Fractal dimension

11.5 x 200 = 2300 km

28 x 100 = 2800 km

70 x 50 = 3500 km

Figure 1. As the length of the measuring stick is scaled smaller and smaller, the total length of the coastline measured increases (See Coastline paradox).

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. 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]}

Contents

• Introduction
• History
• Role of scaling
• D is not a unique descriptor
• Fractal surface structures
• Examples
• Estimating from real-world data
• See also
• Notes
• References
• Further reading
• External links

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.

Introduction

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 approximately 1.2619.

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 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 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. ^{[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

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 ${\displaystyle N}$ stands for the number of measurement units (sticks, squares, etc.), ${\displaystyle \varepsilon }$ for the scaling factor, and ${\displaystyle D}$ for the fractal dimension:

${\displaystyle {N=\varepsilon ^{-D}}.}$

(1)

This scaling rule typifies conventional rules about geometry and dimension – referring to the examples above, it quantifies that ${\displaystyle D=1}$ for lines because ${\displaystyle N=3}$ when ${\displaystyle \varepsilon ={\tfrac {1}{3}}}$, and that ${\displaystyle D=2}$ for squares because ${\displaystyle N=9}$ when ${\displaystyle \varepsilon ={\tfrac {1}{3}}.}$

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 be 4 times as many scaled sticks long rather than the expected 3 (see Fig. 5). In this case, ${\displaystyle N=4}$ when ${\displaystyle \varepsilon ={\tfrac {1}{3}},}$ and the value of ${\displaystyle D}$ can be found by rearranging Equation 1:

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

(2)

That is, for a fractal described by ${\displaystyle N=4}$ when ${\displaystyle \varepsilon ={\tfrac {1}{3}}}$, such as the Koch snowflake, ${\displaystyle D=1.26185\ldots }$, a non-integer value that suggests the fractal has a dimension not equal to the space it resides in. ^[3]

Of note, images shown in this page are not true fractals because the scaling described by ${\displaystyle D}$ 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]}

D is not a unique descriptor

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.

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. ^[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 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. 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]

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 compact sets with exact affine self-similarity all these dimensions coincide, in general they are not equivalent:

• Box counting dimension: D is estimated as the exponent of a power law.

${\displaystyle D_{0}=\lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log {\frac {1}{\varepsilon }}}}.}$

• Information dimension: D considers how the average information needed to identify an occupied box scales with box size; ${\displaystyle p}$ is a probability.

${\displaystyle D_{1}=\lim _{\varepsilon \to 0}{\frac {-\langle \log p_{\varepsilon }\rangle }{\log {\frac {1}{\varepsilon }}}}}$

• Correlation dimension: D is based on ${\displaystyle M}$ 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.

${\displaystyle D_{2}=\lim _{M\to \infty }\lim _{\varepsilon \to 0}{\frac {\log(g_{\varepsilon }/M^{2})}{\log \varepsilon }}}$^{[ 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:

${\displaystyle D_{\alpha }=\lim _{\varepsilon \to 0}{\frac {{\frac {1}{\alpha -1}}\log(\sum _{i}p_{i}^{\alpha })}{\log \varepsilon }}}$

• Higuchi dimension ^[32]

${\displaystyle D={\frac {d\ \log(L(k))}{d\ \log(k)}}}$

• 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 ${\displaystyle S}$ of a metric space ${\displaystyle X}$ and ${\displaystyle d\geq 0}$, the d-dimensional Hausdorff content of S is defined by

${\displaystyle C_{H}^{d}(S):=\inf {\Bigl \{}\sum _{i}r_{i}^{d}:{\text{ there is a cover of }}S{\text{ by balls with radii }}r_{i}>0{\Bigr \}}.}$

The Hausdorff dimension of S is defined by

${\displaystyle \dim _{\operatorname {H} }(X):=\inf\{d\geq 0:C_{H}^{d}(X)=0\}.}$

• Packing dimension
• Assouad dimension
• Local connected dimension ^[33]
• Degree dimension: describes the fractal nature of the degree distribution of graphs. ^[34]

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, ^[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]

See also

• List of fractals by Hausdorff dimension
• Lacunarity  – Term in geometry and fractal analysis
• Fractal derivative  – Generalization of derivative to fractals

Notes

Related Research Articles

Benoit B.Mandelbrot was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.

