Fractal dimension

In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.[1][2][3]

The essential 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.

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 a rectifiable curve: 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 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 about 1.262.

Introduction

Figure 2. A 32-segment quadric fractal scaled and viewed through boxes of different sizes. The pattern illustrates self similarity. The theoretical fractal dimension for this fractal is log32/log8 = 1.67; its empirical fractal dimension from box counting analysis is ±1%[8] 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.[5]:1 Several types of fractal dimension can be measured theoretically and empirically (see Fig. 2).[3][9] 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.[14] The essential idea of fractional or fractal dimensions has a long history in mathematics that can be traced back to the 1600s,[5]:19[15] but the terms fractal and fractal dimension were coined by mathematician Benoit Mandelbrot in 1975.[1][2][5][9][14][16]

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]

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 1.26.

a Koch curve animation
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.
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 or slope, as with ordinary lines. But we cannot do either of these things, because fractal curves have complexity in the form of self-similarity and detail that ordinary lines 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 undefined because the curves are theoretical constructs that never stop repeating themselves.[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 characterized by finding their lengths or slopes. 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 3-d 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.[3][14] The example of the coast line of Britain, for instance, exhibits self-similarity of an approximate pattern with approximate scaling.[5]:26 Overall, fractals show several types and degrees of self-similarity and detail that may not be easily visualized. 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.[20] 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.[5]:197; 262

History
The terms fractal dimension and fractal were coined by Mandelbrot in 1975,[16] about a decade after he published his paper on self-similarity in the coastline of Britain. 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.[15][21][22] The earliest roots of what Mandelbrot synthesized as the fractal dimension have been traced clearly back to writings about undifferentiable, infinitely self-similar functions, which are important in the mathematical definition of fractals, around the time that calculus was discovered in the mid-1600s.[5]:405 There was a lull in the published work on such functions for a time after that, then a renewal starting in the late 1800s with the publishing of mathematical functions and sets that are today called canonical fractals (such as the eponymous works of von Koch,[19] Sierpiński, and Julia), but at the time of their formulation were often considered antithetical mathematical "monsters".[15][22] 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.[4][5]:44[17][21]

See Fractal history for more information

Role of scaling

Lines, squares, and cubes.
Figure 4. Traditional notions of geometry for defining scaling and dimension.
The concept of a fractal dimension rests in unconventional views of scaling and dimension.[23] 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 {\displaystyle N}N stands for the number of sticks, {\displaystyle \varepsilon }\varepsilon for the scaling factor, and {\displaystyle D}D for the fractal dimension:

{\displaystyle {N\propto \varepsilon ^{-D}},}{\displaystyle {N\propto \varepsilon ^{-D}},}

(1)

where {\displaystyle \propto }\propto denotes proportionality. This scaling rule typifies conventional rules about geometry and dimension – for lines, it quantifies that, because {\displaystyle N=3}N=3 when {\displaystyle \varepsilon ={\tfrac {1}{3}}}{\displaystyle \varepsilon ={\tfrac {1}{3}}} as in the example above, {\displaystyle D=1,}{\displaystyle D=1,} and for squares, because {\displaystyle N=9}{\displaystyle N=9} when {\displaystyle \varepsilon ={\tfrac {1}{3}},D=2.}{\displaystyle \varepsilon ={\tfrac {1}{3}},D=2.}

A fractal contour of a koch snowflake
Figure 5. The first four iterations of the Koch snowflake, which has an approximate Hausdorff dimension of 1.2619.
The same rule applies to fractal geometry but less intuitively. To elaborate, a fractal line measured at first to be one length, when remeasured using a new stick scaled by 1/3 of the old may not be the expected 3 but instead 4 times as many scaled sticks long. In this case, {\displaystyle N=4}N=4 when {\displaystyle \varepsilon ={\tfrac {1}{3}},}{\displaystyle \varepsilon ={\tfrac {1}{3}},} and the value of {\displaystyle D}D can be found by rearranging Equation 1:

