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{{anchor|coastline}}
{{multiple image
   | width     = 80
   | footer    = ''Figure 1.'' As the length of the measuring stick is scaled smaller and smaller, the total length of the coastline measured increases.
   | align     = right
   | image1    = britain-fractal-coastline-200km.png
   | alt1      = Coastline of Britain measured using a 200 km scale
   | caption1  = <small>11.5 x 200 = 2300 km</small>
   | image2    = britain-fractal-coastline-100km.png
   | alt2      = Coastline of Britain measured using a 100 km scale
   | caption2  = <small>28 x 100 = 2800 km</small>
   | image3    = britain-fractal-coastline-50km.png
   | alt3      = Coastline of Britain measured using a 50 km scale
   | caption3  = <small>70 x 50 = 3500 km</small>
}}
In [[mathematics]], more specifically in [[fractal geometry]], a '''fractal dimension''' is a ratio providing a statistical index of [[complexity]] comparing how detail in a [[pattern]] (strictly speaking, a [[fractal]] pattern) changes with the [[Scaling (geometry)|scale]] at which it is measured. It has also been characterized as a measure of the [[Space-filling curve|space-filling]] capacity of a pattern that tells how a fractal scales differently from the [[space]] it is embedded in; a fractal dimension does not have to be an integer.<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 essential 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 = 636–8 | year = 1967 | pmid =  17837158 |bibcode = 1967Sci...156..636M }}</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 ([[#coastline|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.<ref name="Mandelbrot1983"/> There are several formal [[#specific definitions|mathematical definitions]] of fractal dimension that build on this basic concept of change in detail with change in scale.

Ultimately, the term ''fractal dimension'' became the phrase with which Mandelbrot himself became most comfortable with respect to encapsulating the meaning of the word ''fractal'', a term he created.  After several iterations over years, Mandelbrot settled on this use of the language:  "...to use ''fractal'' without a pedantic definition, to use ''fractal dimension'' as a generic term applicable to ''all'' the variants."<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 a [[rectifiable curve]]: the [[arc length|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 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 | publisher = Chapman & Hall | year = 2001 | isbn = 978-1-58488-154-4 |pages=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 about 1.262.

== Introduction ==
{{Anchor|32seg}}[[Image:32 segment fractal.jpg|thumb|right|''Figure 2.'' A 32-segment [[fractal#iterated|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%<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 
| accessdate = 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]] ([[#32seg|see Fig. 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 }}</ref><ref>{{cite web|url=http://library.thinkquest.org/26242/full/ap/ap.html|title=Applications|accessdate=2007-10-21|url-status=dead|archiveurl=https://web.archive.org/web/20071012223212/http://library.thinkquest.org/26242/full/ap/ap.html|archivedate=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 | publisher = Wiley | year = 2003 | isbn = 978-0-470-84862-3|page=308}}</ref><ref name="space filling">
{{cite book
| last = Sagan
| first = Hans
| title = Space-Filling Curves
| publisher = Springer-Verlag
| year = 1994
| isbn = 0-387-94265-3
| page=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
|accessdate=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
|accessdate=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
| publisher = AK Peters
| year = 2008
| isbn = 978-1-56881-340-0
| page = 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|a graphic representation of different fractal dimensions]]</ref> This general relationship can be seen in the two images of fractal curves in [[#32seg|Fig.2]] and [[#kline|Fig. 3]] – the 32-segment contour in Fig. 2, convoluted and space filling, has a fractal dimension of 1.67, compared to the perceptibly less complex Koch curve in Fig. 3, which has a fractal dimension of 1.26.

{{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 a theoretical construct that is made by iteratively scaling a starting segment. As shown, each new segment is scaled by 1/3 into 4 new pieces laid end to end with 2 middle pieces leaning toward each other between the other two pieces, so that if they were a triangle its base would be the length of the middle piece, so that the whole new segment fits across the traditionally measured length between the endpoints of the previous segment. Whereas the animation only shows a few iterations, the theoretical curve is scaled in this way infinitely. Beyond about 6 iterations on an image this small, the detail is lost.]]

The relationship of an increasing fractal dimension with space-filling might be taken to mean fractal dimensions measure density, but that is not so; the two are not strictly correlated.<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">[[Fractal#characteristics|See 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 or [[slope]], as with ordinary lines. But we cannot do either of these things, because fractal curves have complexity in the form of self-similarity and detail that ordinary lines lack.<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 [[Arc length#Curves with infinite length|undefined]] because the curves are theoretical constructs that never stop repeating themselves.<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 they cannot be measured by being broken down into many segments approximating their respective lengths. They cannot be characterized by finding their lengths or slopes. However, their fractal dimensions can be determined, which shows that both fill space more than ordinary lines but less than surfaces, and allows them to be compared in this regard.

