, Fractal dimensions were first applied as an index characterizing complicated geometric forms for which the details seemed more important than the gross picture. indicating that a set fills its space qualitatively and quantitatively differently from how an ordinary geometrical set does. 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. is a classic iterated fractal curve. It is made by starting from a line segment, and then iteratively scaling each segment by 1/3 into 4 new pieces laid end to end with 2 middle pieces leaning toward each other along an equilateral triangle, so that the whole new segment spans the distance between the endpoints of the original segment. The animation only shows a few iterations, but the theoretical curve is scaled in this way infinitely. 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. These features are evident in the two examples of fractal curves. Both are curves with topological dimension of 1, so one might hope to be able to measure their length and derivative in the same way as with ordinary curves. 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. Every smaller piece is composed of an infinite number of scaled segments that look exactly like the first iteration. These are not rectifiable curves, meaning that they cannot be measured by being broken down into many segments approximating their respective lengths. They cannot be meaningfully characterized by finding their lengths and derivatives. However, their fractal dimensions can be determined, which shows that both fill space more than ordinary lines but less than surfaces, and allows them to be compared in this regard. 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. This sort of structure can be extended to other spaces (e.g., a fractal that extends the Koch curve into 3D space has a theoretical D = 2.5849). However, such neatly countable complexity is only one example of the self-similarity and detail that are present in fractals. 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. == History ==

The terms fractal dimension and fractal were coined by Mandelbrot in 1975, These works were accompanied by perhaps the most pivotal point in the development of the concept of a fractal dimension through the work of Hausdorff in the early 1900s who defined a "fractional" dimension that has come to be named after him and is frequently invoked in defining modern fractals. See Fractal history for more information == Mathematical definition ==

The mathematical definition of fractal dimension can be derived by observing and then generalizing the effect of traditional dimension on measurement-changes under scaling. For example, say you have a line and a measuring-stick of equal length. Now shrink the stick to 1/3 its size; you can now fit 3 sticks into the line. Similarly, in two dimensions, say you have a square and an identical "measuring-square". Now shrink the measuring-square's side to 1/3 its length; you can now fit 3^2 = 9 measuring-squares into the square. Such familiar scaling relationships obey equation , where \varepsilon is the scaling factor, D the dimension, and N the resulting number of units (sticks, squares, etc.) in the measured object:{{NumBlk|:|N = \varepsilon^{-D}.|}} In the line example, the dimension D = 1 because there are N = 3 units when the scaling factor \varepsilon = 1/3. In the square example, D = 2 because N = 9 when \varepsilon = 1/3. s of the Koch snowflake, which has a Hausdorff dimension of approximately 1.2619. Fractal dimension generalizes traditional dimension in that it can be fractional, but it has exactly the same relationship with scaling that traditional dimension does; in fact, it is derived by simply rearranging equation : {{NumBlk|:|D = -\log_\varepsilon N = -\frac{\log N}{\log \varepsilon}.|}} D can be thought of as the power of the scaling factor of an object's measure given some scaling of its "radius". For example, the Koch snowflake has D = 1.26185\ldots, indicating that lengthening its radius grows its measure faster than if it were a one-dimensional shape (such as a polygon), but slower than if it were a two-dimensional shape (such as a filled polygon). Of note, images shown in this page are not true fractals because the scaling described by D cannot continue past the point of their smallest component, a pixel. However, the theoretical patterns that the images represent have no discrete pixel-like pieces, but rather are composed of an infinite number of infinitely scaled segments and do indeed have the claimed fractal dimensions. == D is not a unique descriptor ==

branching fractals that are made by producing 4 new parts for every 1/3 scaling, thus having the same theoretical D as the Koch curve, and for which the empirical box counting D has been demonstrated with 2% accuracy. 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 Fig. 6. For examples of how fractal patterns can be constructed, see Fractal, Sierpinski triangle, Mandelbrot set, Diffusion-limited aggregation, L-system. == Fractal surface structures ==

The concept of fractality is applied increasingly in the field of surface science, providing a bridge between surface characteristics and functional properties. Numerous surface descriptors are used to interpret the structure of nominally flat surfaces, which often exhibit self-affine features across multiple length-scales. Mean surface roughness, usually denoted RA, is the most commonly applied surface descriptor, however, numerous other descriptors including mean slope, root-mean-square roughness (RRMS) and others are regularly applied. 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. The fractal dimensions of surfaces have been employed to explain and better understand phenomena in areas of contact mechanics, frictional behavior, electrical contact resistance and transparent conducting oxides. == Examples ==

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–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: • Box-counting dimension is estimated as the exponent of a power law: • : D_0 = \lim_{\varepsilon \to 0} \frac{\log N(\varepsilon)}{\log\frac{1}{\varepsilon}}. • Information dimension considers how the average information needed to identify an occupied box scales with box size (p is a probability): • : D_1 = \lim_{\varepsilon \to 0} \frac{-\langle \log p_\varepsilon \rangle}{\log\frac{1}{\varepsilon}}. • Correlation dimension is based on M as the number of points used to generate a representation of a fractal and gε, the number of pairs of points closer than ε to each other: • : D_2 = \lim_{M \to \infty} \lim_{\varepsilon \to 0} \frac{\log (g_\varepsilon / M^2)}{\log \varepsilon}. • Generalized, or Rényi dimensions: the box-counting, information, and correlation dimensions can be seen as special cases of a continuous spectrum of generalized dimensions of order α, defined by • : D_\alpha = \lim_{\varepsilon \to 0} \frac{\frac{1}{\alpha - 1} \log(\sum_i p_i^\alpha)}{\log\varepsilon}. • Higuchi dimension • : D = \frac{d\log L(k)}{d \log k}. • Lyapunov dimension • Multifractal dimensions: a special case of Rényi dimensions where scaling behaviour varies in different parts of the pattern. • Uncertainty exponent • Hausdorff dimension: For any subset S of a metric space X and d \geq 0, the d-dimensional Hausdorff content of S is defined by C_H^d(S) := \inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii }r_i > 0\Bigr\}. The Hausdorff dimension of S is defined by • : \dim_{\operatorname{H}}(X) := \inf\{d \ge 0: C_H^d(X) = 0\}. • Packing dimension • Assouad dimension • Local connected dimension • Degree dimension describes the fractal nature of the degree distribution of graphs. • Parabolic Hausdorff dimension == Estimating from real-world data ==

Many real-world phenomena exhibit limited or statistical fractal properties and fractal dimensions that have been estimated from sampled data using computer-based fractal analysis techniques. Practically, measurements of fractal dimension are affected by various methodological issues and are sensitive to numerical or experimental noise and limitations in the amount of data. Nonetheless, the field is rapidly growing as estimated fractal dimensions for statistically self-similar phenomena may have many practical applications in various fields, including astronomy, acoustics, architecture, geology and earth sciences, diagnostic imaging, ecology, electrochemical processes, image analysis, biology and medicine, neuroscience, and Riemann zeta zeros. Fractal dimension estimates have also been shown to correlate with Lempel–Ziv complexity in real-world data sets from psychoacoustics and neuroscience. == See also ==