Editing Fractal dimension

{{short description|Ratio providing a statistical index of complexity variation with scale}}
{{anchor|coastline}}
{{multiple image
   | width     = 100
   | footer    = 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]]).
   | align     = right
   | image1    = britain-fractal-coastline-200km.png
   | alt1      = Coastline of Britain measured using a 200 km scale
   | caption1  = 11.5 × 200 km = 2300 km
   | image2    = britain-fractal-coastline-100km.png
   | alt2      = Coastline of Britain measured using a 100 km scale
   | caption2  = 28 × 100 km = 2800 km
   | image3    = britain-fractal-coastline-50km.png
   | alt3      = Coastline of Britain measured using a 50 km scale
   | caption3  = 70 × 50 km = 3500 km
}}
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 [[Scaling (geometry)|scale]] at which it is measured. 
It is also a measure of the [[Space-filling curve|space-filling]] capacity of a pattern and tells how a fractal scales differently, in a fractal (non-integer) dimension.<ref name="Falconer" /><ref name="space filling"/><ref name="vicsek">{{cite book | last = Vicsek | first = Tamás | title = Fractal growth phenomena | publisher = World Scientific | year = 1992 | isbn = 978-981-02-0668-0 | page=10}}</ref>

The main idea of "fractured" [[Hausdorff dimension|dimensions]] has a long history in mathematics, but the term itself was brought to the fore by [[Benoit Mandelbrot]] based on [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|his 1967 paper]] on [[self-similarity]] in which he discussed ''fractional dimensions''.<ref name="coastline">{{Cite journal | last1 = Mandelbrot | first1 = B. | title = How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension | doi = 10.1126/science.156.3775.636 | journal = Science | volume = 156 | issue = 3775 | pages = 636–638 | year = 1967 | pmid = 17837158 | bibcode = 1967Sci...156..636M | s2cid = 15662830 | url = http://ena.lp.edu.ua:8080/handle/ntb/52473 | access-date = 2020-11-12 | archive-date = 2021-10-19 | archive-url = https://web.archive.org/web/20211019193011/http://ena.lp.edu.ua:8080/handle/ntb/52473 | url-status = dead }}</ref> 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 [[#coastline|Fig.&nbsp;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.<ref name="Mandelbrot1983"/> There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale, see {{slink|#Examples}} below.

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".<ref>{{cite book |first=Gerald |last=Edgar |title=Measure, Topology, and Fractal Geometry |url=https://books.google.com/books?id=dk2vruTv0_gC&pg=PR7 |date=2007 |publisher=Springer |isbn=978-0-387-74749-1 |pages=7}}</ref>

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 [[Arc length|rectifiable]]: the length of the curve between any two points on the Koch snowflake is [[Arc length#Curves with infinite length|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.<ref>{{cite book | last = Harte | first = David | title = Multifractals | url = https://archive.org/details/multifractalsthe00hart_175 | url-access = limited | publisher = Chapman & Hall | year = 2001 | isbn = 978-1-58488-154-4 |pages=[https://archive.org/details/multifractalsthe00hart_175/page/n55 3]–4}}</ref> 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 ==
{{Anchor|32seg}}
[[Image:32 segment fractal.jpg|thumb|right|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 5/3 ≈ 1.67; its empirical fractal dimension from [[box counting]] analysis is ±1%<ref name="empirical fractal dimension">{{cite book
| first = Audrey
| last = Balay-Karperien
| title = Defining Microglial Morphology: Form, Function, and Fractal Dimension
| publisher = Charles Sturt University
| url = http://trove.nla.gov.au/work/162139699 
| access-date = 9 July 2013
| year = 2004
| page = 86
}}</ref> using [[fractal analysis]] software.]]

