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{{General geometry|expanded=branches}}{{Short description|Infinitely detailed mathematical structure}}
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[[File:sierpinski-carpet.gif|thumb|[[Sierpiński carpet|Sierpiński Carpet]] - Infinite perimeter and zero area]]
[[File:Mandel zoom 14 satellite julia island.jpg|thumb|Highly magnified area on the boundary of the [[Mandelbrot set]]]]
[[File:Mandel zoom 00 mandelbrot set.jpg|thumb|The [[Mandelbrot set]]: its boundary is a fractal curve with [[Hausdorff dimension]] 2. (Note that the colored sections of the image are not actually part of the Mandelbrot Set, but rather they are based on how quickly the function that produces it diverges.)|200x200px]]
[[File:Mandelbrot 12 Encirclements.jpg|thumb|Mandelbrot set with 12 encirclements]]
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{{Anchor|Mandelbrot zooms}}
[[File:Mandelbrot sequence new.gif|thumb|Zooming into the boundary of the Mandelbrot set|200x200px]]

In [[mathematics]], a '''fractal''' is a [[Shape|geometric shape]] containing detailed structure at arbitrarily small scales, usually having a [[fractal dimension]] strictly exceeding the [[topological dimension]]. Many fractals appear similar at various scales, as illustrated in successive magnifications of the [[Mandelbrot set]].<ref name="Mandelbrot1983" /><ref name="Falconer" /><ref name="patterns" /><ref name="vicsek"/> This exhibition of similar patterns at increasingly smaller scales is called [[self-similarity]], also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the [[Menger sponge]], the shape is called [[affine geometry|affine]] self-similar.<ref name="Gouyet" /> Fractal geometry relates to the mathematical branch of [[measure theory]] by their [[Hausdorff dimension]].

One way that fractals are different from finite [[geometric figures]] is how they [[Scaling (geometry)|scale]]. Doubling the edge lengths of a filled [[polygon]] multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the [[radius]] of a filled sphere is doubled, its [[volume]] scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the conventional dimension of the filled sphere). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an [[integer]] and is in general greater than its conventional dimension.<ref name="Mandelbrot1983">{{cite book |last=Mandelbrot |first=Benoît B. |title=The fractal geometry of nature |url=https://books.google.com/books?id=0R2LkE3N7-oC |year=1983 |publisher=Macmillan |isbn=978-0-7167-1186-5}}</ref> This power is called the [[fractal dimension]] of the geometric object, to distinguish it from the conventional dimension (which is formally called the [[topological dimension]]).<ref name="Mandelbrot Chaos">{{cite book | last=Mandelbrot | first=Benoît B. | title=Fractals and Chaos| publisher=Springer | location=Berlin | 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>

Analytically, many fractals are nowhere [[Differentiable function|differentiable]].<ref name="Mandelbrot1983" /><ref name="vicsek" /> An infinite [[fractal curve]] can be conceived of as winding through space differently from an ordinary line – although it is still [[topological dimension|topologically 1-dimensional]], its fractal dimension indicates that it locally fills space more efficiently than an ordinary line.<ref name="Mandelbrot1983" /><ref name="Mandelbrot Chaos" />
[[File:Sierpinski carpet 6.svg|200x200px|thumb|[[Sierpinski carpet]] (to level 6), a fractal with a [[topological dimension]] of 1 and a [[Hausdorff dimension]] of 1.893]]
[[File:LineSegment selfSimilar svg.svg|thumb|200px|A [[line segment]] is [[similarity (geometry)|similar]] to a proper part of itself, but hardly a fractal.]]

Starting in the 17th century with notions of [[recursion]], fractals have moved through increasingly rigorous mathematical treatment to the study of [[Continuous function|continuous]] but not [[Differentiable function|differentiable]] functions in the 19th century by the seminal work of [[Bernard Bolzano]], [[Bernhard Riemann]], and [[Karl Weierstrass]],<ref>{{cite journal |last1=Segal |first1=S. L. |title=Riemann's example of a continuous 'nondifferentiable' function continued |journal=The Mathematical Intelligencer |date=June 1978 |volume=1 |issue=2 |pages=81–82 |doi=10.1007/BF03023065|s2cid=120037858 }}</ref> and on to the coining of the word ''[[wikt:fractal|fractal]]'' in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century.<ref name="classics" /><ref name="MacTutor" />

There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals."<ref>{{cite web |last=Mandelbrot |first=Benoit |title=24/7 Lecture on Fractals |url=https://www.youtube.com/watch?v=5e7HB5Oze4g#t=70 | archive-url=https://ghostarchive.org/varchive/youtube/20211211/5e7HB5Oze4g| archive-date=2021-12-11 | url-status=live|work=2006 Ig Nobel Awards |date=July 8, 2013 |publisher=Improbable Research}}{{cbignore}}</ref> More formally, in 1982 Mandelbrot defined ''fractal'' as follows: "A fractal is by definition a set for which the [[Hausdorff dimension|Hausdorff–Besicovitch dimension]] strictly exceeds the [[topological dimension]]."<ref>Mandelbrot, B. B.: The Fractal Geometry of Nature. W. H. Freeman and Company, New York (1982); p. 15.</ref> Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a rough or fragmented [[Shape|geometric shape]] that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole."<ref name="Mandelbrot1983" /> Still later, Mandelbrot proposed "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 Science & Business Media | isbn = 978-0-387-74749-1 | page = 7}}</ref>

The consensus among mathematicians is that theoretical fractals are infinitely self-similar [[iteration|iterated]] and detailed mathematical constructs, of which many [[List of fractals by Hausdorff dimension|examples]] have been formulated and studied.<ref name="Mandelbrot1983" /><ref name="Falconer" /><ref name="patterns">{{Cite book |title=Fractals:The Patterns of Chaos |last=Briggs |first=John |year= 1992 |publisher= Thames and Hudson |location= London |isbn=978-0-500-27693-8 |page=148 }}</ref> Fractals are not limited to geometric patterns, but can also describe processes in time.<ref name="Gouyet" /><ref name="vicsek" /><ref name="time series" /> Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical, and aural media<ref name="music">{{Cite journal | last1=Brothers | first1=Harlan J. | doi=10.1142/S0218348X0700337X | title=Structural Scaling in Bach's Cello Suite No. 3 | journal=Fractals | volume=15 | issue=1 | pages=89–95 | year=2007 }}</ref> and found in [[#fractals in nature|nature]],<ref name="heart" /><ref name="cerebellum">{{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 name="neuroscience">{{Cite journal | last1=Karperien | first1=Audrey L. | last2=Jelinek | first2=Herbert F. | last3=Buchan | first3=Alastair M. | doi=10.1142/S0218348X08003880 | title=Box-Counting Analysis of Microglia Form in Schizophrenia, Alzheimer's Disease and Affective Disorder | journal=Fractals | volume=16 | issue=2 | page=103 | year=2008 }}</ref><ref name="branching" /> [[#fractals in technology|technology]],<ref name="soil">{{Cite journal | last1=Hu | first1=Shougeng | last2=Cheng | first2=Qiuming | last3=Wang | first3=Le | last4=Xie | first4=Shuyun | title=Multifractal characterization of urban residential land price in space and time | doi=10.1016/j.apgeog.2011.10.016 | journal=Applied Geography | volume=34 | pages=161–170 | year=2012 | bibcode=2012AppGe..34..161H }}</ref><ref name="diagnostic imaging">{{Cite journal | last1=Karperien | first1=Audrey | last2=Jelinek | first2=Herbert F. | last3=Leandro | first3=Jorge de Jesus Gomes| last4=Soares | first4=João V. B. | last5=Cesar Jr | first5=Roberto M. | last6=Luckie | first6=Alan | title=Automated detection of proliferative retinopathy in clinical practice | journal=Clinical Ophthalmology | volume=2 | issue=1 | pages=109–122 | year=2008 | pmid=19668394 | pmc=2698675| doi=10.2147/OPTH.S1579 | doi-access=free }}</ref><ref name="medicine">{{cite book|first1=Gabriele A. |last1=Losa |first2=Theo F. |last2=Nonnenmacher |title=Fractals in biology and medicine |url=https://books.google.com/books?id=t9l9GdAt95gC |year=2005 |publisher=Springer|isbn=978-3-7643-7172-2}}</ref><ref name="seismology" /> [[#In creative works|art]],<ref name="novel" /><ref name="African art" /> and [[architecture]].<ref name="springer.com 9783319324241">Ostwald, Michael J., and Vaughan, Josephine (2016) ''[[The Fractal Dimension of Architecture]]'' Birhauser, Basel. {{doi|10.1007/978-3-319-32426-5}}.</ref> Fractals are of particular relevance in the field of [[chaos theory]] because they show up in the geometric depictions of most chaotic processes (typically either as attractors or as boundaries between basins of attraction).<ref>{{cite web |url=http://necsi.edu/projects/baranger/cce.pdf| first=Michael |last=Baranger |title=Chaos, Complexity, and Entropy: A physics talk for non-physicists}}</ref>

==Etymology==
The term "fractal" was coined by the mathematician [[Benoît Mandelbrot]] in 1975.<ref>Benoît Mandelbrot, ''Objets fractals'', 1975, p. 4</ref> Mandelbrot based it on the Latin {{lang|la|[[wikt:fractus#Latin|frāctus]]}}, meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional [[fractal dimension|dimensions]] to geometric [[patterns in nature]].<ref name="Mandelbrot1983" /><ref name="Mandelbrot quote">{{cite book |title=Mathematical people: profiles and interviews |last1=Albers |first1=Donald J. |last2=Alexanderson |first2=Gerald L. |publisher=AK Peters |year=2008 |isbn=978-1-56881-340-0 |location=Wellesley, MA |page=214 |chapter=Benoît Mandelbrot: In his own words |author2-link=Gerald L. Alexanderson}}</ref><ref>{{Cite OED|id=74094|term=fractal}}</ref>

==Introduction==
[[File:Simple Fractals.png|thumb|right| A simple fractal tree|200x200px]]
[[File:FractalTree.gif|thumb|A fractal "tree" to eleven iterations]]

The word "fractal" often has different connotations for mathematicians and the general public, where the public is more likely to be familiar with [[fractal art]] than the mathematical concept. The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with a little mathematical background.

