Editing Fractal dimension (section)

== 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}}