A '''[[fractal]] dimension''' is an index for characterizing [[fractal]] patterns or [[Set (mathematics)|sets]] by quantifying their [[complexity]] as a ratio of the change in detail to the change in scale.<ref name="Mandelbrot1983"/>{{rp|1}} Several types of fractal dimension can be measured theoretically and [[fractal analysis|empirically]] (see [[#32seg|Fig.&nbsp;2]]).<ref name="vicsek"/><ref name="medicine"/> Fractal dimensions are used to characterize a broad spectrum of objects ranging from the abstract<ref name="Falconer"/><ref name="vicsek"/> to practical phenomena, including turbulence,<ref name="Mandelbrot1983"/>{{rp|97–104}} river networks,{{rp|246–247}} urban growth,<ref>{{Cite journal | last1 = Chen | first1 = Yanguang <!-- editor is irrelevant here| editor1-last = Hernández Montoya | editor1-first = Alejandro Raúl -->| title = Modeling Fractal Structure of City-Size Distributions Using Correlation Functions | doi = 10.1371/journal.pone.0024791 | journal = PLOS ONE | volume = 6 | issue = 9 | pages = e24791 | year = 2011 | pmid = 21949753 | pmc = 3176775|arxiv = 1104.4682 |bibcode = 2011PLoSO...624791C | doi-access = free}}</ref><ref>{{cite web |url=http://library.thinkquest.org/26242/full/ap/ap.html |title=Applications |access-date=2007-10-21 |url-status=dead |archive-url=https://web.archive.org/web/20071012223212/http://library.thinkquest.org/26242/full/ap/ap.html |archive-date=2007-10-12}}</ref> human physiology,<ref name="doi10.1364/boe.1.000268">{{Cite journal | last1 = Popescu | first1 = D. P. | last2 = Flueraru | first2 = C. | last3 = Mao | first3 = Y. | last4 = Chang | first4 = S. | last5 = Sowa | first5 = M. G. | title = Signal attenuation and box-counting fractal analysis of optical coherence tomography images of arterial tissue | doi = 10.1364/boe.1.000268 | journal = Biomedical Optics Express | volume = 1 | issue = 1 | pages = 268–277 | year = 2010 | pmid = 21258464 | pmc = 3005165}}</ref><ref name="doi10.1007/s11682-008-9057-9">{{Cite journal | last1 = King | first1 = R. D. | last2 = George | first2 = A. T. | last3 = Jeon | first3 = T. | last4 = Hynan | first4 = L. S. | last5 = Youn | first5 = T. S. | last6 = Kennedy | first6 = D. N. | last7 = Dickerson | first7 = B. | author8 = the Alzheimer's Disease Neuroimaging Initiative | doi = 10.1007/s11682-008-9057-9 | title = Characterization of Atrophic Changes in the Cerebral Cortex Using Fractal Dimensional Analysis | journal = Brain Imaging and Behavior | volume = 3 | issue = 2 | pages = 154–166 | year = 2009 | pmid =  20740072| pmc =2927230 }}</ref> medicine,<ref name="medicine"/> and market trends.<ref name="time series"/> The essential idea of ''fractional'' or ''fractal'' [[Hausdorff dimension|dimensions]] has a long history in mathematics that can be traced back to the 1600s,<ref name="Mandelbrot1983"/>{{rp|19}}<ref name="classics"/> but the terms ''fractal'' and ''fractal dimension'' were coined by mathematician Benoit Mandelbrot in 1975.<ref name="Falconer">{{cite book | last = Falconer | first = Kenneth | title = Fractal Geometry | url = https://archive.org/details/fractalgeometrym00falc | url-access = limited | publisher = Wiley | year = 2003 | isbn = 978-0-470-84862-3 |page=[https://archive.org/details/fractalgeometrym00falc/page/n336 308]}}</ref><ref name="space filling">
{{cite book
| last = Sagan
| first = Hans
| title = Space-Filling Curves
| url = https://archive.org/details/spacefillingcurv00saga_539
| url-access = limited
| publisher = Springer-Verlag
| year = 1994
| isbn = 0-387-94265-3
| page=[https://archive.org/details/spacefillingcurv00saga_539/page/n170 156] }}</ref><ref name="Mandelbrot1983">
{{cite book
|author=Benoit B. Mandelbrot
|title=The fractal geometry of nature
|url=https://books.google.com/books?id=0R2LkE3N7-oC
|access-date=1 February 2012
|year=1983
|publisher=Macmillan
|isbn=978-0-7167-1186-5}}</ref><ref name="medicine">{{cite book
|editor1-last = Losa
|editor1-first= Gabriele A.
|editor2-last= Nonnenmacher
|editor2-first= Theo F.
|title=Fractals in biology and medicine
|url=https://books.google.com/books?id=t9l9GdAt95gC
|access-date=1 February 2012
|year=2005
|publisher=Springer
|isbn=978-3-7643-7172-2}}</ref><ref name="time series">
{{cite book
| last = Peters
| first = Edgar
| title = Chaos and order in the capital markets : a new view of cycles, prices, and market volatility
| publisher = Wiley
| year = 1996
| isbn = 0-471-13938-6 }}</ref><ref name="Mandelbrot quote">{{cite book
| last1 = Albers
| last2 = Alexanderson | author2-link = Gerald L. Alexanderson
| title = Mathematical people : profiles and interviews
| url = https://archive.org/details/mathematicalpeop00djal
| url-access = limited
| publisher = AK Peters
| year = 2008
| isbn = 978-1-56881-340-0
| page = [https://archive.org/details/mathematicalpeop00djal/page/n242 214]
| chapter = Benoit Mandelbrot: In his own words}}
</ref>

''Fractal dimensions'' were first applied as an index characterizing complicated geometric forms for which the details seemed more important than the gross picture.<ref name="Mandelbrot quote"/> For sets describing ordinary geometric shapes, the theoretical fractal dimension equals the set's familiar [[Euclidean geometry|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.<ref name="Mandelbrot Chaos"/>

Unlike topological dimensions, the fractal index can take non-[[integer]] values,<ref>{{Cite journal | last1 = Sharifi-Viand | first1 = A. | last2 = Mahjani | first2 = M. G. | last3 = Jafarian | first3 = M. | title = Investigation of anomalous diffusion and multifractal dimensions in polypyrrole film | doi = 10.1016/j.jelechem.2012.02.014 | journal = Journal of Electroanalytical Chemistry | volume = 671 | pages = 51–57 | year = 2012 }}</ref> indicating that a set fills its space qualitatively and quantitatively differently from how an ordinary geometrical set does.<ref name="Falconer"/><ref name="space filling"/><ref name="vicsek"/> 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.<ref name="Mandelbrot Chaos"/>{{rp|48}}<ref group=notes>See [[List of fractals by Hausdorff dimension]] for a graphic representation of different fractal dimensions.</ref> This general relationship can be seen in the two images of [[fractal curves]] in [[#32seg|Fig.&nbsp;2]] and [[#kline|Fig.&nbsp;3]]{{snd}} the 32-segment contour in Fig.&nbsp;2, convoluted and space-filling, has a fractal dimension of 1.67, compared to the perceptibly less complex Koch curve in Fig.&nbsp;3, which has a fractal dimension of approximately 1.2619.