The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in the [[infinite regress]] in parallel mirrors or the [[homunculus]], the little man inside the head of the little man inside the head ...). The difference for fractals is that the pattern reproduced must be detailed.<ref name="Mandelbrot1983" />{{rp|166; 18}}<ref name="Falconer" /><ref name="Mandelbrot quote" />

This idea of being detailed relates to another feature that can be understood without much mathematical background: Having a [[fractal dimension]] greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric [[shapes]] are usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure is [[rep-tile]]d into pieces each 1/3 the length of the original, then there are always three equal pieces. A solid square is understood to be two-dimensional; if such a figure is rep-tiled into pieces each scaled down by a factor of 1/3 in both dimensions, there are a total of 3<sup>2</sup> = 9 pieces.

We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/''r'', there are a total of ''r''<sup>''n''</sup> pieces. Now, consider the [[Koch curve]]. It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So, strictly by analogy, we can consider the "dimension" of the Koch curve as being the unique real number ''D'' that satisfies 3<sup>''D''</sup> = 4. This number is called the ''fractal dimension'' of the Koch curve; it is not the conventionally perceived dimension of a curve. In general, a key property of fractals is that the fractal dimension differs from the ''conventionally understood'' dimension (formally called the topological dimension).

[[File:3D Computer Generated Fractal.png|thumb|3D computer-generated fractal]]
This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere [[Differentiable function|differentiable]]". In a concrete sense, this means fractals cannot be measured in traditional ways.<ref name="Mandelbrot1983" /><ref name="vicsek" /><ref name="Gordon" /> To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of [[rectifiable curve|measuring]] with a tape measure. But in measuring an infinitely "wiggly" fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. The result is that one must need infinite tape to perfectly cover the entire curve, i.e. the snowflake has an infinite perimeter.<ref name="Mandelbrot1983" />

{{anchor|history|koch}}

==History==
[[File:Von Koch curve.gif|thumb|A [[Koch snowflake]] is a fractal that begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral bump|alt=|208x208px]]
[[File:Cantor set in seven iterations.svg|thumb|Cantor (ternary) set]]
The history of fractals traces a path from chiefly theoretical studies to modern applications in [[computer graphics]], with several notable people contributing canonical fractal forms along the way.<ref name="classics">{{cite book | last=Edgar | first=Gerald | title=Classics on Fractals | publisher=Westview Press | location=Boulder, CO | 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 
 |last=Trochet 
 |first=Holly 
 |archive-url=https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html
 |url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/fractals.html 
 |archive-date=March 12, 2012
 |year=2009 
 }}</ref> 
A common theme in traditional [[Architecture of Africa|African architecture]] is the use of fractal scaling, whereby small parts of the structure tend to look similar to larger parts, such as a circular village made of circular houses.<ref>{{cite book| last = Eglash| first = Ron| title = African Fractals Modern Computing and Indigenous Design| year = 1999| publisher = Rutgers University Press| isbn = 978-0-8135-2613-3 }}</ref>
According to [[Clifford A. Pickover|Pickover]], the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher [[Gottfried Leibniz]] pondered [[recursion|recursive]] [[self-similarity]] (although he made the mistake of thinking that only the [[straight line]] was self-similar in this sense).<ref name="Pickover">{{cite book |page=310 |url=https://books.google.com/books?id=JrslMKTgSZwC&q=fractal+koch+curve+book&pg=PA310 |first=Clifford A. |last=Pickover 
|title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics |year=2009 |publisher=Sterling |isbn=978-1-4027-5796-9 }}</ref>

In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them.<ref name="Mandelbrot1983" />{{rp|405}} Indeed, according to various historical accounts, after that point few mathematicians tackled the issues and the work of those who did remained obscured largely because of resistance to such unfamiliar emerging concepts, which were sometimes referred to as mathematical "monsters".<ref name="Gordon" /><ref name="classics" /><ref name="MacTutor" /> Thus, it was not until two centuries had passed that on July 18, 1872 [[Karl Weierstrass]] presented the first definition of a [[Weierstrass function|function]] with a [[Graph of a function|graph]] that would today be considered a fractal, having the non-[[intuition (knowledge)|intuitive]] property of being everywhere [[continuous function|continuous]] but [[nowhere differentiable]] at the Royal Prussian Academy of Sciences.<ref name="classics" />{{rp|7}}<ref name="MacTutor" />

In addition, the quotient difference becomes arbitrarily large as the summation index increases.<ref>{{Cite web|url=http://www-history.mcs.st-and.ac.uk/HistTopics/fractals.html|title=Fractal Geometry|website=www-history.mcs.st-and.ac.uk|access-date=April 11, 2017}}</ref> Not long after that, in 1883, [[Georg Cantor]], who attended lectures by Weierstrass,<ref name="MacTutor" /> published examples of [[subset]]s of the real line known as [[Cantor set]]s, which had unusual properties and are now recognized as fractals.<ref name="classics" />{{rp|11–24}} Also in the last part of that century, [[Felix Klein]] and [[Henri Poincaré]] introduced a category of fractal that has come to be called "self-inverse" fractals.<ref name="Mandelbrot1983" />{{rp|166}}

{{anchor|julia}}
[[File:Julia set (indigo).png|thumb|A [[Julia set]], a fractal related to the Mandelbrot set|alt=|200x200px]]
[[File:Fractal tree.gif|thumb|A [[Sierpinski gasket]] can be generated by a fractal tree.|200x200px]]
One of the next milestones came in 1904, when [[Helge von Koch]], extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand-drawn images of a similar function, which is now called the [[Koch snowflake]].<ref name="classics" />{{rp|25}}<ref name="MacTutor" /> Another milestone came a decade later in 1915, when [[Wacław Sierpiński]] constructed his famous [[Sierpinski triangle|triangle]] then, one year later, his [[Sierpinski carpet|carpet]]. By 1918, two French mathematicians, [[Pierre Fatou]] and [[Gaston Julia]], though working independently, arrived essentially simultaneously at results describing what is now seen as fractal behaviour associated with mapping [[complex numbers]] and iterative functions and leading to further ideas about [[strange attractors|attractors and repellors]] (i.e., points that attract or repel other points), which have become very important in the study of fractals.<ref name="vicsek" /><ref name="classics" /><ref name="MacTutor" />

Very shortly after that work was submitted, by March 1918, [[Felix Hausdorff]] expanded the definition of "dimension", significantly for the evolution of the definition of fractals, to allow for sets to have non-integer dimensions.<ref name="MacTutor" /> The idea of self-similar curves was taken further by [[Paul Lévy (mathematician)|Paul Lévy]], who, in his 1938 paper ''Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole'', described a new fractal curve, the [[Lévy C curve]].<ref group="notes" name="levy note" />

{{anchor|multifractal|Strange attractor}}
[[File:Karperien Strange Attractor 200.gif|thumb|A [[strange attractor]] that exhibits [[multifractal]] scaling|200x200px]]
[[File:Uniform Triangle Mass Center grade 5 fractal.gif|thumb|Uniform mass center triangle fractal|200x200px]]
[[File:60 degrees 2x recursive IFS.jpg|thumb|2x 120 degrees recursive [[Iterated function systems|IFS]]|200x200px]]

Different researchers have postulated that without the aid of modern computer graphics, early investigators were limited to what they could depict in manual drawings, so lacked the means to visualize the beauty and appreciate some of the implications of many of the patterns they had discovered (the Julia set, for instance, could only be visualized through a few iterations as very simple drawings).<ref name="Mandelbrot1983" />{{rp|179}}<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><ref name="MacTutor" /> That changed, however, in the 1960s, when [[Benoit Mandelbrot]] started writing about self-similarity in papers such as ''[[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension]]'',<ref>{{cite journal|author=Mandelbrot, B.|title=How Long Is the Coast of Britain?|journal=Science|date=1967|volume=156|issue=3775|pages=636–638|doi=10.1126/science.156.3775.636|pmid=17837158|bibcode=1967Sci...156..636M|s2cid=15662830|url=http://ena.lp.edu.ua:8080/handle/ntb/52473|access-date=October 31, 2020|archive-date=October 19, 2021|archive-url=https://web.archive.org/web/20211019193011/http://ena.lp.edu.ua:8080/handle/ntb/52473}}</ref><ref>{{cite journal |journal=New Scientist |date=April 4, 1985 |title=Fractals – Geometry Between Dimensions |first=Michael |last=Batty |page=31 |volume=105 |issue=1450 }}</ref> which built on earlier work by [[Lewis Fry Richardson]].

