Editing Fractal (section)

{{Short description|Infinitely detailed mathematical structure}}
{{Other uses}}
[[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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[[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 lies within the mathematical branch of [[measure theory]].

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 | pages=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>