{{anchor|kline}}
[[Image:blueklineani2.gif|right|thumb|alt=a Koch curve animation|Figure 3. The [[Koch curve]] is a classic [[iteration|iterated]] fractal curve. It is made by starting from a line segment, and then iteratively scaling each segment by 1/3 into 4 new pieces laid end to end with 2 middle pieces leaning toward each other along an equilateral triangle, so that the whole new segment spans the distance between the endpoints of the original segment. The animation only shows a few iterations, but the theoretical curve is scaled in this way infinitely.]]

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.<ref name="empirical fractal dimension"/> Instead, a fractal dimension measures complexity, a concept related to certain key features of fractals: [[self-similarity]] and [[Fractal#characteristics|detail or irregularity]].<ref group="notes">See {{slink|Fractal#Characteristics}}.</ref> 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.<ref name="Mandelbrot1983"/> The ''self-similarity'' lies in the infinite scaling, and the ''detail'' in the defining elements of each set. The [[arc length|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.<ref name="von Koch paper">Helge von Koch, "On a continuous curve without tangents constructible from elementary geometry" In {{harvnb|Edgar|2004|pp=25–46}}.</ref> Every smaller piece is composed of an infinite number of scaled segments that look exactly like the first iteration. These are not [[rectifiable curve]]s, meaning that 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 [[List of fractals by Hausdorff dimension|fractal]] that extends the Koch curve into 3D 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.<ref name="vicsek"/><ref name="time series"/> The example of the coast line of Britain, for instance, exhibits self-similarity of an approximate pattern with approximate scaling.<ref name="Mandelbrot1983"/>{{rp|26}} Overall, [[fractal]]s show several [[Fractal#characteristics|types and degrees of self-similarity]] and detail that may not be easily visualized. These include, as examples, [[Attractor|strange attractors]], for which the detail has been described as in essence, smooth portions piling up,<ref name = "Mandelbrot Chaos"/>{{rp|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.<ref name="heart">{{Cite journal | last1 = Tan | first1 = Can Ozan | last2 = Cohen | first2 = Michael A. | last3 = Eckberg | first3 = Dwain L. | last4 = Taylor | first4 = J. Andrew | title = Fractal properties of human heart period variability: Physiological and methodological implications | doi = 10.1113/jphysiol.2009.169219 | journal = The Journal of Physiology | volume = 587 | issue = 15 | pages = 3929–3941 | year = 2009 | pmid = 19528254| pmc = 2746620}}</ref> 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.<ref name="Mandelbrot1983"/>{{rp|197;262}}

== History ==
The terms ''fractal dimension'' and ''fractal'' were coined by Mandelbrot in 1975,<ref name="Mandelbrot quote"/> 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.<ref name="classics"/><ref name="Gordon"/><ref name="MacTutor"/> 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.<ref name="Mandelbrot1983" />{{rp|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 [[Helge von Koch|von Koch]],<ref name="von Koch paper"/> [[Sierpiński]], and [[Gaston Julia|Julia]]), but at the time of their formulation were often considered antithetical mathematical "monsters".<ref name="classics">
{{cite book
| editor-last = Edgar
| editor-first = Gerald
| title = Classics on Fractals
| publisher = Westview Press
| year = 2004| isbn = 978-0-8133-4153-8 }}
</ref><ref name="MacTutor">{{cite web
 |title=A History of Fractal Geometry 
 |work=MacTutor History of Mathematics 
 |author=Trochet, Holly 
 |archive-url=https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html
 |archive-date=12 March 2012
 |url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/fractals.html 
 |year=2009 
 |url-status=dead
}}
</ref> These works were accompanied by perhaps the most pivotal point in the development of the concept of a fractal dimension through the work of [[Felix Hausdorff|Hausdorff]] in the early 1900s who defined a "fractional" [[Hausdorff dimension|dimension]] that has come to be named after him and is frequently invoked in defining modern [[fractals]].<ref name="coastline"/><ref name="Mandelbrot1983"/>{{rp|44}}<ref name="Mandelbrot Chaos">
{{cite book
| last = Mandelbrot
| first = Benoit
| title = Fractals and Chaos
| publisher = Springer
| year = 2004
| isbn = 978-0-387-20158-0
| quote = A fractal set is one for which the fractal (Hausdorff-Besicovitch) dimension strictly exceeds the topological dimension
| page= 38}}</ref><ref name = "Gordon">{{cite book
| last = Gordon
| first = Nigel
| title = Introducing fractal geometry
| publisher = Icon
| location = Duxford
| year = 2000
| isbn = 978-1-84046-123-7
| page = [https://archive.org/details/introducingfract0000lesm/page/71 71]
| url = https://archive.org/details/introducingfract0000lesm/page/71
}}</ref>