In 1975,<ref name="Mandelbrot quote" /> Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations. These images, such as of his canonical [[Mandelbrot set]], captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".<ref>{{cite book |url=https://books.google.com/books?id=qDQjyuuDRxUC&pg=PA1 |page=1 |title=Fractal surfaces |volume= 1 |first=John C. |last=Russ |access-date=February 5, 2011 |year=1994 |publisher=Springer |isbn=978-0-306-44702-0 }}</ref><ref name="Gordon" /><ref name="classics" /><ref name="Pickover" />

In 1980, [[Loren Carpenter]] gave a presentation at the [[SIGGRAPH]] where he introduced his software for generating and rendering fractally generated landscapes.<ref>{{Cite web|title=Vol Libre, an amazing CG film from 1980|url=https://kottke.org/09/07/vol-libre-an-amazing-cg-film-from-1980|access-date=2023-02-12|website=kottke.org|date=July 29, 2009 |language=en}}</ref>

{{anchor|characteristics}}

==Definition and characteristics==
One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmented [[Shape|geometric shape]] that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole";<ref name="Mandelbrot1983" /> this is generally helpful but limited. Authors disagree on the exact definition of ''fractal'', but most usually elaborate on the basic ideas of self-similarity and the unusual relationship fractals have with the space they are embedded in.<ref name="Mandelbrot1983" /><ref name="Gouyet">{{cite book | last=Gouyet | first=Jean-François | title=Physics and fractal structures | publisher=Masson Springer | location=Paris/New York | year=1996 | isbn=978-0-387-94153-0 }}</ref><ref name="Falconer">{{Cite book | last=Falconer | first=Kenneth | title=Fractal Geometry: Mathematical Foundations and Applications | publisher=John Wiley & Sons | year=2003 | page=xxv | isbn= 978-0-470-84862-3 | no-pp=true }}</ref><ref name="vicsek" /><ref>{{cite book | last=Edgar | first=Gerald | title=Measure, topology, and fractal geometry | publisher=Springer-Verlag | location=New York | year=2008 | isbn=978-0-387-74748-4 |page=1 }}</ref>

One point agreed on is that fractal patterns are characterized by [[fractal dimension]]s, but whereas these numbers quantify [[complexity]] (i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns.<ref>{{cite book |last=Karperien |first=Audrey |title=Defining microglial morphology: Form, Function, and Fractal Dimension |publisher=Charles Sturt University |year= 2004 |doi=10.13140/2.1.2815.9048 }}</ref> In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose [[Hausdorff–Besicovitch dimension]] is greater than its [[Lebesgue covering dimension|topological dimension]].<ref name="Mandelbrot quote" /> However, this requirement is not met by [[space-filling curve]]s such as the [[Hilbert curve]].<ref group=notes name="space filling note" />

Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. According to [[Kenneth Falconer (mathematician)|Falconer]], fractals should be only generally characterized by a [[wikt:gestalt|gestalt]] of the following features;<ref name="Falconer" />

* Self-similarity, which may include:
:* Exact self-similarity: identical at all scales, such as the [[#koch|Koch snowflake]]
:* Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms; e.g., the [[Mandelbrot set]]'s satellites are approximations of the entire set, but not exact copies.
:* Statistical self-similarity: repeats a pattern [[stochastic]]ally so numerical or statistical measures are preserved across scales; e.g., [[#random|randomly generated fractals]] like the well-known example of the [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|coastline of Britain]] for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines fractals like the Koch snowflake.<ref name="vicsek" />
:* Qualitative self-similarity: as in a time series<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 | location=New York | year=1996 | isbn=978-0-471-13938-6 }}</ref>
:* [[Multifractal]] scaling: characterized by more than one fractal dimension or scaling rule
* Fine or detailed structure at arbitrarily small scales. A consequence of this structure is fractals may have [[emergent properties]]<ref>{{cite book | first1=John |last1=Spencer |first2=Michael S. C. |last2=Thomas |first3=James L. |last3=McClelland |title=Toward a unified theory of development: connectionism and dynamic systems theory re-considered |publisher=Oxford University Press |location=Oxford/New York |year=2009 |isbn=978-0-19-530059-8 }}</ref> (related to the next criterion in this list).
* Irregularity locally and globally that cannot easily be described in the language of traditional [[Euclidean geometry]] other than as the limit of a [[recursion|recursively]] defined sequence of stages. For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up surfaces" and "swirls upon swirls";<ref name="Mandelbrot Chaos" />''see [[#algorithms|Common techniques for generating fractals]]''.

As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar without having other typically fractal features. A straight line, for instance, is self-similar but not fractal because it lacks detail, and is easily described in Euclidean language without a need for recursion.<ref name="Mandelbrot1983" /><ref name="vicsek" />

When Mandelbrot introduced the term fractal, he excluded magnification range as a defining characteristic in order to accommodate physical fractals with more limited ranges than their mathematical counterparts.<ref> B.B. Mandelbrot, "The Fractal Geometry of Nature" New York: W.H. Freeman & Company 1982 </ref>

==Common techniques for generating fractals==
{{See also|Fractal-generating software}}
{{anchor|L-system}}
[[File:KarperienFractalBranch.jpg|thumb|Self-similar branching pattern modeled [[in silico]] using [[L-systems]] principles<ref name="branching">{{Cite book |editor=Sarker, Ruhul |title=Workshop proceedings: the Sixth Australia-Japan Joint Workshop on Intelligent and Evolutionary Systems, University House, ANU |oclc=224846454|chapter=MicroMod-an L-systems approach to neural modelling |last1=Jelinek |first1=Herbert F. |last2=Karperien |first2=Audrey |last3=Cornforth |first3=David |last4=Cesar |first4=Roberto |last5=Leandro |first5=Jorge de Jesus Gomes |url=https://books.google.com/books?id=FFSUGQAACAAJ |access-date=February 3, 2012 |year=2002 |publisher=University of New South Wales |isbn=978-0-7317-0505-4 |quote=Event location: Canberra, Australia}}</ref>|alt=|201x201px]]
{{anchor|algorithms}}
Images of fractals can be created by [[Fractal-generating software|fractal generating programs]]. Because of the [[butterfly effect]], a small change in a single variable can have an [[Predictability|unpredictable]] outcome.

* ''[[Iterated function systems]] (IFS)'' – use fixed geometric replacement rules; may be stochastic or deterministic;<ref name="IFS">{{cite book |editor=Pickover, Clifford A. |title=Chaos and fractals: a computer graphical journey: ten year compilation of advanced research |url=https://books.google.com/books?id=A51ARsapVuUC |access-date=February 4, 2012 |date=August 3, 1998 |publisher=Elsevier |isbn=978-0-444-50002-1 |last=Frame |first=Angus |chapter=Iterated Function Systems |pages=349–351 }}</ref> e.g., [[Koch snowflake]], [[Cantor set]], Haferman carpet,<ref>{{cite web |title=Haferman Carpet |url=https://www.wolframalpha.com/input/?i=Haferman+carpet |access-date=October 18, 2012 |publisher=WolframAlpha }}</ref> [[Sierpinski carpet]], [[Sierpinski gasket]], [[Peano curve]], [[dragon curve|Harter-Heighway dragon curve]], [[T-square (fractal)|T-square]], [[Menger sponge]]
* ''[[Strange attractor]]s'' – use iterations of a map or solutions of a system of initial-value differential or difference equations that exhibit chaos (e.g., see [[#multifractal|multifractal]] image, or the [[logistic map]])
* ''[[L-system]]s'' – use string rewriting; may resemble branching patterns, such as in plants, biological cells (e.g., neurons and immune system cells<ref name="branching" />), blood vessels, pulmonary structure,<ref name="modeling vasculature" /> etc. or [[turtle graphics]] patterns such as [[space-filling curves]] and tilings
* ''Escape-time fractals'' – use a [[formula]] or [[recurrence relation]] at each point in a space (such as the [[complex plane]]); usually quasi-self-similar; also known as "orbit" fractals; e.g., the [[Mandelbrot set]], [[Julia set]], [[Burning Ship fractal]], [[Nova fractal]] and [[Lyapunov fractal]]. The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly.
* {{anchor|random}}''Random fractals'' – use stochastic rules; e.g., [[Lévy flight]], [[Percolation theory|percolation clusters]], [[Self-avoiding walk|self avoiding walks]], [[fractal landscapes]], trajectories of [[Brownian motion]] and the [[Brownian tree]] (i.e., dendritic fractals generated by modeling [[diffusion-limited aggregation]] or reaction-limited aggregation clusters).<ref name="vicsek">{{cite book |last=Vicsek |first=Tamás | title=Fractal growth phenomena | publisher=World Scientific | location=Singapore/New Jersey | year=1992 | isbn=978-981-02-0668-0|pages=31; 139–146 }}</ref>

[[File:Finite subdivision of a radial link.png|thumb|A fractal generated by a [[finite subdivision rule]] for an [[alternating link]]|202x202px]]
*''[[Finite subdivision rule]]s'' – use a recursive [[topological]] algorithm for refining tilings<ref name="finite">J. W. Cannon, W. J. Floyd, W. R. Parry. ''Finite subdivision rules''. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153–196.</ref> and they are similar to the process of [[cell division]].<ref name="biol">{{Cite book|last1=Carbone|first1=Alessandra|url=https://books.google.com/books?id=qZHyqUli9y8C&dq=%2522james+w.+cannon%2522+maths&pg=PA65|title=Pattern Formation in Biology, Vision and Dynamics|last2=Gromov|first2=Mikhael|last3=Prusinkiewicz|first3=Przemyslaw|date=2000|publisher=World Scientific|isbn=978-981-02-3792-9|language=en}}</ref> The iterative processes used in creating the [[Cantor set]] and the [[Sierpinski carpet]] are examples of finite subdivision rules, as is [[barycentric subdivision]].

