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Figure 1. 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%.

Figure 2. The Koch curve is a classic iterated fractal curve. It is a theoretical construct that is made by infinitely 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.

11.5 x 200 = 2300 km

28 x 100 = 2800 km

70 x 50 = 3500 km

Figure 3. As the length of the measuring stick is scaled smaller and smaller, the total length of the coastline measured increases.

Figure 4. Traditional notions of geometry for defining scaling and dimension.

Figure 5. The first four iterations of the Koch snowflake, which has an approximate Hausdorff dimension of 1.2619.

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 $D$ as the Koch curve and for which the empirical box counting $D$ has been demonstrated with 2% accuracy 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 with change in scale. There are several types of fractal dimension that can be determined theoretically and empirically (see Figure 1). The sets that fractal dimensions are used for characterizing come from a broad spectrum ranging from the abstract to a host of practical phenomena, including turbulence, river networks, urban growth, human physiology, medicine, and market trends. The essential idea of "fractional" or "fractal" dimensions has a long history in mathematics that can be traced as far back as the 1600s, but the term itself was brought to the fore by mathematician Benoît Mandelbrot who, in 1975, coined the terms fractal and fractal dimension.

Fractal dimensions were first applied as an index characterizing certain complex geometric forms for which the details seemed more important than the gross picture. To elaborate, 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. Moreover, unlike topological dimensions, the fractal index can fall between integer values, attesting that a set fills its space qualitatively and quantitatively differently than an ordinary geometrical set does. For instance, a curve with fractal dimension very near to 1, say 1.10, behaves quite like ordinary lines, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface; in turn, a surface with fractal dimension of 2.1 fills space very much like ordinary surfaces, but one with a fractal dimension of 2.9 folds and flows to fill space rather nearly like a volume.

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. Rather, what a fractal dimension measures is complexity, a concept tied up in certain key features of fractals: self-similarity and detail or irregularity. These features are evident in the exemplary fractal Koch curve illustrated in Figure 2. It is a curve with a topological dimension of 1, so one might hope to be able to measure its length or slope, as with ordinary lines. But we cannot do either of these things, because the fractal curve has complexity in the form of self-similarity and detail that ordinary lines lack but necessarily define fractals. The self-similarity lies in the infinite scaling, and the detail in the defining element of the Koch set. The length between any two points on a Koch curve is infinitely unmeasurable because the curve is a theoretical construct that never stops repeating itself. Every smaller piece of it is composed of an infinite number of scaled segments that look exactly like the first iteration. It is by no means a rectifiable curve, meaning it cannot be measured by being broken down into many segments approximating its length. Thus, we cannot characterize it by finding its length or slope, but we can determine its fractal dimension, which turns out to be 1.2619 (see calculations), and tells us that the Koch curve fills space somewhat more than ordinary lines, but notably less than if it were a surface.

History

The term fractal dimension that Mandelbrot coined in 1975 was not a strictly new concept. Mandelbrot tells how it had been brewing since the invention of calculus in the mid 1600s. Indeed, his notion of fractal dimension relied directly on a type of "fractional" dimension known as the Hausdorff dimension that mathematicians had been working with since the early 1900s. The novelty and brilliance were that Mandelbrot brought this and several other ideas together and applied them in a new way to study complex geometries that defied description in usual linear terms. He formally coined the term about a decade after publishing a preliminary 1967 paper on self-similarity in which he discussed fractional dimensions. 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 Figure 3). 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.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 Fractal page for more details of the history of fractals

Role of scaling in fractal dimensions

The calculation of any type of fractal dimension rests in nonconventional views of scaling and dimension. It is perhaps easiest to understand from a geometric perspective. As Figure 4 illustrates, traditional notions of geometry dictate that shapes scale predictably according to intuitive and familiar ideas about the space they are contained within. Consider the intuitive idea 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 intuitive knowledge holds in 2 dimensions, as well. If one measures the area of a square then measures again with a box 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 $N$ stands for the number of new sticks, $\epsilon$ for the scaling factor, and $D$ for the fractal dimension:

${N\propto \epsilon ^{-D}}$ (1)

This scaling rule typifies conventional rules about geometry and dimension - for lines, it quantifies that, because $N$=3 when $\epsilon$=1/3 as in the example above, $D$=1, and for squares, because $N$=9 when $\epsilon$=1/3, $D$=2.