''See [[fractal#history|Fractal history]] for more information''

{{anchor|calculations}}

== Mathematical definition ==
{{anchor|unity}}
[[Image:Fractaldimensionexample-2.png|right|thumb|alt=Lines, squares, and cubes.|Figure 4. Traditional notions of geometry for defining scaling and dimension.<br/>
<math>1</math>, <math>1^2 = 1</math>, <math>1^3 = 1;</math><br/>
<math>2</math>, <math>2^2 = 4</math>, <math>2^3 = 8;</math><br/>
<math>3</math>, <math>3^2 = 9</math>, <math>3^3 = 27.</math><ref>Appignanesi, Richard; ed. (2006). ''Introducing Fractal Geometry'', p.&nbsp;28<!--shows division by 2 and 3-->. Icon. {{ISBN|978-1840467-13-0}}.</ref>]]The mathematical definition of fractal dimension can be derived by observing and then generalizing the effect of traditional dimension on measurement-changes under scaling.<ref name = fil>{{cite book |author=Iannaccone, Khokha |year=1996 |title=Fractal Geometry in Biological Systems |publisher=CRC Press |isbn=978-0-8493-7636-8}}</ref> For example, say you have a line and a measuring-stick of equal length. Now shrink the stick to 1/3 its size; you can now fit 3&nbsp;sticks into the line. Similarly, in two dimensions, say you have a square and an identical "measuring-square". Now shrink the measuring-square's side to 1/3 its length; you can now fit 3^2 = 9&nbsp;measuring-squares into the square. Such familiar scaling relationships obey equation&nbsp;{{EqNote|1}}, where  <math>\varepsilon</math> is the scaling factor, <math>D</math> the dimension, and <math>N</math> the resulting number of units (sticks, squares, etc.) in the measured object:{{NumBlk|:|<math>N = \varepsilon^{-D}.</math>|{{EquationRef|1}}}}

In the line example, the dimension <math>D = 1</math> because there are <math>N = 3</math> units when the scaling factor <math>\varepsilon = 1/3</math>. In the square example, <math>D = 2</math> because <math>N = 9</math> when <math>\varepsilon = 1/3</math>.

{{anchor|koch}}
[[Image:KochFlake.svg|right|thumb|alt=A fractal contour of a koch snowflake|Figure 5. The first four [[iteration]]s of the [[Koch snowflake]], which has a [[Hausdorff dimension]] of approximately 1.2619.]]

Fractal dimension generalizes traditional dimension in that it can be fractional, but it has exactly the same relationship with scaling that traditional dimension does; in fact, it is derived by simply rearranging equation {{EqNote|1}}:

{{NumBlk|:|<math>D = -\log_\varepsilon N = -\frac{\log N}{\log \varepsilon}.</math>|{{EquationRef|2}}}}

<math>D</math> can be thought of as the power of the scaling factor of an object's measure given some scaling of its "radius". 

For example, the [[Koch snowflake]] has <math>D = 1.26185\ldots</math>, indicating that lengthening its radius grows its measure faster than if it were a one-dimensional shape (such as a polygon), but slower than if it were a two-dimensional shape (such as a filled polygon).<ref name="vicsek" />

Of note, images shown in this page are not true fractals because the scaling described by <math>D</math> 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 [[Infinity|infinite]] number of infinitely scaled segments and do indeed have the claimed fractal dimensions.<ref name="Mandelbrot1983" /><ref name="fil" />

== ''D'' is not a unique descriptor ==
{{anchor|statistical koch like scaling image}}
[[File:onetwosix.png|thumb|upright=1.5|right|Figure 6. Two [[L-systems]] branching fractals that are made by producing 4 new parts for every 1/3 [[self similarity|scaling]], thus having the same theoretical <math>D</math> as the Koch curve, and for which the empirical [[box counting]] <math>D</math> has been demonstrated with 2% accuracy.<ref name="empirical fractal dimension"/>]]

As is the case with dimensions determined for lines, squares, and cubes, fractal dimensions are general descriptors that do not uniquely define patterns.<ref name="fil"/><ref name=Vicsek>{{cite book |author=Vicsek, Tamás |title=Fluctuations and scaling in biology |publisher=Oxford University Press |year=2001 |isbn=0-19-850790-9 }}</ref> 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 [[#statistical koch like scaling image|Fig.&nbsp;6]].

{{anchor|specific definitions}}
For examples of how fractal patterns can be constructed, see [[Fractal]], [[Sierpinski triangle]], [[Mandelbrot set]], [[Diffusion-limited aggregation]], [[L-system]].