==Applications==

===Simulated fractals===
[[File:Julia-Set z2+c ani.gif|thumb|[[Fractal art]] made from a [[Julia set|Julia-Set]]]]
Fractal patterns have been modeled extensively, albeit within a range of scales rather than infinitely, owing to the practical limits of physical time and space. Models may simulate theoretical fractals or [[#fractals in nature|natural phenomena with fractal features]]. The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks for [[fractal analysis]]. Some specific applications of fractals to technology are listed [[#fractals in technology|elsewhere]]. Images and other outputs of modelling are normally referred to as being "fractals" even if they do not have strictly fractal characteristics, such as when it is possible to zoom into a region of the fractal image that does not exhibit any fractal properties. Also, these may include calculation or display [[Artifact (error)|artifacts]] which are not characteristics of true fractals.

Modeled fractals may be sounds,<ref name="music" /> digital images, electrochemical patterns, [[circadian rhythm]]s,<ref>{{Cite journal | last1=Fathallah-Shaykh | first1=Hassan M. | title=Fractal Dimension of the Drosophila Circadian Clock | doi=10.1142/S0218348X11005476 | journal=Fractals | volume=19 | issue=4 | pages=423–430 | year=2011 }}</ref> etc.
Fractal patterns have been reconstructed in physical 3-dimensional space<ref name="medicine" />{{rp|10}} and virtually, often called "[[in silico]]" modeling.<ref name="modeling vasculature" /> Models of fractals are generally created using [[fractal-generating software]] that implements techniques such as those outlined above.<ref name="vicsek" /><ref name="time series" /><ref name="medicine" /> As one illustration, trees, ferns, cells of the nervous system,<ref name="branching" /> blood and lung vasculature,<ref name="modeling vasculature">{{cite book |chapter=Fractal aspects of three-dimensional vascular constructive optimization 
| first1=Horst K. |last1=Hahn |first2=Manfred |last2=Georg |first3=Heinz-Otto |last3=Peitgen| 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 | year=2005 | publisher=Springer | isbn=978-3-7643-7172-2 | pages=55–66 }}</ref> and other branching [[patterns in nature]] can be modeled on a computer by using recursive [[algorithm]]s and [[L-systems]] techniques.<ref name="branching" />

The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a [[frond]] from a [[fern]] is a miniature replica of the whole: not identical, but similar in nature. Similarly, random fractals have been used to describe/create many highly irregular real-world objects, such as coastlines and mountains. A limitation of modeling fractals is that resemblance of a fractal model to a natural phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithms.

{{anchor|fractals in nature}}

===Natural phenomena with fractal features===
{{Further|Patterns in nature}}

Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connection between fractals and leaves, for instance, is currently being used to determine how much carbon is contained in trees.<ref>"Hunting the Hidden Dimensional". ''Nova''. PBS. WPMB-Maryland. October 28, 2008.</ref> Phenomena known to have fractal features include: <!-- Please provide citations for these entries, or those lacking them will be removed -->

{{div col|colwidth=20em}}
* [[Actin cytoskeleton]]<ref>{{Cite journal |last=Sadegh |first=Sanaz |date=2017 |title=Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Meshwork |journal=Physical Review X |volume=7 |issue=1 |article-number=011031 |doi=10.1103/PhysRevX.7.011031 |pmc=5500227 |pmid=28690919|arxiv=1702.03997 |bibcode=2017PhRvX...7a1031S }}</ref>
* [[Algae]]
* [[Animal coloration]] patterns
* [[Blood vessel]]s and [[pulmonary vessels]]<ref name="modeling vasculature" />
* Brownian motion (generated by a one-dimensional [[Wiener process]]).<ref>{{Cite book |first1=Kenneth |last1=Falconer |title=Fractals, A Very Short Introduction |publisher=Oxford University Press |year=2013}}</ref>
* Clouds and rainfall areas<ref>{{Cite journal |last=Lovejoy |first=Shaun |date=1982 |title=Area-perimeter relation for rain and cloud areas|journal=Science |volume=216 |issue=4542 |pages=185–187|doi=10.1126/science.216.4542.185|pmid=17736252 |bibcode=1982Sci...216..185L |s2cid=32255821 }}</ref>
* Coastlines <ref>{{cite journal | last1=Boffetta | first1=G. | last2=Celani | first2=A. | last3=Dezzani | first3=D. | last4=Seminara | first4=A. | title=How winding is the coast of Britain? Conformal invariance of rocky shorelines | journal=Geophysical Research Letters | volume=35 | issue=3 | date=2008 | issn=0094-8276 | doi=10.1029/2007GL033093 | doi-access=free | arxiv=0712.3076 | bibcode=2008GeoRL..35.3615B }}</ref>
* [[Impact crater|Craters]]
* Crystals<ref name="crystal">{{cite book |chapter-url=https://books.google.com/books?id=qZHyqUli9y8C&q=crystal+fractals+book&pg=PA78 |first1=James W. |last1=Cannon |first2=William J. |last2=Floyd |first3=Walter R. |last3=Perry |chapter=Crystal growth, biological cell growth and geometry |pages=65–82 |title=Pattern formation in biology, vision and dynamics |year=2000 |publisher=World Scientific |isbn=978-981-02-3792-9 |editor-last1=Carbone|editor-first1=Alessandra|editor-last2=Gromov|editor-first2=Mikhael|editor-last3=Prusinkiewicz|editor-first3=Przemyslaw}}</ref>
* [[DNA]]
* Dust grains<ref>{{citation|title=Electrification in granular gases leads to constrained fractal growth|author=Singh, Chamkor|author2=Mazza, Marco|journal=Scientific Reports|volume=9|year=2019|issue=1|page=9049|doi=10.1038/s41598-019-45447-x|publisher=Nature Publishing Group|pmid=31227758|pmc=6588598|arxiv=1812.06073|bibcode=2019NatSR...9.9049S|doi-access=free}}</ref>
* [[Earthquakes]]<ref name="seismology">{{Cite journal | last1=Vannucchi | first1=Paola |author1-link=Paola Vannucchi| last2=Leoni | first2=Lorenzo | doi=10.1016/j.epsl.2007.07.056 | title=Structural characterization of the Costa Rica décollement: Evidence for seismically-induced fluid pulsing | journal=Earth and Planetary Science Letters | volume=262 | issue=3–4 | page=413 | year=2007 |bibcode=2007E&PSL.262..413V | hdl=2158/257208 | s2cid=128467785 | hdl-access=free }}</ref><ref>{{cite book |pages=128–140 |title=Critical phenomena in natural sciences: chaos, fractals, selforganization, and disorder: concepts and tools |first=Didier |last=Sornette |year=2004 |publisher=Springer |isbn=978-3-540-40754-6 }}</ref>
* [[Fault line]]s
* Geometrical optics<ref name="geomopt">{{citation |first1=D. |last1=Sweet |first2=E. |last2=Ott |first3=J. A. |last3=Yorke |title=Complex topology in Chaotic scattering: A Laboratory Observation |year=1999 |journal=Nature |volume=399 |page=315 |doi=10.1038/20573 |issue=6734 |bibcode = 1999Natur.399..315S |s2cid=4361904 }}</ref>
* Heart rates<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>
* [[Heart sounds]]
* [[Lake]] shorelines and areas<ref>{{cite journal |author1=D. Seekell |author2=B. Cael |author3=E. Lindmark |author4=P. Byström |title=The fractal scaling relationship for river inlets to lakes |journal=Geophysical Research Letters |date=2021 |volume=48 |issue=9 |article-number=e2021GL093366 |doi=10.1029/2021GL093366| issn=0094-8276 |bibcode=2021GeoRL..4893366S |s2cid=235508504 |url=https://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-183511 }}</ref><ref>{{cite journal |author1=D. Seekell |author2=M. L. Pace |author3=L. J. Tranvik |author4=C. Verpoorter |title=A fractal-based approach to lake size-distributions |journal=Geophysical Research Letters |date=2013 |volume=40 |issue=3 |pages=517–521 |doi=10.1002/grl.50139 |bibcode=2013GeoRL..40..517S |s2cid=14482711 |url=https://hal.archives-ouvertes.fr/hal-00932495/file/grl.50139.pdf }}</ref><ref>{{cite journal |author1=B. B. Cael |author2=D. A. Seekell |title=The size-distribution of Earth's lakes |journal=Scientific Reports |date=2016 |volume=6 |article-number=29633 |doi=10.1038/srep29633 |pmid=27388607 |pmc=4937396 |bibcode=2016NatSR...629633C }}</ref>
* [[Lightning]] bolts
* Mountain-goat horns
* [[Neuron]]s
* Polymers
* Percolation
* [[Mountain|Mountain ranges]]
* [[Wind wave|Ocean waves]]<ref name="nature">{{cite book |url=https://books.google.com/books?id=l2E4ciBQ9qEC&q=lightning+fractals+book&pg=PA45 |pages=44–46 |title=Fractals and chaos: an illustrated course |first=Paul S. |last=Addison |year=1997 |publisher=CRC Press |access-date=February 5, 2011 |isbn=978-0-7503-0400-9 }}</ref>
* Pineapple
* [[Proteins]]<ref>{{cite journal|last1=Enright|first1=Matthew B.|last2=Leitner|first2=David M.|title=Mass fractal dimension and the compactness of proteins|journal=Physical Review E|date=January 27, 2005|volume=71|issue=1|article-number=011912|doi=10.1103/PhysRevE.71.011912|pmid=15697635|bibcode = 2005PhRvE..71a1912E |url=https://zenodo.org/record/895378}}</ref>
* [[Psychedelic experience]]<ref> {{cite journal |author1=Thomas F. Varley |author2= Robin Carhart-Harris |author3=Leor Roseman |author4=David K. Menon |author5=Emmanuel A. Stamatakis |title=Serotonergic psychedelics LSD & psilocybin increase the fractal dimension of cortical brain activity in spatial and temporal domains |journal=[[NeuroImage]] |volume=220 |year=2020 |doi=10.1016/j.neuroimage.2020.117049|doi-access=free }}</ref>
*[[Purkinje cell]]s<ref>{{cite journal|last1=Takeda|first1=T|last2=Ishikawa|first2=A|last3=Ohtomo|first3=K|last4=Kobayashi|first4=Y|last5=Matsuoka|first5=T|title=Fractal dimension of dendritic tree of cerebellar Purkinje cell during onto- and phylogenetic development|journal=Neurosci Research|date=February 1992|volume=13|issue=1|pages=19–31|doi=10.1016/0168-0102(92)90031-7|pmid=1314350|s2cid=4158401}}</ref>
* [[Rings of Saturn]]<ref>{{cite book|last1=Takayasu|first1=H.|title=Fractals in the physical sciences|date=1990|publisher=Manchester University Press|location=Manchester|isbn=978-0-7190-3434-3|page=[https://archive.org/details/fractalsinphysic0000taka_s1f9/page/36 36]|url=https://archive.org/details/fractalsinphysic0000taka_s1f9/page/36}}</ref><ref>{{Cite journal|last1=Jun|first1=Li|author-link2=Martin Ostoja-Starzewski|last2=Ostoja-Starzewski|first2=Martin|title=Edges of Saturn's Rings are Fractal|journal=SpringerPlus|date=April 1, 2015|volume=4,158|page=158|doi=10.1186/s40064-015-0926-6|pmid=25883885|pmc=4392038 |doi-access=free }}</ref>
* [[river|River networks]]<ref>{{cite journal |last1=Tarboton |first1=David G |last2=Bras |first2=Rafael L |last3=Rodriguez-Iturbe |first3=Ignacio |title=The fractal nature of river networks |journal=Water Resources Research |date=1988 |volume=24 |issue=8 |page=1317|doi=10.1029/WR024i008p01317 |bibcode=1988WRR....24.1317T }}</ref>
* [[Romanesco broccoli]]
* Snowflakes<ref name="snowflake">{{cite book
|url=https://books.google.com/books?id=aHux78oQbbkC&q=snowflake+fractals+book&pg=PA25 |page=25 |first1=Yves |last1=Meyer |first2=Sylvie |last2=Roques |title=Progress in wavelet analysis and applications: proceedings of the International Conference "Wavelets and Applications", Toulouse, France – June 1992 |year=1993 |publisher=Atlantica Séguier Frontières |access-date=February 5, 2011 |isbn=978-2-86332-130-0 }}</ref>
* Soil pores<ref>Ozhovan M. I., Dmitriev I. E., Batyukhnova O. G. Fractal structure of pores of clay soil. Atomic Energy, 74, 241–243 (1993).</ref>
*Surfaces in [[Turbulence|turbulent]] flows<ref>{{cite journal |last1=Sreenivasan |first1=K. R. |first2=C. |last2=Meneveau |title=The Fractal Facets of Turbulence |journal=Journal of Fluid Mechanics |date=1986 |volume=173 |pages=357–386 |doi=10.1017/S0022112086001209 |bibcode=1986JFM...173..357S|s2cid=55578215 }}</ref><ref>{{cite journal |last1=de Silva |first1=C. M. |first2=J. |last2=Philip |first3=K. |last3=Chauhan |first4=C. |last4=Meneveau |first5=I. |last5=Marusic |title=Multiscale Geometry and Scaling of the Turbulent–Nonturbulent Interface in High Reynolds Number Boundary Layers |journal=Phys. Rev. Lett. |date=2013 |volume=111 |issue=6039 |pages=192–196 |doi=10.1126/science.1203223 |pmid=21737736 |bibcode=2011Sci...333..192A|s2cid=22560587 }}</ref>
* Trees <ref>{{cite journal |last1=Mandelbrot |first1=Benoit B |title=The fractal geometry of trees and other natural phenomena |journal=Geometrical Probability and Biological Structures: Buffon's 200th Anniversary: Proceedings of the Buffon Bicentenary Symposium on Geometrical Probability, Image Analysis, Mathematical Stereology, and Their Relevance to the Determination of Biological Structures |date=1978 |pages=235–249}}</ref>
* Protein complexes<ref>{{cite journal| journal=Nature | vauthors=((Sendker, F. L.)), ((Lo, Y. K.)), ((Heimerl, T.)), ((Bohn, S.)), ((Persson, L. J.)), ((Mais, C.-N.)), ((Sadowska, W.)), ((Paczia, N.)), ((Nußbaum, E.)), ((Sánchez Olmos, M. del C.)), ((Forchhammer, K.)), ((Schindler, D.)), ((Erb, T. J.)), ((Benesch, J. L. P.)), ((Marklund, E. G.)), ((Bange, G.)), ((Schuller, J. M.)), ((Hochberg, G. K. A.)) | title=Emergence of fractal geometries in the evolution of a metabolic enzyme | date=2024 |volume=628 |pages=894–900 |url=https://www.nature.com/articles/s41586-024-07287-2}} </ref>
{{div col end}}