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, $N$=4 when $\epsilon$=1/3, and the value of $D$ can be found by rearranging Equation 1:

${\log {\epsilon }{N}={-D}={\frac {\log {N}}{log{\epsilon }}}}$ (2)

That is, for a fractal described by $N$=4 when $\epsilon$=1/3, $D$=1.2619, a non-integer dimension that suggests the fractal line or curve has a dimension not equal to the 1-dimensional space it resides in. This is the scaling relationship in the well-known Koch curve discussed earlier (see Figure 3). Of note, the image itself is not a true fractal because the scaling described by the value of $D$ cannot continue infinitely for the simple reason that the image only exists to the point of its smallest component, a pixel. The theoretical Koch flake pattern that the digital image represents, 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. Another point that should be clarified is that the Koch fractal has a dimension between 1 and 2 related to the space it exists in, but fractal dimensions can also be assigned to other spaces (e.g., a 3-dimensional fractal that extends the Koch curve into 3-d space has a theoretical D=2.5849).

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. The value of D for the Koch fractal discussed above, for instance, quantifies the pattern's inherent scaling, but does not uniquely describe it. Many fractal structures or patterns could be found or constructed that have the same scaling relationship but are dramatically different from the Koch curve, as is illustrated in Figure 5. In addition, fractal dimensions found theoretically or empirically do not tell what the underlying process was for creating the dataset or structure nor do they reveal whether that process was fractal; they also do not provide enough information to reconstruct it.

For examples of how fractal patterns can be constructed, see Fractal, Sierpinski triangle, Mandelbrot set, Diffusion limited aggregation.

Specific Fractal Dimensions

The concept of fractal dimension described in this article is a basic view of a complicated construct. There are several formal mathematical definitions of different types of fractal dimension. 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 using box counting techniques.

$D_{0}=\lim _{\epsilon \rightarrow 0}{\frac {\log N(\epsilon )}{\log {\frac {1}{\epsilon }}}}.$

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

$D_{1}=\lim _{\epsilon \rightarrow 0}{\frac {-\langle \log p_{\epsilon }\rangle }{\log {\frac {1}{\epsilon }}}}$

• Correlation dimension D is based on $M$ 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.

$D_{2}=\lim _{\epsilon \rightarrow 0,M\rightarrow \infty }{\frac {\log(g_{\epsilon }/M^{2})}{\log \epsilon }}$

• 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:

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

• Multifractal dimensions: a special case of Rényi dimensions where scaling behaviour varies in different parts of the pattern.
• Uncertainty exponent
• Hausdorff dimension
• Packing dimension
• Local connected dimension

Estimating fractal dimensions of real-world data

The fractal dimensions described in this article are for formally-defined theoretical fractals. However, many real-world phenomena also exhibit limited or statistical fractal properties and fractal dimensions have been estimated for sampled data from many such phenomena using computer based fractal analysis techniques. Fractal dimensions in practise are estimated from regression lines over $logvslog$ plots of size vs scale. Practical dimension estimates 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 and as evidenced by searching databases such as PubMed, the past decade has seen methods develop from being largely theoretical to the point where estimated fractal dimensions for statistically self-similar phenomena have many practical applications in multifarious fields including:

• diagnostic imaging
• physiology
• neuroscience
• medicine,
• physics

• image analysis
• acoustics
• Riemann zeta zeros
• electrochemical processes.

See also

• List of fractals by Hausdorff dimension
• Lacunarity
• Fractal analysis
• Box counting
• Multifractal analysis

References

• Mandelbrot, Benoît B., The (Mis)Behavior of Markets, A Fractal View of Risk, Ruin and Reward (Basic Books, 2004)

External links

• [1] TruSoft's Benoit - Fractal Analysis Software product calculates fractal dimensions and hurst exponents.
• [2] Fractal Dimension Estimator Java Applet
• [3] Fractal Analysis Software for Biologists; free from NIH ImageJ website

Category:Chaos theory Category:Dynamical systems Category:Dimension theory Category:Fractals