== Fractal surface structures ==
[[File:Wiki df figure.png|thumb|upright=1.5|Figure 7: Illustration of increasing surface fractality. Self-affine surfaces (left) and corresponding surface profiles (right) showing increasing fractal dimension ''D<sub>f</sub>''.]]
The concept of fractality is applied increasingly in the field of [[Surface Science|surface science]], providing a bridge between surface characteristics and functional properties.<ref>{{Citation |last=Pfeifer |first=Peter |chapter=Fractals in Surface Science: Scattering and Thermodynamics of Adsorbed Films |date=1988 |volume=10 |pages=283–305 |editor-last=Vanselow |editor-first=Ralf |publisher=Springer Berlin Heidelberg |doi=10.1007/978-3-642-73902-6_10 |isbn=9783642739040 |editor2-last=Howe |editor2-first=Russell |title=Chemistry and Physics of Solid Surfaces VII |series=Springer Series in Surface Sciences}}.</ref> 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<sub>A</sub>, is the most commonly applied surface descriptor, however, numerous other descriptors including mean slope, [[root-mean-square]] roughness (R<sub>RMS</sub>) 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.<ref>{{Cite journal |last1=Milanese |first1=Enrico |last2=Brink |first2=Tobias |last3=Aghababaei |first3=Ramin |last4=Molinari |first4=Jean-François |date=December 2019 |title=Emergence of self-affine surfaces during adhesive wear |journal=Nature Communications |volume=10 |issue=1 |pages=1116 |doi=10.1038/s41467-019-09127-8 |issn=2041-1723 |pmc=6408517 |pmid=30850605 |bibcode=2019NatCo..10.1116M}}</ref> The fractal dimensions of surfaces have been employed to explain and better understand phenomena in areas of [[contact mechanics]],<ref>[https://www.researchgate.net/publication/318345969_Contact_stiffness_of_multiscale_surfaces_by_truncation_analysis Contact stiffness of multiscale surfaces], In the International Journal of Mechanical Sciences (2017), 131.</ref> [[friction | frictional behavior]],<ref>[https://www.researchgate.net/publication/283675011_Static_friction_at_fractal_interfaces Static Friction at Fractal Interfaces], Tribology International (2016), vol.&nbsp;93.</ref> [[electrical contact resistance]]<ref>{{cite journal |first1=Zhai |last1=Chongpu |first2=Hanaor |last2=Dorian |first3=Proust |last3=Gwénaëlle |first4=Gan |last4=Yixiang |title=Stress-Dependent Electrical Contact Resistance at Fractal Rough Surfaces |journal=Journal of Engineering Mechanics |volume=143 |issue=3 |pages=B4015001 |year=2017 |doi=10.1061/(ASCE)EM.1943-7889.0000967 }}</ref> and [[transparent conducting oxide]]s.<ref>{{Cite journal |last1=Kalvani |first1=Payam Rajabi |last2=Jahangiri |first2=Ali Reza |last3=Shapouri |first3=Samaneh |last4=Sari |first4=Amirhossein |last5=Jalili |first5=Yousef Seyed |date=August 2019 |title=Multimode AFM analysis of aluminum-doped zinc oxide thin films sputtered under various substrate temperatures for optoelectronic applications |journal=Superlattices and Microstructures |volume=132 |pages=106173 |doi=10.1016/j.spmi.2019.106173 |s2cid=198468676 }}</ref>

== 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 [[Limit (mathematics)|limits]] estimated from [[regression line]]s over [[log–log plot]]s 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]] is [[box counting|estimated]] as the exponent of a [[power law#Estimating the exponent from empirical data|power law]]:
*: <math>D_0 = \lim_{\varepsilon \to 0} \frac{\log N(\varepsilon)}{\log\frac{1}{\varepsilon}}.</math>
* [[Information dimension]] considers how the average [[information entropy|information]] needed to identify an occupied box scales with box size (<math>p</math> is a probability):
*: <math>D_1 = \lim_{\varepsilon \to 0} \frac{-\langle \log p_\varepsilon \rangle}{\log\frac{1}{\varepsilon}}.</math>
* [[Correlation dimension]] is based on <math>M</math> as the number of points used to generate a representation of a fractal and ''g''<sub>ε</sub>, the number of pairs of points closer than ε to each other:{{Citation needed|reason=Limit was incorrect, not sure if correction is|date=June 2017}}
*: <math>D_2 = \lim_{M \to \infty} \lim_{\varepsilon \to 0} \frac{\log (g_\varepsilon / M^2)}{\log \varepsilon}.</math> 
* Generalized, or Rényi dimensions: the box-counting, information, and correlation dimensions can be seen as special cases of a continuous spectrum of [[Rényi entropy|generalized dimensions]] of order α, defined by
*: <math>D_\alpha = \lim_{\varepsilon \to 0} \frac{\frac{1}{\alpha - 1} \log(\sum_i p_i^\alpha)}{\log\varepsilon}.</math>
* [[Higuchi dimension]]<ref>{{cite journal |first=T. |last=Higuchi |title=Approach to an irregular time-series on the basis of the fractal theory |journal=Physica D |volume=31 |issue=2 |pages=277–283 |year=1988 |doi=10.1016/0167-2789(88)90081-4 |bibcode=1988PhyD...31..277H }}</ref>
*: <math>D = \frac{d\log L(k)}{d \log k}.</math>
* [[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 <math>S</math> of a metric space <math>X</math> and <math>d \geq 0</math>, the ''d''-dimensional ''Hausdorff content'' of ''S'' is defined by <math display="block">
 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\}.
</math> The Hausdorff dimension of ''S'' is defined by
*: <math>\dim_{\operatorname{H}}(X) := \inf\{d \ge 0: C_H^d(X) = 0\}.</math>
* [[Packing dimension]]
* [[Assouad dimension]]
* [[Local connected dimension]]<ref>{{Cite journal | last1 = Jelinek | first1 = A. | doi = 10.2147/OPTH.S1579 | last2 = Jelinek | first2 = H. F. | last3 = Leandro | first3 = J. J. | last4 = Soares | first4 = J. V. | last5 = Cesar Jr | first5 = R. M. | last6 = Luckie | first6 = A. | title = Automated detection of proliferative retinopathy in clinical practice | journal = Clinical Ophthalmology | pages = 109–122 | year = 2008 | pmid =  19668394 | pmc = 2698675 | volume=2 | issue=1 | doi-access = free }}</ref>
* Degree dimension describes the fractal nature of the degree distribution of graphs.<ref>{{Cite journal | last1 = Li | first1 = N. Z. | last2 = Britz | first2 = T. | title = On the scale-freeness of random colored substitution networks | journal = Proceedings of the American Mathematical Society | pages = 1377–1389 | year = 2024 | volume=152 | number=4 | doi = 10.1090/proc/16604 | arxiv = 2109.14463 }}</ref>
* [[Parabolic Hausdorff dimension]]