<gallery mode="packed" heights="140">
File:Frost patterns 2.jpg|Frost crystals occurring naturally on cold glass form fractal patterns
File:Optical Billiard Spheres dsweet.jpeg|Fractal basin boundary in a geometrical optical system<ref name="geomopt"/>
File:Glue1 800x600.jpg|A fractal is formed when pulling apart two glue-covered [[Acryloyl group|acrylic]] sheets
File:Square1.jpg|High-voltage breakdown within a {{convert|4|in|abbr=on}} block of acrylic glass creates a fractal [[Lichtenberg figure]]
File:Romanesco broccoli (Brassica oleracea).jpg|[[Romanesco broccoli]], showing [[self-similar]] form approximating a natural fractal
File:Fractal defrosting patterns on Mars.jpg|Fractal defrosting patterns, polar Mars. The patterns are formed by sublimation of frozen CO<sub>2</sub>. Width of image is about a kilometer.
File:Brefeldia maxima plasmodium on wood.jpg|[[Slime mold]] ''[[Brefeldia maxima]]'' growing fractally on wood
File:Dendrit.jpg|[[Psilomelane]] [[Dendrite (crystal)|dendrite]]s in the [[Solnhofen Limestone]]
</gallery>

===Fractals in cell biology===
Fractals often appear in the realm of living organisms where they arise through branching processes and other complex pattern formation. Richard Taylor and co-workers have shown that the dendritic branches of neurons form fractal patterns.<ref> J.H. Smith, C. Rowland, B. Harland, S. Moslehi, K. Schobert, R.M. Montgomery, W.J. Watterson, J. Dalrymple-Alford, R.P. Taylor, "How Neurons Exploit Fractal Geometry to Maximize Physical Connectivity", Scientific Reports, 11, 2332 (2021) </ref> Ian Wong and co-workers have shown that migrating cells can form fractals by clustering and [[Branching process|branching]].<ref>{{Cite journal|last1=Leggett|first1=Susan E.|last2=Neronha|first2=Zachary J.|last3=Bhaskar|first3=Dhananjay|last4=Sim|first4=Jea Yun|last5=Perdikari|first5=Theodora Myrto|last6=Wong|first6=Ian Y.|date=2019-08-27|title=Motility-limited aggregation of mammary epithelial cells into fractal-like clusters|journal=Proceedings of the National Academy of Sciences|language=en|volume=116|issue=35|pages=17298–17306|doi=10.1073/pnas.1905958116|issn=0027-8424|pmc=6717304|pmid=31413194|bibcode=2019PNAS..11617298L|doi-access=free}}</ref> [[Neuron|Nerve cells]] function through processes at the cell surface, with phenomena that are enhanced by largely increasing the surface to volume ratio. As a consequence nerve cells often are found to form into fractal patterns.<ref>{{Cite journal|last1=Jelinek|first1=Herbert F|last2=Fernandez|first2=Eduardo|date=June 1998|title=Neurons and fractals: how reliable and useful are calculations of fractal dimensions?|url=https://linkinghub.elsevier.com/retrieve/pii/S0165027098000211|journal=Journal of Neuroscience Methods|language=en|volume=81|issue=1–2|pages=9–18|doi=10.1016/S0165-0270(98)00021-1|pmid=9696304|s2cid=3811866|url-access=subscription}}</ref> These processes are crucial in cell [[physiology]] and different [[Pathology|pathologies]].<ref>{{Cite journal|last=Cross|first=Simon S.|date=1997|title=Fractals in Pathology|journal=The Journal of Pathology|language=en|volume=182|issue=1|pages=1–8|doi=10.1002/(SICI)1096-9896(199705)182:1<1::AID-PATH808>3.0.CO;2-B|pmid=9227334|s2cid=23274235 |issn=1096-9896|doi-access=free}}</ref>

Multiple subcellular structures also are found to assemble into fractals. [[Diego Krapf]] has shown that through branching processes the [[actin]] filaments in human cells assemble into fractal patterns.<ref>{{Cite journal |last1=Sadegh |first1=Sanaz |last2=Higgins |first2=Jenny L. |last3=Mannion |first3=Patrick C. |last4=Tamkun |first4=Michael M. |last5=Krapf |first5=Diego |date=2017-03-09 |title=Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Meshwork |journal=Physical Review X |language=en |volume=7 |issue=1 |article-number=011031 |doi=10.1103/PhysRevX.7.011031 |issn=2160-3308 |pmc=5500227 |pmid=28690919|arxiv=1702.03997 |bibcode=2017PhRvX...7a1031S }}</ref> Similarly Matthias Weiss showed that the [[endoplasmic reticulum]] displays fractal features.<ref>{{Cite journal|last1=Speckner|first1=Konstantin|last2=Stadler|first2=Lorenz|last3=Weiss|first3=Matthias|date=2018-07-09|title=Anomalous dynamics of the endoplasmic reticulum network|url=https://link.aps.org/doi/10.1103/PhysRevE.98.012406|journal=Physical Review E|language=en|volume=98|issue=1|article-number=012406|doi=10.1103/PhysRevE.98.012406|pmid=30110830|bibcode=2018PhRvE..98a2406S|s2cid=52010780|issn=2470-0045|url-access=subscription}}</ref> The current understanding is that fractals are ubiquitous in cell biology, from [[protein]]s, to [[organelle]]s, to whole cells.

