Fractal Dimension

[Julia Dodoo]

Inmathematics, 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. 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]

The main idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions.[4] In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used (see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick.[5] There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale: see the section Examples.

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]

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.

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]

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.[8] Instead, a fractal dimension measures complexity, a concept related to certain key features of fractals: self-similarity and detail or irregularity.[notes 2] 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.[5] The self-similarity lies in the infinite scaling, and the detail in the defining elements of each set. 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.[19] 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 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 concept of a fractal dimension rests in unconventional views of scaling and dimension.[24] As Fig. 4 illustrates, traditional notions of geometry dictate that shapes scale predictably according to intuitive and familiar ideas about the space they are contained within, such that, for instance, measuring a line using first one measuring stick then another 1/3 its size, will give for the second stick a total length 3 times as many sticks long as with the first. This holds in 2 dimensions, as well. If one measures the area of a square then measures again with a box of side length 1/3 the size of the original, one will find 9 times as many squares as with the first measure. Such familiar scaling relationships can be defined mathematically by the general scaling rule in Equation 1, where the variable N \displaystyle N stands for the number of measurement units (sticks, squares, etc.), ε \displaystyle \varepsilon for the scaling factor, and D \displaystyle D for the fractal dimension:

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. 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.[5][24]

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.

The concept of fractality is applied increasingly in the field of surface science, providing a bridge between surface characteristics and functional properties.[26] 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.[27] The fractal dimensions of surfaces have been employed to explain and better understand phenomena in areas of contact mechanics,[28] frictional behavior,[29] electrical contact resistance[30] and transparent conducting oxides.[31]

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:

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 includingastronomy,[35] acoustics,[36][37] geology and earth sciences,[38] diagnostic imaging,[39][40][41]ecology,[42]electrochemical processes,[43]image analysis,[44][45][46][47]biology and medicine,[48][49][50]neuroscience,[51][13]network analysis,physiology,[12]physics,[52][53] and Riemann zeta zeros.[54] Fractal dimension estimates have also been shown to correlate with Lempel-Ziv complexity in real-world data sets from psychoacoustics and neuroscience.[55][36]

An alternative to a direct measurement, is considering a mathematical model that resembles formation of a real-world fractal object. In this case, a validation can also be done by comparing other than fractal properties implied by the model, with measured data. In colloidal physics, systems composed of particles with various fractal dimensions arise. To describe these systems, it is convenient to speak about a distribution of fractal dimensions, and eventually, a time evolution of the latter: a process that is driven by a complex interplay between aggregation and coalescence.[56]

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