== Estimating from real-world data ==
Many real-world phenomena exhibit limited or statistical fractal properties and fractal dimensions that have been estimated from [[Sample (statistics)|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,<ref>{{Cite journal |last1=Caicedo-Ortiz |first1=H. E. |last2=Santiago-Cortes |first2=E. |last3=López-Bonilla |first3=J. |last4=Castañeda |first4=H. O. |year=2015 |title=Fractal dimension and turbulence in Giant HII Regions |journal=Journal of Physics: Conference Series |volume=582 |issue=1 |pages=1–5 |doi=10.1088/1742-6596/582/1/012049 |arxiv=1501.04911 |bibcode=2015JPhCS.582a2049C |doi-access=free}}</ref> acoustics,<ref name="pubmed.ncbi.nlm.nih.gov">{{cite journal |title=A Mathematical Approach to Correlating Objective Spectro-Temporal Features of Non-linguistic Sounds With Their Subjective Perceptions in Humans |year=2019 |pmid=31417350 |last1=Burns |first1=T. |last2=Rajan |first2=R. |journal=Frontiers in Neuroscience |volume=13 |page=794 |doi=10.3389/fnins.2019.00794 |pmc=6685481 |doi-access=free}}</ref><ref>{{Cite journal |last1=Maragos |first1=P. |last2=Potamianos |first2=A. |year=1999 |title=Fractal dimensions of speech sounds: Computation and application to automatic speech recognition |journal=The Journal of the Acoustical Society of America |volume=105 |issue=3 |pages=1925–1932 |bibcode=1999ASAJ..105.1925M |doi=10.1121/1.426738 |pmid=10089613}}</ref> architecture,<ref>{{cite book |last1=Ostwald |first1=Michael J. |last2=Vaughan |first2=Josephine |last3=Tucker |first3=Chris |title=Characteristic visual complexity: Fractal dimensions in the architecture of frank lloyd wright and le corbusier. In: Architecture and Mathematics from Antiquity to the Future: Volume II: The 1500s to the Future |date=2015 |publisher=Springer International Publishing |isbn=978-331900143-2 |pages=339–354 |url=https://www.researchgate.net/publication/271197753}}</ref><ref>{{Cite book |last=Bovill |first=Carl |title=Fractal geometry in architecture and design |date=1996 |publisher=Birkhäuser |isbn=978-0-8176-3795-8 |series=Design science collection |location=Boston}}</ref><ref>{{cite journal |last1=Vaughan |first1=Josephine |last2=Ostwald |first2=Michael j. |date=2014 |title=Measuring the significance of façade transparency in Australian regionalist architecture: A computational analysis of 10 designs by Glenn Murcutt |journal=Architectural Science Review |volume=57 |issue=4 |pages=249–259 |doi=10.1080/00038628.2014.940273 |hdl=1959.13/1293729|hdl-access=free }}</ref>   geology and earth sciences,<ref>{{Cite journal |last=Avşar |first=Elif |date=2020-09-01 |title=Contribution of fractal dimension theory into the uniaxial compressive strength prediction of a volcanic welded bimrock |url=https://doi.org/10.1007/s10064-020-01778-y |journal=Bulletin of Engineering Geology and the Environment |language=en |volume=79 |issue=7 |pages=3605–3619 |doi=10.1007/s10064-020-01778-y |bibcode=2020BuEGE..79.3605A |s2cid=214648440 |issn=1435-9537}}</ref> diagnostic imaging,<ref>{{Cite journal
| last1 = Landini | first1 = G.
| last2 = Murray | first2 = P. I.
| last3 = Misson | first3 = G. P.
| title = Local connected fractal dimensions and lacunarity analyses of 60 degrees fluorescein angiograms
| journal = Investigative Ophthalmology & Visual Science
| volume = 36
| issue = 13
| pages = 2749–2755
| year = 1995
| pmid = 749909
}}</ref><ref>{{Cite journal | last1 = Cheng | first1 = Qiuming | author-link = Qiuming Cheng| title = Multifractal Modeling and Lacunarity Analysis | journal = Mathematical Geology | volume = 29 | issue = 7 | pages = 919–932 | doi = 10.1023/A:1022355723781 | year = 1997 | bibcode = 1997MatG...29..919C | s2cid = 118918429 }}</ref><ref>{{Cite journal |last1=Santiago-Cortés |first1=E.| last2=Martínez Ledezma |first2=J. L. |year=2016 |title=Fractal dimension in human retinas |url=https://jci.uniautonoma.edu.co/2016/2016-8.pdf |journal=Journal de Ciencia e Ingeniería |volume=8 |pages=59–65 |issn=2145-2628 |eissn=2539-066X}}</ref>
ecology,<ref>{{Cite journal |last1=Wildhaber |first1=Mark L. |last2=Lamberson |first2=Peter J. |last3=Galat |first3=David L. |date=2003-05-01 |title=A Comparison of Measures of Riverbed Form for Evaluating Distributions of Benthic Fishes |journal=North American Journal of Fisheries Management |volume=23 |issue=2 |pages=543–557 |doi=10.1577/1548-8675(2003)023<0543:acomor>2.0.co;2 |bibcode=2003NAJFM..23..543W |issn=1548-8675}}</ref> electrochemical processes,<ref>{{Cite journal |last1=Eftekhari |first1=A. |year=2004 |title=Fractal Dimension of Electrochemical Reactions |journal=Journal of the Electrochemical Society |volume=151 |issue=9 |pages=E291–E296 |doi=10.1149/1.1773583 |bibcode=2004JElS..151E.291E}}</ref>