===In creative works===
{{Further|Fractal art|Mathematics and art}}

'''Fractal expressionism''' is used to distinguish [[fractal art]] generated directly by artists from fractal art generated using mathematics and/or computers.<ref>R.P.Taylor, A.P. Micolich and D. Jonas, Fractal Expressionism, Physics World, 25, October 1999.</ref>

Since 1999 numerous scientific groups have performed fractal analysis on over 50 paintings created by [[Jackson Pollock]] by pouring paint directly onto horizontal canvasses, see for example.<ref>Taylor, Richard P., Adam P. Micolich, and David Jonas. "Fractal Analysis of Pollock's Drip Paintings." Nature 399.6735 (1999): 422. Print.</ref><ref>J.R. Mureika, C.C. Dyer, G.C. Cupchik, "Multifractal Structure in Nonrepresentational Art", Physical Review E, vol. 72, 046101-1-15 (2005).</ref><ref name="C. Redies, J 2007">C. Redies, J. Hasenstein and J. Denzler, "Fractal-Like Image Statistics in Visual Art: Similar to Natural Scenes", Spatial Vision, vol. 21, 137-148 (2007).</ref><ref name="S. Lee, S 2007">S. Lee, S. Olsen and B. Gooch, "Simulating and Analyzing Jackson Pollock's Paintings" Journal of Mathematics and the Arts, vol.1, 73-83 (2007).</ref><ref>J. Alvarez-Ramirez, C. Ibarra-Valdez, E. Rodriguez and L. Dagdug, "1/f-Noise Structure in Pollock's Drip Paintings", Physica A, vol. 387, 281-295 (2008).</ref><ref name="D.J. Field, 2008">D.J. Graham and D.J. Field, "Variations in Intensity for Representative and Abstract Art, and for Art from Eastern and Western Hemispheres" Perception, vol. 37, 1341-1352 (2008).</ref><ref>J. Alvarez-Ramirez, J. C. Echeverria, E. Rodriguez "Performance of a High-Dimensional R/S Analysis Method for Hurst Exponent Estimation" Physica A, vol. 387, 6452-6462 (2008).</ref><ref>J. Coddington, J. Elton, D. Rockmore and Y. Wang, "Multi-fractal Analysis and Authentication of Jackson Pollock Paintings", Proceedings SPIE, vol. 6810, 68100F 1-12 (2008).</ref><ref>M. Al-Ayyoub, M. T. Irfan and D.G. Stork, "Boosting Multi-Feature Visual Texture Classifiers for the Authentification of Jackson Pollock's Drip Paintings", SPIE proceedings on Computer Vision and Image Analysis of Art II, vol. 7869, 78690H (2009).</ref><ref>J.R. Mureika and R.P. Taylor, "The Abstract Expressionists and Les Automatistes: multi-fractal depth", Signal Processing, vol. 93 573 (2013).</ref><ref>R.P. Taylor et al, "Authenticating Pollock Paintings Using Fractal Geometry", Pattern Recognition Letters, vol. 28, 695-702 (2005).</ref><ref>K Zheng et al Vis Comput. DOI 10.1007/s00371-014-0985-7</ref><ref>E De la Calleja et al Annals of Physics, Vol. 371, 313 (2016)</ref> In 2015, fractal analysis was used to achieve a 93% success rate in distinguishing real from imitation Pollocks.<ref>{{cite journal |first1=L. |last1=Shamar |title=What Makes a Pollock Pollock: A Machine Vision Approach |journal=International Journal of Arts and Technology |volume=8 |pages=1–10 |year=2015 |doi=10.1504/IJART.2015.067389 |url=http://vfacstaff.ltu.edu/lshamir/publications/wm_pollock.pdf |citeseerx=10.1.1.647.365 |access-date=October 24, 2017 |archive-date=October 25, 2017 |archive-url=https://web.archive.org/web/20171025022609/http://vfacstaff.ltu.edu/lshamir/publications/wm_pollock.pdf }}</ref> A 2024 study used an artificial intelligence technique based on fractals to achieve a 99% success rate.<ref> J.H. Smith, C. Holt, N.H. Smith, R.P. Taylor "Using Machine Learning to Distinguish Between Authentic and Imitation Jackson Pollock Poured Paintings: A Tile Driven Approach to Computeraon", "n anOl ne" vol. 19, e0302962 (2024) </ref>

[[Decalcomania]], a technique used by artists such as [[Max Ernst]], can produce fractal-like patterns.<ref>Frame, Michael; and Mandelbrot, Benoît B.; [http://classes.yale.edu/Fractals/Panorama/ ''A Panorama of Fractals and Their Uses''] {{Webarchive|url=https://web.archive.org/web/20071223090421/http://classes.yale.edu/Fractals/Panorama/ |date=December 23, 2007 }}</ref> It involves pressing paint between two surfaces and pulling them apart.

Cyberneticist [[Ron Eglash]] has suggested that fractal geometry and mathematics are prevalent in [[African art]], games, [[divination]], trade, and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles.<ref name="African art">{{cite web |url=http://www.rpi.edu/~eglash/eglash.dir/afractal/afractal.htm |first=Ron |last=Eglash |title=African Fractals: Modern Computing and Indigenous Design |location=New Brunswick |publisher=Rutgers University Press |year=1999 |access-date=October 17, 2010 |archive-url=https://web.archive.org/web/20180103005701/http://homepages.rpi.edu/~eglash/eglash.dir/afractal/afbook.htm |archive-date=January 3, 2018 }}</ref><ref>{{Cite web|last=Nelson|first=Bryn|date=2000-02-23|title=Sophisticated Mathematics Behind African Village Designs / Fractal patterns use repetition on large, small scale|url=https://www.sfgate.com/education/article/Sophisticated-Mathematics-Behind-African-Village-2774181.php|access-date=2023-02-12|website=SFGATE|language=en-US}}</ref> [[Hokky Situngkir]] also suggested the similar properties in Indonesian traditional art, [[batik]], and [[ornament (art)|ornaments]] found in traditional houses.<ref>Situngkir, Hokky; Dahlan, Rolan (2009). ''Fisika batik: implementasi kreatif melalui sifat fraktal pada batik secara komputasional''. Jakarta: Gramedia Pustaka Utama. {{ISBN|978-979-22-4484-7}}</ref><ref>{{cite news |last=Rulistia |first=Novia D. |date=October 6, 2015 |title=Application maps out nation's batik story |url=http://www.thejakartapost.com/news/2015/10/06/application-maps-out-nation-s-batik-story.html |newspaper=The Jakarta Post |access-date=September 25, 2016}}</ref>

Ethnomathematician Ron Eglash has discussed the planned layout of [[Benin city]] using fractals as the basis, not only in the city itself and the villages but even in the rooms of houses. He commented that "When Europeans first came to Africa, they considered the architecture very disorganised and thus primitive. It never occurred to them that the Africans might have been using a form of mathematics that they hadn't even discovered yet."<ref>Koutonin, Mawuna (March 18, 2016). "Story of cities #5: Benin City, the mighty medieval capital now lost without trace". Retrieved April 2, 2018.</ref>

In a 1996 interview with [[Michael Silverblatt]], [[David Foster Wallace]] explained that the structure of the first draft of ''[[Infinite Jest]]'' he gave to his editor Michael Pietsch was inspired by fractals, specifically the [[Sierpinski triangle]] (a.k.a. Sierpinski gasket), but that the edited novel is "more like a lopsided Sierpinsky Gasket".<ref name="novel">{{cite web |url=http://www.kcrw.com/etc/programs/bw/bw960411david_foster_wallace |first=David Foster |last=Wallace |title=Bookworm on KCRW |date=August 4, 2006 |publisher=Kcrw.com |access-date=October 17, 2010 |archive-date=November 11, 2010 |archive-url=https://web.archive.org/web/20101111033857/http://www.kcrw.com/etc/programs/bw/bw960411david_foster_wallace }}</ref>

Some works by the Dutch artist [[M. C. Escher]], such as [[Circle Limit III]], contain shapes repeated to infinity that become smaller and smaller as they get near to the edges, in a pattern that would always look the same if zoomed in.

{{anchor|fractals in technology}}

<gallery heights="140" mode="packed">
File:Animated fractal mountain.gif|A fractal that models the surface of a mountain (animation)
File:FRACTAL-3d-FLOWER.jpg|3D recursive image
File:Fractal-BUTTERFLY.jpg|Recursive fractal butterfly image
File:Apophysis-100303-104.jpg|A [[fractal flame]]
</gallery>

Biophilic fractals are patterns designed to induce the health and well-being benefits associated with exposure to nature's scenery.<ref>{{cite book|last=Wilson|first=Edward O.|title=Biophilia|year=1984|publisher=[[Harvard University Press]]|location=[[Cambridge, Massachusetts|Cambridge, MA]]|isbn=0-674-07442-4|url-access=registration|url=https://archive.org/details/biophilia00wils}}</ref> These include stress-reduction and enhanced cognitive capacity. Designers and architects incorporate biophilic fractals into the built environment to counter the fact that people spend 92% of their time indoors and away from nature's scenery. The Fractal Chapel designed by INNOCAD architecture in the state hospital in Graz, Austria, is a prominent example and recipient of the 2025 IIDA (International Interior Design Association) Best of Competition Award.

