Editing Fractal (section)

==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 
 |url-status=dead
}}</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}}

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[[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|url-status=dead}}</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>

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