image analysis,<ref>{{cite journal |author=Al-Kadi O. S., Watson D. |year=2008 |title=Texture Analysis of Aggressive and non-Aggressive Lung Tumor CE CT Images |url=http://sro.sussex.ac.uk/1919/1/tbme.pdf |url-status=dead |journal=IEEE Transactions on Biomedical Engineering |volume=55 |issue=7 |pages=1822–1830 |doi=10.1109/tbme.2008.919735 |pmid=18595800 |s2cid=14784161 |archive-url=https://web.archive.org/web/20140413124458/http://sro.sussex.ac.uk/1919/1/tbme.pdf |archive-date=2014-04-13 |access-date=2014-04-10}}</ref><ref>{{cite journal |author=Pierre Soille and Jean-F. Rivest |year=1996 |title=On the Validity of Fractal Dimension Measurements in Image Analysis |url=http://mdigest.jrc.ec.europa.eu/soille/soille-rivest96.pdf |url-status=dead |journal=Journal of Visual Communication and Image Representation |volume=7 |issue=3 |pages=217–229 |doi=10.1006/jvci.1996.0020 |issn=1047-3203 |archive-url=https://web.archive.org/web/20110720161245/http://mdigest.jrc.ec.europa.eu/soille/soille-rivest96.pdf |archive-date=2011-07-20}}</ref><ref>{{Cite journal |last1=Tolle |first1=C. R. |last2=McJunkin |first2=T. R. |last3=Gorsich |first3=D. J. |year=2003 |title=Suboptimal minimum cluster volume cover-based method for measuring fractal dimension |url=https://zenodo.org/record/1282294 |journal=IEEE Transactions on Pattern Analysis and Machine Intelligence |volume=25 |pages=32–41 |citeseerx=10.1.1.79.6978 |doi=10.1109/TPAMI.2003.1159944}}</ref><ref>{{Cite book |last1=Gorsich |first1=D. J. |last2=Tolle |first2=C. R. |last3=Karlsen |first3=R. E. |last4=Gerhart |first4=G. R. |editor-first1=Michael A. |editor-first2=Akram |editor-first3=Andrew F. |editor-last1=Unser |editor-last2=Aldroubi |editor-last3=Laine |year=1996|contribution=Wavelet and fractal analysis of ground-vehicle images |url=https://zenodo.org/record/1235586 |title=Wavelet Applications in Signal and Image Processing IV |series=Proceedings of the SPIE |volume=2825 |pages=109–119 |bibcode=1996SPIE.2825..109G |doi=10.1117/12.255224 |s2cid=121560110}}</ref> biology and medicine,<ref>{{Cite journal | last1 = Liu | first1 = Jing Z. | last2 = Zhang | first2 = Lu D. | last3 = Yue | first3 = Guang H. | doi = 10.1016/S0006-3495(03)74817-6 | title = Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging | journal = Biophysical Journal | volume = 85 | issue = 6 | pages = 4041–4046 | year = 2003 | pmid = 14645092 | pmc = 1303704 |bibcode = 2003BpJ....85.4041L }}</ref><ref>{{Cite journal | last1 = Smith | first1 = T. G. | last2 = Lange | first2 = G. D. | last3 = Marks | first3 = W. B. | doi = 10.1016/S0165-0270(96)00080-5 | title = Fractal methods and results in cellular morphology — dimensions, lacunarity and multifractals | journal = Journal of Neuroscience Methods | volume = 69 | issue = 2 | pages = 123–136 | year = 1996 | pmid = 8946315 | s2cid = 20175299 | url = https://zenodo.org/record/1259855 }}</ref><ref>{{Cite journal | last1 = Li | first1 = J. | last2 = Du | first2 = Q. | last3 = Sun | first3 = C. | doi = 10.1016/j.patcog.2009.03.001 | title = An improved box-counting method for image fractal dimension estimation | journal = Pattern Recognition | volume = 42 | issue = 11 | pages = 2460–9 | year = 2009 | bibcode = 2009PatRe..42.2460L }}</ref> neuroscience,<ref name="10.12688/f1000research.6590.1">{{cite journal |title=Combining complexity measures of EEG data: multiplying measures reveal previously hidden information |year=2015 |pmc=4648221 |last1=Burns |first1=T. |last2=Rajan |first2=R. |journal=F1000Research |volume=4 |page=137 |doi=10.12688/f1000research.6590.1 |pmid=26594331 |doi-access=free }}</ref><ref name="doi10.1007/s11682-008-9057-9" /> [[Fractal dimension on networks|network analysis]], physiology,<ref name="doi10.1364/boe.1.000268"/> physics,<ref>{{Cite journal | last1 = Dubuc | first1 = B. | last2 = Quiniou | first2 = J. | last3 = Roques-Carmes | first3 = C. | last4 = Tricot | first4 = C. | last5 = Zucker | first5 = S. | title = Evaluating the fractal dimension of profiles | doi = 10.1103/PhysRevA.39.1500 | journal = Physical Review A | volume = 39 | issue = 3 | pages = 1500–1512 | year = 1989 | pmid = 9901387 |bibcode = 1989PhRvA..39.1500D }}</ref><ref>{{Cite journal | last1 = Roberts | first1 = A. | last2 = Cronin | first2 = A. | doi = 10.1016/S0378-4371(96)00165-3 | title = Unbiased estimation of multi-fractal dimensions of finite data sets | journal = Physica A: Statistical Mechanics and Its Applications | volume = 233 | issue = 3–4 | pages = 867–878 | year = 1996 |bibcode = 1996PhyA..233..867R | arxiv = chao-dyn/9601019 | s2cid = 14388392 }}</ref> and [[Riemann hypothesis|Riemann zeta zeros]].<ref name="Shanker">{{Cite journal | last1 = Shanker | first1 = O. | title = Random matrices, generalized zeta functions and self-similarity of zero distributions | doi = 10.1088/0305-4470/39/45/008 | journal = Journal of Physics A: Mathematical and General | volume = 39 | issue = 45 | pages = 13983–13997 | year = 2006 |bibcode = 2006JPhA...3913983S }}</ref> Fractal dimension estimates have also been shown to correlate with [[Lempel–Ziv complexity]] in real-world data sets from psychoacoustics and neuroscience.<ref name="10.12688/f1000research.6590.1"/><ref name="pubmed.ncbi.nlm.nih.gov"/>