===Physiological responses: Fractal Fluency===

Fractal fluency is a [[neuroscience]] model that proposes that, through exposure to nature's fractal scenery, people's [[visual system]]s have adapted to efficiently process fractals with ease. This adaptation occurs at many stages of the visual system, from the way people's eyes move to which regions of the brain get activated.<ref> R.P. Taylor, C. Viengkham, J.H. Smith, C. Rowland, S. Moslehi, S. Stadlober, A. Lesjak, M. Lesjak, B. Spehar, "Fractal Fluency: Processing of Fractal Stimuli Across Sight, Sound and Touch", The Fractal Geometry of the Brain, Edition II, Advances in Neurobiology, vol 36. 907-934, Springer, 2024. </ref> Fluency puts the viewer in a 'comfort zone' so inducing an aesthetic experience. Neuroscience experiments have shown that Jackson Pollock's fractal paintings induce the same positive [[Physiology|physiological]] responses in the observer as nature's fractals and mathematical fractals.<ref name="R.P. Taylor, B. Spehar 2011">R.P. Taylor, B. Spehar, P. Van Donkelaar and C.M. Hagerhall, "Perceptual and Physiological Responses to Jackson Pollock's Fractals," Frontiers in Human Neuroscience, vol. 5 1- 13 (2011).</ref> This shows that fractal expressionism is related to fractal fluency by providing motivation for artists, such as Pollock, to use Fractal Expressionism in their art to appeal to people.

Humans appear to be especially well-adapted to processing fractal patterns with [[fractal dimension]] between 1.3 and 1.5.<ref> R.P. Taylor, "The Potential of Biophilic Fractal Designs to Promote Health and Performance: A Review of Experiments and Applications" Journal of Sustainability: Special edition "Architecture and Salutogenesis: Beyond Indoor Environmental Quality" vol. 13, 823 (2021) </ref> When humans view fractal patterns with fractal dimensions in this range, these fractals reduce physiological stress and boost cognitive abilities.<ref name="Taylor 2006">{{cite journal | last=Taylor | first=Richard P. | title=Reduction of Physiological Stress Using Fractal Art and Architecture | journal=Leonardo | volume=39 | issue=3 | year=2006 | pages=245–251 | doi=10.1162/leon.2006.39.3.245| s2cid=8495221 | url=https://zenodo.org/record/894740 }}</ref>

===Applications in technology===
{{Main|Fractal analysis}}

{{div col|colwidth=25em}}
*Fractal Bionics<ref> R.P. Taylor, "The Interdisciplinary Journey to Fractal Bionics" Invited Article for Physics Today, 44-50 December 2024 </ref>
* [[Fractal antenna]]s<ref name="antenna">{{cite journal |last1=Hohlfeld |first1=Robert G. |last2=Cohen |first2=Nathan |title=Self-similarity and the geometric requirements for frequency independence in Antennae |journal=Fractals |volume=7 |issue=1 |pages=79–84 |year=1999 |doi=10.1142/S0218348X99000098 }}</ref>
*Fractal transistor<ref name="Fractal transistor">{{cite book| last1=Reiner |first1=Richard |first2=Patrick |last2=Waltereit |first3=Fouad |last3=Benkhelifa |first4=Stefan |last4=Müller |first5=Herbert |last5=Walcher |first6=Sandrine |last6=Wagner |first7=Rüdiger |last7=Quay |first8=Michael |last8=Schlechtweg |first9=Oliver | last10=Ambacher| first10=O.|last9=Ambacher |date=2012 | title=2012 24th International Symposium on Power Semiconductor Devices and ICs | chapter=Fractal structures for low-resistance large area AlGaN/GaN power transistors |isbn=978-1-4577-1596-9 |doi=10.1109/ISPSD.2012.6229091 |pages=341–344 | s2cid=43053855 }}</ref>
* Fractal heat exchangers<ref>{{cite web|author1=Zhiwei Huang|author2=Yunho Hwang|author3=Vikrant Aute|author4=Reinhard Radermacher|title=Review of Fractal Heat Exchangers|date=2016|url=http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=2724&context=iracc|format=PDF|postscript='' International Refrigeration and Air Conditioning Conference''. Paper 1725}}</ref>
* Digital imaging
* Architecture<ref name="springer.com 9783319324241"/>
* Urban growth<ref>{{Cite journal | last1=Chen | first1=Yanguang | title=Modeling Fractal Structure of City-Size Distributions Using Correlation Functions | doi=10.1371/journal.pone.0024791| journal=PLOS ONE | volume=6 | issue=9 | article-number=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=October 21, 2007 |archive-url=https://web.archive.org/web/20071012223212/http://library.thinkquest.org/26242/full/ap/ap.html |archive-date=October 12, 2007 }}</ref>
* [[Categorisation|Classification]] of [[histopathology]] slides
* [[Fractal landscape]] or [[Coast]]line [[complexity]]
* Detecting 'life as we don't know it' by fractal analysis<ref>{{cite journal| url = http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9012687&fileId=S1473550413000177| title = "Detecting 'life as we don't know it' by fractal analysis"| journal = International Journal of Astrobiology| date = October 2013| volume = 12| issue = 4| pages = 314–320| doi = 10.1017/S1473550413000177| last1 = Azua-Bustos| first1 = Armando| last2 = Vega-Martínez| first2 = Cristian| hdl = 11336/26238| s2cid = 122793675| hdl-access = free}}</ref>
* Enzymes ([[Michaelis–Menten kinetics]])
* [[Algorithmic composition|Generation of new music]]
* [[Signal (information theory)|Signal]] and [[fractal compression|image compression]]
* Creation of digital photographic enlargements
* [[Fractal in soil mechanics]]
* [[Game Design|Computer and video game design]]
* [[Computer graphics]]
* [[Life|Organic]] environments
* [[Procedural generation]]
* [[Fractography]] and [[fracture mechanics]]
* [[SAXS|Small angle scattering theory of fractally rough systems]]
* [[T-shirt]]s and other fashion
* Generation of patterns for camouflage, such as [[MARPAT]]
* [[Digital sundial]]
* Technical analysis of price series
* [[Fractal dimension on networks|Fractals in networks]]
* Medicine<ref name="medicine" />
* [[Neuroscience]]<ref name="cerebellum" /><ref name="neuroscience" />
* [[Diagnostic imaging]]<ref name="diagnostic imaging" />
* [[Pathology]]<ref name="pathology">{{Cite journal | last1=Smith | first1=Robert F. | last2=Mohr | first2=David N. | last3=Torres | first3=Vicente E. | last4=Offord | first4=Kenneth P. | last5=Melton III | first5=L. Joseph 
| title=Renal insufficiency in community patients with mild asymptomatic microhematuria | journal=Mayo Clinic Proceedings | volume=64 | issue=4 | pages=409–414 | year=1989 | pmid=2716356 |doi=10.1016/s0025-6196(12)65730-9}}</ref><ref>{{Cite journal | last1=Landini | first1=Gabriel | title=Fractals in microscopy | doi=10.1111/j.1365-2818.2010.03454.x | journal=Journal of Microscopy | volume=241 | issue=1 | pages=1–8 | year=2011 | pmid= 21118245| s2cid=40311727 }}</ref>
* Geology<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>
* [[Geography]]<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 | article-number=e24791 | year=2011 | pmid=21949753 | pmc=3176775|arxiv = 1104.4682 |bibcode = 2011PLoSO...624791C | doi-access=free}}</ref>
* [[Archaeology]]<ref name="archaeology">{{cite journal | last1=Burkle-Elizondo | first1=Gerardo
| last2=Valdéz-Cepeda | first2=Ricardo David | title=Fractal analysis of Mesoamerican pyramids | journal=Nonlinear Dynamics, Psychology, and Life Sciences | volume=10 | issue=1 | pages=105–122 | year=2006 
| pmid=16393505}}</ref><ref>{{Cite journal | last1=Brown | first1=Clifford T. | last2=Witschey | first2=Walter R. T. | last3=Liebovitch | first3=Larry S. | title=The Broken Past: Fractals in Archaeology | doi=10.1007/s10816-005-2396-6 | journal=Journal of Archaeological Method and Theory | volume=12 | pages=37–78 | year=2005 | s2cid=7481018 }}</ref>
* [[Soil mechanics]]<ref name="soil" />
* [[Seismology]]<ref name="seismology" />
* [[Search and rescue]]<ref name="search and rescue">{{cite journal| title=An Algorithmic Approach to Generate After-disaster Test Fields for Search and Rescue Agents| first1=Panteha |last1=Saeedi |first2=Soren A. |last2=Sorensen | journal=Proceedings of the World Congress on Engineering 2009 | year=2009 |pages=93–98 | isbn=978-988-17-0125-1|url=https://www.iaeng.org/publication/WCE2009/WCE2009_pp93-98.pdf}}</ref>
* [[Morton order#Applications|Morton order]] space filling curves for [[GPU]] [[cache coherency]] in [[texture mapping]],<ref>{{cite web|title=GPU internals|url=https://fileadmin.cs.lth.se/cs/Personal/Michael_Doggett/pubs/doggett12-tc.pdf}}</ref><ref>{{cite web|title=sony patents|url=https://www.google.ch/patents/US20150287166?dq=morton+order+texture+swizzling}}</ref><ref>{{cite web|title = description of swizzled and hybrid tiled swizzled textures|url=https://news.ycombinator.com/item?id=2239173}}</ref> [[rasterisation]]<ref>{{cite web | title=US8773422B1 - System, method, and computer program product for grouping linearly ordered primitives | website=Google Patents | date=December 4, 2007 | url=https://patents.google.com/patent/US8773422 | access-date=December 28, 2019}}</ref><ref>{{cite web | title=US20110227921A1 - Processing of 3D computer graphics data on multiple shading engines | website=Google Patents | date=December 15, 2010 | url=https://www.google.ch/patents/US20110227921 | access-date=December 27, 2019}}</ref> and indexing of turbulence data.<ref>{{cite web |url=https://turbulence.pha.jhu.edu |title=Johns Hopkins Turbulence Databases}}</ref><ref>{{cite journal|last1=Li|first1=Y.|first2=E.|last2=Perlman|first3=M.|last3=Wang|first4=y.|last4=Yang|first5=C.|last5=Meneveau|first6=R.|last6=Burns|first7=S.|last7=Chen|first8=A.|last8=Szalay|first9=G.|last9=Eyink|title=A Public Turbulence Database Cluster and Applications to Study Lagrangian Evolution of Velocity Increments in Turbulence|journal=Journal of Turbulence|date=2008|volume=9|pages=N31|doi=10.1080/14685240802376389|arxiv=0804.1703|bibcode=2008JTurb...9...31L|s2cid=15768582}}</ref>
{{div col end}}

