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]

The main idea of "fractured"

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

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.

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,

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

scaling

so have the same theoretical ${\displaystyle D}$ as the Koch curve and for which the empirical box counting ${\displaystyle D}$ 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.^[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

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

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

• 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

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

References

Further reading

• 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.