The two fractal curves described above show a type of self-similarity that is exact with a repeating unit of detail that is readily visualized. This sort of structure can be extended to other spaces (e.g., a [[List of fractals by Hausdorff dimension|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.<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 = 3929–41 | 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 undifferentiable, infinitely self-similar functions, which are important in the mathematical definition of fractals, around the time that [[calculus]] was discovered in the mid-1600s.<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 | ref=harv
| 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 
 |archiveurl=https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html
 |archivedate=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=71}}</ref>

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

{{anchor|calculations}}

== Role of scaling ==
{{anchor|unity}}
[[Image:Fractaldimensionexample.PNG|right|thumb|width=300px|alt=Lines, squares, and cubes.|''Figure 4.'' Traditional notions of geometry for defining scaling and dimension.]]

The concept of a fractal dimension rests in unconventional views of scaling and dimension.<ref name = fil>{{cite book|author=Iannaccone, Khokha|year=1996|title=Fractal Geometry in Biological Systems|isbn=978-0-8493-7636-8}}</ref> As [[#unity|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 <math>N</math> stands for the number of sticks, <math>\varepsilon</math> for the scaling factor, and <math>D</math> for the fractal dimension:

{{NumBlk|:|<math>{{N \propto \varepsilon^{-D}}},</math>|{{EquationRef|1}}}}

where <math>\propto</math> denotes [[Proportionality (mathematics)|proportionality]]. This scaling rule typifies conventional rules about geometry and dimension – for lines, it quantifies that, because <math>N=3</math> when <math>\varepsilon =\tfrac{1}{3}</math> as in the example above, <math>D=1,</math> and for squares, because <math>N=9</math> when <math>\varepsilon= \tfrac{1}{3}, D=2.</math>

{{anchor|koch}}
[[Image:KochFlake.svg|width=300|right|thumb|alt=A fractal contour of a koch snowflake|''Figure 5.'' The first four [[iteration]]s of the [[Koch snowflake]], which has an approximate [[Hausdorff dimension]] of 1.2619.]]

The same rule applies to fractal geometry but less intuitively. To elaborate, a fractal line measured at first to be one length, when remeasured using a new stick scaled by 1/3 of the old may not be the expected 3 but instead 4 times as many scaled sticks long. In this case, <math>N=4</math> when <math>\varepsilon = \tfrac{1}{3},</math> and the value of <math>D</math> can be found by rearranging Equation 1:

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

That is, for a fractal described by <math>N=4</math> when <math>\varepsilon =\tfrac{1}{3}, D=1.2619,</math> a non-integer dimension that suggests the fractal has a dimension not equal to the space it resides in.<ref name="vicsek" /> The scaling used in this example is the same scaling of the [[#kline|Koch curve]] and [[#koch|snowflake]]. Of note, the images shown are not true fractals because the scaling described by the value of <math>D</math> cannot continue infinitely for the simple reason that the images only exist to the point of their smallest component, a pixel. The theoretical pattern that the digital images represent, however, has no discrete pixel-like pieces, but rather is composed of an [[Infinity|infinite]] number of infinitely scaled segments joined at different angles and does indeed have a fractal dimension of 1.2619.<ref name="Mandelbrot1983"/><ref name=fil/>

== ''D'' is not a unique descriptor ==
{{anchor|statistical koch like scaling image}}[[File:onetwosix.png|300|thumb|right|''Figure 6''. Two [[L-systems]] branching fractals that are made by producing 4 new parts for every 1/3 [[self similarity|scaling]] so have 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|Figure 6]].

''For examples of how fractal patterns can be constructed, see [[Fractal]], [[Sierpinski triangle]], [[Mandelbrot set]], [[Diffusion limited aggregation]], [[L-System]].''