==See also==
{{Portal|Mathematics|Systems science}}
<!-- alphabetical order please [[WP:SEEALSO]] -->
<!-- please add a short description [[WP:SEEALSO]], via {{subst:AnnotatedListOfLinks}} or {{Annotated link}} -->
{{div col|colwidth=20em|small=yes}}
*{{Annotated link|Banach fixed point theorem}} 
*{{Annotated link|Bifurcation theory}}
*{{Annotated link|Box counting}}
*{{Annotated link|Cymatics}}
*{{Annotated link|Determinism}}
*{{Annotated link|Diamond-square algorithm}}
*{{Annotated link|Droste effect}}
*{{Annotated link|Feigenbaum function}}
*{{Annotated link|Form constant}}
*{{Annotated link|Fractal cosmology}}
*{{Annotated link|Fractal derivative}}
*{{Annotated link|Fractalgrid}}
*{{Annotated link|Fractal sequence}}
*{{Annotated link|Fractal string}}
*{{Annotated link|Fracton}}
*{{Annotated link|Graftal}}
*{{Annotated link|Greeble}}
*{{Annotated link|H tree}}
*{{Annotated link|Infinite regress}}
*{{Annotated link|Lacunarity}}
*{{Annotated link|List of fractals by Hausdorff dimension}} 
*{{Annotated link|Mandelbulb}}
*{{Annotated link|Mandelbox}}
*{{Annotated link|Macrocosm and microcosm}}
*{{Annotated link|Matryoshka doll}}
*{{Annotated link|Menger Sponge}}
*{{Annotated link|Multifractal system}}
*{{Annotated link|Newton fractal}}
*{{Annotated link|Percolation}}
*{{Annotated link|Power law}}
*{{Annotated link|List of important publications in mathematics#Geometry|Publications in fractal geometry}}
*{{Annotated link|Random walk}}
*{{Annotated link|Self-reference}}
*{{Annotated link|Self-similarity}}
*{{Annotated link|Systems theory}}
*{{Annotated link|Strange loop}}
*{{Annotated link|Turbulence}}
*{{Annotated link|Wiener process}}
{{div col end}}
<!-- alphabetical order please [[WP:SEEALSO]] -->

==Notes==
{{Reflist
|group=notes
|refs=
<ref name="space filling note">The Hilbert curve map is not a [[homeomorphism]], so it does not preserve topological dimension. The topological dimension and Hausdorff dimension of the image of the Hilbert map in '''R'''<sup>2</sup> are both 2. Note, however, that the topological dimension of the ''graph'' of the Hilbert map (a set in '''R'''<sup>3</sup>) is 1.</ref>
<ref name="levy note">The original paper, {{cite journal|last=Lévy |first=Paul |title=Les Courbes planes ou gauches et les surfaces composées de parties semblables au tout |journal=Journal de l'École Polytechnique |year=1938 |pages=227–247, 249–291 }}, is translated in [[#classics|Edgar]], pages 181–239.</ref>
}}

==References==
{{Reflist}}

==Further reading==

* Stanley, Eugene H, Ostrowsky, N. (editors); On Growth and Fractal Form <small>Fractal and Non-Fractal Patterns in Physics,</small>  Martinus Nijhoff Publisher, 1986. {{ISBN|0-89838-850-3}}
* Barnsley, Michael F.; and Rising, Hawley; ''Fractals Everywhere''. Boston: Academic Press Professional, 1993. {{isbn|0-12-079061-0}}
* Duarte, German A.; ''Fractal Narrative. About the Relationship Between Geometries and Technology and Its Impact on Narrative Spaces''. Bielefeld: Transcript, 2014. {{isbn|978-3-8376-2829-6}}
* Falconer, Kenneth; ''Techniques in Fractal Geometry''. John Wiley and Sons, 1997. {{isbn|0-471-92287-0}}
* Jürgens, Hartmut; [[Heinz-Otto Peitgen|Peitgen, Heinz-Otto]]; and Saupe, Dietmar; ''Chaos and Fractals: New Frontiers of Science''. New York: Springer-Verlag, 1992. {{isbn|0-387-97903-4}}
* [[Benoit Mandelbrot|Mandelbrot, Benoit B.]]; ''[[The Fractal Geometry of Nature]]''. New York: W. H. Freeman and Co., 1982. {{isbn|0-7167-1186-9}}
* Peitgen, Heinz-Otto; and Saupe, Dietmar; eds.; ''The Science of Fractal Images''. New York: Springer-Verlag, 1988. {{isbn|0-387-96608-0}}
* [[Clifford A. Pickover|Pickover, Clifford A.]]; ed.; ''Chaos and Fractals: A Computer Graphical Journey – A 10 Year Compilation of Advanced Research''. Elsevier, 1998. {{isbn|0-444-50002-2}}
* Jones, Jesse; ''Fractals for the Macintosh'', Waite Group Press, Corte Madera, CA, 1993. {{isbn|1-878739-46-8}}.
* Lauwerier, Hans; ''Fractals: Endlessly Repeated Geometrical Figures'', Translated by Sophia Gill-Hoffstadt, Princeton University Press, Princeton NJ, 1991. {{isbn|0-691-08551-X}}, cloth. {{isbn|0-691-02445-6}} paperback. "This book has been written for a wide audience..." Includes sample BASIC programs in an appendix.
* {{Cite book | last=Sprott | first=Julien Clinton | title=Chaos and Time-Series Analysis | publisher=Oxford University Press | year=2003 | isbn=978-0-19-850839-7 }}
* Wahl, Bernt; Van Roy, Peter; Larsen, Michael; and Kampman, Eric; [https://www.fractalexplorer.com ''Exploring Fractals on the Macintosh''], Addison Wesley, 1995. {{isbn|0-201-62630-6}}
* Lesmoir-Gordon, Nigel; ''The Colours of Infinity: The Beauty, The Power and the Sense of Fractals''. 2004. {{isbn|1-904555-05-5}} (The book comes with a related DVD of the [[Arthur C. Clarke]] documentary introduction to the fractal concept and the [[Mandelbrot set]].)
* Liu, Huajie; ''Fractal Art'', Changsha: Hunan Science and Technology Press, 1997, {{isbn|9787535722348}}.
* Gouyet, Jean-François; ''Physics and Fractal Structures'' (Foreword by B. Mandelbrot); Masson, 1996. {{isbn|2-225-85130-1}}, and New York: Springer-Verlag, 1996. {{isbn|978-0-387-94153-0}}. Out-of-print. Available in PDF version at.{{cite web |url=http://www.jfgouyet.fr/fractal/fractauk.html |title=Physics and Fractal Structures |language=fr |publisher=Jfgouyet.fr |access-date=October 17, 2010 }}
* {{Cite book |first1=Kenneth |last1=Falconer |title=Fractals, A Very Short Introduction |publisher=Oxford University Press |year=2013 }}

==External links==
{{Commons}}
{{Wikibooks|Fractals}}
* {{webarchive |url=http://webarchive.loc.gov/all/20011116035117/http://dmoz.org/science/math/chaos_and_fractals/ |title=Fractals |date=November 16, 2001}}
* "[https://www.pbs.org/wgbh/nova/physics/hunting-hidden-dimension.html Hunting the Hidden Dimension]", [[PBS]] ''[[Nova (American TV series)|NOVA]]'', first aired August 24, 2011
* [https://www.ted.com/talks/benoit_mandelbrot_fractals_and_the_art_of_roughness Benoit Mandelbrot: Fractals and the Art of Roughness] ([https://www.ted.com/talks/benoit_mandelbrot_fractals_and_the_art_of_roughness]), [[TED (conference)|TED]], February 2010
* [https://adsabs.harvard.edu/abs/2007PrGeo..22..451Y Equations of self-similar fractal measure based on the fractional-order calculus]（2007）
* [https://www.youtube.com/watch?v=Ixhg5FgVqGs Oriental Five Elements Fractal]

{{Fractals}}
{{Chaos theory}}
{{Mathematical art}}
{{Patterns in nature}}{{Geometry}}{{Authority control}}

[[Category:Fractals| ]]
[[Category:Computational fields of study]]
[[Category:Mathematical structures]]
[[Category:Topology]]