{\displaystyle {\log _{\varepsilon }{N}={-D}={\frac {\log {N}}{\log {\varepsilon }}}}}{\displaystyle {\log _{\varepsilon }{N}={-D}={\frac {\log {N}}{\log {\varepsilon }}}}}

(2)

That is, for a fractal described by {\displaystyle N=4}N=4 when {\displaystyle \varepsilon ={\tfrac {1}{3}},D=1.2619,}{\displaystyle \varepsilon ={\tfrac {1}{3}},D=1.2619,} a non-integer dimension that suggests the fractal has a dimension not equal to the space it resides in.[3] The scaling used in this example is the same scaling of the Koch curve and snowflake. Of note, the images shown are not true fractals because the scaling described by the value of {\displaystyle D}D cannot continue infinitely for the simple reason that the images only exist to the point of their smallest component, a pixel. The theoretical pattern that the digital images represent, however, has no discrete pixel-like pieces, but rather is composed of an infinite number of infinitely scaled segments joined at different angles and does indeed have a fractal dimension of 1.2619.[5][23]

D is not a unique descriptor

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 {\displaystyle D}D as the Koch curve and for which the empirical box counting {\displaystyle D}D has been demonstrated with 2% accuracy.[8]
As is the case with dimensions determined for lines, squares, and cubes, fractal dimensions are general descriptors that do not uniquely define patterns.[23][24] 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.

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.[25] 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.[26] The fractal dimensions of surfaces have been employed to explain and better understand phenomena in areas of contact mechanics,[27] frictional behavior,[28] electrical contact resistance[29] and transparent conducting oxides.[30]

Figure 7: Illustration of increasing surface fractality. Self-affine surfaces (left) and corresponding surface profiles (right) showing increasing fractal dimension Df
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 vs log plots of size vs scale. Several formal mathematical definitions of different types of fractal dimension are listed below. Although for some classic fractals all these dimensions coincide, in general they are not equivalent:

Box counting dimension: D is estimated as the exponent of a power law.
{\displaystyle D_{0}=\lim {\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log {\frac {1}{\varepsilon }}}}.}{\displaystyle D{0}=\lim {\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log {\frac {1}{\varepsilon }}}}.}
Information dimension: D considers how the average information needed to identify an occupied box scales with box size; {\displaystyle p}p is a probability.
{\displaystyle D{1}=\lim {\varepsilon \to 0}{\frac {-\langle \log p{\varepsilon }\rangle }{\log {\frac {1}{\varepsilon }}}}}{\displaystyle D_{1}=\lim {\varepsilon \to 0}{\frac {-\langle \log p{\varepsilon }\rangle }{\log {\frac {1}{\varepsilon }}}}}
Correlation dimension: D is based on {\displaystyle M}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.
{\displaystyle D_{2}=\lim {M\to \infty }\lim {\varepsilon \to 0}{\frac {\log(g{\varepsilon }/M^{2})}{\log \varepsilon }}}{\displaystyle D{2}=\lim {M\to \infty }\lim {\varepsilon \to 0}{\frac {\log(g{\varepsilon }/M^{2})}{\log \varepsilon }}}[citation needed]
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:
{\displaystyle D{\alpha }=\lim {\varepsilon \to 0}{\frac {{\frac {1}{\alpha -1}}\log(\sum {i}p{i}^{\alpha })}{\log \varepsilon }}}{\displaystyle D{\alpha }=\lim {\varepsilon \to 0}{\frac {{\frac {1}{\alpha -1}}\log(\sum {i}p{i}^{\alpha })}{\log \varepsilon }}}
Higuchi dimension[31]
{\displaystyle D={\frac {d\ \log(L(k))}{d\ \log(k)}}}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 {\displaystyle S}S of a metric space {\displaystyle X}X and {\displaystyle d\geq 0}{\displaystyle d\geq 0}, the d-dimensional Hausdorff content of S is defined by
{\displaystyle 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 }}.}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
{\displaystyle \dim {\operatorname {H} }(X):=\inf{d\geq 0:C{H}^{d}(X)=0}.}\dim {{\operatorname {H}}}(X):=\inf{d\geq 0:C{H}^{d}(X)=0}.