{{anchor|specific definitions}}

== Fractal surface structures ==
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|last=Milanese|first=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}}</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 93</ref> [[electrical contact resistance]]<ref>{{cite journal |first=Zhai |last=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 |url=}}</ref> and [[transparent conducting oxide]]s.<ref>{{Cite journal|last=Kalvani|first=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}}</ref>

[[File:Wiki df figure.png|thumb|''Figure 7:'' Illustration of increasing surface fractality. Self-affine surfaces (left) and corresponding surface profiles (right) showing increasing fractal dimension ''D<sub>f</sub>'']]

== 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)|limit]]s estimated from [[regression line]]s over [[logarithm|log vs log]] plots of size vs scale. Several formal mathematical definitions of different types of fractal dimension are listed below. Although for some classic fractals all these dimensions coincide, in general they are not equivalent:

* [[box-counting dimension|Box counting dimension]]: ''D'' 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]]: ''D'' 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]]: ''D'' 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.
::<math>D_2 = \lim_{M \to \infty} \lim_{\varepsilon \to 0} \frac{\log (g_\varepsilon / M^2)}{\log \varepsilon}</math> {{Citation needed|reason=Limit was incorrect, not sure if correction is|date=June 2017}}

* 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 |url=}}</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>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}}</ref>

== 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
diagnostic imaging,<ref>{{Cite journal 
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| 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 = 7499097
}}</ref><ref>{{Cite journal | last1 = Cheng | first1 = Qiuming | authorlink = 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 }}</ref>
physiology,<ref name="doi10.1364/boe.1.000268"/>
neuroscience,<ref name="doi10.1007/s11682-008-9057-9"/>
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 = 4041–6 | 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 | pmc = | 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 }}</ref>
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 = 1500–12 | 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 }}</ref>
image analysis,<ref>{{cite journal|author=Al-Kadi O.S, Watson D.|title=Texture Analysis of Aggressive and non-Aggressive Lung Tumor CE CT Images|url=http://sro.sussex.ac.uk/1919/1/tbme.pdf| journal=IEEE Transactions on Biomedical Engineering|volume=55 |issue=7|pages=1822–30 |year=2008|doi=10.1109/tbme.2008.919735|pmid=18595800}}</ref><ref>{{cite journal
 |title       = On the Validity of Fractal Dimension Measurements in Image Analysis
 |journal     = Journal of Visual Communication and Image Representation
 |volume      = 7
 |pages       = 217–229
 |year        = 1996
 |issn        = 1047-3203
 |doi         = 10.1006/jvci.1996.0020
 |url         = http://mdigest.jrc.ec.europa.eu/soille/soille-rivest96.pdf
 |author      = Pierre Soille and Jean-F. Rivest
 |issue       = 3
 |url-status     = dead
 |archiveurl  = https://web.archive.org/web/20110720161245/http://mdigest.jrc.ec.europa.eu/soille/soille-rivest96.pdf
 |archivedate = 2011-07-20
}}</ref><ref>{{Cite journal | last1 = Tolle | first1 = C. R. | last2 = McJunkin | first2 = T. R. | last3 = Gorsich | first3 = D. J. | title = Suboptimal minimum cluster volume cover-based method for measuring fractal dimension | doi = 10.1109/TPAMI.2003.1159944 | journal = IEEE Transactions on Pattern Analysis and Machine Intelligence | volume = 25 | pages = 32–41 | year = 2003 | url = https://zenodo.org/record/1282294 | citeseerx = 10.1.1.79.6978 }}</ref><ref>{{Cite journal | doi = 10.1117/12.255224| title = Wavelet and fractal analysis of ground-vehicle images| volume = 2825| pages = 109–119| series = Wavelet Applications in Signal and Image Processing IV| year = 1996| last1 = Gorsich | first1 = D. J. | last2 = Tolle | first2 = C. R. | last3 = Karlsen | first3 = R. E. | last4 = Gerhart | first4 = G. R. | url = https://zenodo.org/record/1235586}}</ref>
ecology,<ref>{{Cite journal|last=Wildhaber|first=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|issn=1548-8675}}</ref>
acoustics,<ref>{{Cite journal | last1 = Maragos | first1 = P. | last2 = Potamianos | first2 = A. | doi = 10.1121/1.426738 | 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 = 1925–32 | year = 1999 | pmid =  10089613 |bibcode = 1999ASAJ..105.1925M }}</ref>
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 = 13983–97 | year = 2006 |bibcode = 2006JPhA...3913983S }}</ref>
and electrochemical processes.<ref>{{Cite journal | last1 = Eftekhari | first1 = A. | title = Fractal Dimension of Electrochemical Reactions | doi = 10.1149/1.1773583 | journal = Journal of the Electrochemical Society | volume = 151 | issue = 9 | pages = E291–6 | year = 2004 }}</ref>

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|colloidal 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 ==
* [[List of fractals by Hausdorff dimension]]
* [[Lacunarity]]
* [[Fractal derivative]]

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

==References==
{{reflist}}

==Further reading==
*{{cite book |first=Benoit B. |last=Mandelbrot |authorlink=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}}

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