In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.

In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.

The Mandelbrot set is a two-dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal and fractal structures. The set is defined in the complex plane as the complex numbers for which the function does not diverge to infinity when iterated starting at , i.e., for which the sequence , , etc., remains bounded in absolute value.

In mathematics, a self-similar object is exactly or approximately similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.

The Sierpiński triangle, also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński.

The Koch snowflake is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch.

Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still digital images, animations, and media. Fractal art developed from the mid-1980s onwards. It is a genre of computer art and digital art which are part of new media art. The mathematical beauty of fractals lies at the intersection of generative art and computer art. They combine to produce a type of abstract art.

In mathematics, the Menger sponge is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.

In mathematics, a random walk, sometimes known as a drunkard's walk, is a random process that describes a path that consists of a succession of random steps on some mathematical space.

In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set in a Euclidean space , or more generally in a metric space . It is named after the Polish mathematician Hermann Minkowski and the French mathematician Georges Bouligand.

In chaos theory, the correlation dimension is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension.

A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite length — and every subarc longer than a single point has infinite length.

The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal curve–like properties of coastlines; i.e., the fact that a coastline typically has a fractal dimension. Although the "paradox of length" was previously noted by Hugo Steinhaus, the first systematic study of this phenomenon was by Lewis Fry Richardson, and it was expanded upon by Benoit Mandelbrot.

A multifractal system is a generalization of a fractal system in which a single exponent is not enough to describe its dynamics; instead, a continuous spectrum of exponents is needed.

Lacunarity, from the Latin lacuna, meaning "gap" or "lake", is a specialized term in geometry referring to a measure of how patterns, especially fractals, fill space, where patterns having more or larger gaps generally have higher lacunarity. Beyond being an intuitive measure of gappiness, lacunarity can quantify additional features of patterns such as "rotational invariance" and more generally, heterogeneity. This is illustrated in Figure 1 showing three fractal patterns. When rotated 90°, the first two fairly homogeneous patterns do not appear to change, but the third more heterogeneous figure does change and has correspondingly higher lacunarity. The earliest reference to the term in geometry is usually attributed to Benoit Mandelbrot, who, in 1983 or perhaps as early as 1977, introduced it as, in essence, an adjunct to fractal analysis. Lacunarity analysis is now used to characterize patterns in a wide variety of fields and has application in multifractal analysis in particular.

Box counting is a method of gathering data for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at each smaller scale. The essence of the process has been compared to zooming in or out using optical or computer based methods to examine how observations of detail change with scale. In box counting, however, rather than changing the magnification or resolution of a lens, the investigator changes the size of the element used to inspect the object or pattern. Computer based box counting algorithms have been applied to patterns in 1-, 2-, and 3-dimensional spaces. The technique is usually implemented in software for use on patterns extracted from digital media, although the fundamental method can be used to investigate some patterns physically. The technique arose out of and is used in fractal analysis. It also has application in related fields such as lacunarity and multifractal analysis.

An ordinary fractal string is a bounded, open subset of the real number line. Such a subset can be written as an at-most-countable union of connected open intervals with associated lengths written in non-increasing order; we also refer to as a fractal string. For example, is a fractal string corresponding to the Cantor set. A fractal string is the analogue of a one-dimensional "fractal drum," and typically the set has a boundary which corresponds to a fractal such as the Cantor set. The heuristic idea of a fractal string is to study a (one-dimensional) fractal using the "space around the fractal." It turns out that the sequence of lengths of the set itself is "intrinsic," in the sense that the fractal string itself contains information about the fractal to which it corresponds.

The Fibonacci word fractal is a fractal curve defined on the plane from the Fibonacci word.

References

Further reading

• Mandelbrot, Benoit B.; Hudson, Richard L. (2010). The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward. Profile Books. ISBN   978-1-84765-155-6.

External links

Wikimedia Commons has media related to Fractals by dimension .

• TruSoft's Benoit, fractal analysis software product calculates fractal dimensions and hurst exponents.
• A Java Applet to Compute Fractal Dimensions
• Introduction to Fractal Analysis
• Bowley, Roger (2009). "Fractal Dimension". Sixty Symbols. Brady Haran for the University of Nottingham.
• "Fractals are typically not self-similar". 3Blue1Brown.