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 [[colloid]]al physics, systems composed of particles with various fractal dimensions arise. To describe these systems, it is convenient  to speak about a [[Probability distribution|distribution]] of fractal dimensions and, eventually, a time evolution of the latter: a process that is driven by a complex interplay between [[Particle aggregation|aggregation]] and [[Coalescence (chemistry)|coalescence]].<ref>{{Cite journal | doi = 10.1002/mats.201300140 | title = Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems | journal = Macromolecular Theory and Simulations | volume = 23 | issue = 3 | pages = 170–181 | year = 2014 | last1 = Kryven | first1 = I. | last2 = Lazzari | first2 = S. | last3 = Storti | first3 = G. | url = http://dare.uva.nl/personal/pure/en/publications/population-balance-modeling-of-aggregation-and-coalescence-in-colloidal-systems(05340e78-b40a-4bd0-ac6a-7f044fff1617).html}}</ref>

== See also ==
* {{annotated link|List of fractals by Hausdorff dimension}}
* {{annotated link|Lacunarity}}
* {{annotated link|Fractal derivative}}

== Notes ==
{{reflist|group="notes"}}

==References==
{{reflist}}

==Further reading==
*{{cite book |first1=Benoit B. |last1=Mandelbrot |author-link=Benoit Mandelbrot |first2=Richard L. |last2=Hudson |title=The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward |url=https://books.google.com/books?id=zg91TAIs6bgC |date=2010 |publisher=Profile Books |isbn=978-1-84765-155-6}}

==External links==
{{commons category}}
* [http://www.trusoft-international.com TruSoft's Benoit], fractal analysis software product calculates fractal dimensions and hurst exponents.
* [http://www.stevec.org/fracdim/ A Java Applet to Compute Fractal Dimensions]
* [http://rsb.info.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm Introduction to Fractal Analysis]
* {{cite web|last=Bowley|first=Roger|title=Fractal Dimension|url=http://www.sixtysymbols.com/videos/fractal.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|year=2009}}
*"[https://www.youtube.com/watch?v=gB9n2gHsHN4 Fractals are typically not self-similar"]. ''3Blue1Brown''.
{{Fractals}}
{{Dimension topics}}

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