Editing Fractal (section)

==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 |pages=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 | 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 | pages=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 |pages=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 |pages=e2021GL093366 |doi=10.1029/2021GL093366| issn=0094-8276 |bibcode=2021GeoRL..4893366S |s2cid=235508504 |url=http://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 |pages=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|pages=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}}</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|pages=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>
{{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. 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}}</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 |page=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|pages=012406|doi=10.1103/PhysRevE.98.012406|pmid=30110830|bibcode=2018PhRvE..98a2406S|s2cid=52010780|issn=2470-0045}}</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}}

Since 1999 numerous scientific groups have performed fractal analysis on over 50 paintings created by [[Jackson Pollock]] by pouring paint directly onto horizontal canvasses.<ref>{{cite journal |first=R. P. |last=Taylor |display-authors=et al |title=Fractal Analysis of Pollock's Drip Paintings |journal=Nature |volume=399 |issue=6735 |page=422 |year=1999|bibcode=1999Natur.399..422T |doi=10.1038/20833 |s2cid=204993516 |doi-access=free }}</ref><ref>{{cite journal |first1=R. P. |last1=Taylor |display-authors=etal |title=Fractal Analysis: Revisiting Pollock's Paintings (Reply)|journal=Nature |volume=444 |issue=7119 |pages=E10–11 |year=2006 |doi=10.1038/nature05399|bibcode=2006Natur.444E..10T |s2cid=31353634 }}</ref><ref>{{cite journal |first1=S. |last1=Lee |first2=S. |last2=Olsen |first3=B. |last3=Gooch |title=Simulating and Analyzing Jackson Pollock's Paintings |journal=Journal of Mathematics and the Arts |volume=1 |issue=2 |pages=73–83 |year=2007 |doi=10.1080/17513470701451253|citeseerx=10.1.1.141.7470 |s2cid=8529592 }}</ref>

Recently, fractal analysis has been 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 |url-status=dead }}</ref> Cognitive neuroscientists have shown that Pollock's fractals induce the same stress-reduction in observers as computer-generated fractals and Nature's fractals.<ref>{{cite journal |first1=R. P. |last1=Taylor |first2=B. |last2=Spehar |first3=P. |last3=Van Donkelaar |first4=C. M. |last4=Hagerhall |title=Perceptual and Physiological Responses to Jackson Pollock's Fractals |journal=Frontiers in Human Neuroscience |volume=5 |pages=1–13 |year=2011 |doi=10.3389/fnhum.2011.00060|pmid=21734876 |pmc=3124832 |doi-access=free }}</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 |url-status=dead }}</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 |url-status=dead }}</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.

Aesthetics and Psychological Effects of Fractal Based Design:<ref>{{Cite journal |last1=Robles |first1=Kelly E. |last2=Roberts |first2=Michelle |last3=Viengkham |first3=Catherine |last4=Smith |first4=Julian H. |last5=Rowland |first5=Conor |last6=Moslehi |first6=Saba |last7=Stadlober |first7=Sabrina |last8=Lesjak |first8=Anastasija |last9=Lesjak |first9=Martin |last10=Taylor |first10=Richard P. |last11=Spehar |first11=Branka |last12=Sereno |first12=Margaret E. |date=2021 |title=Aesthetics and Psychological Effects of Fractal Based Design |journal=Frontiers in Psychology |volume=12 |doi=10.3389/fpsyg.2021.699962 |pmid=34484047 |pmc=8416160 |issn=1664-1078 |doi-access=free }}</ref> Highly prevalent in nature, fractal patterns possess self-similar components that repeat at varying size scales. The perceptual experience of human-made environments can be impacted with inclusion of these natural patterns. Previous work has demonstrated consistent trends in preference for and complexity estimates of fractal patterns. However, limited information has been gathered on the impact of other visual judgments. Here we examine the aesthetic and perceptual experience of fractal ‘global-forest’ designs already installed in humanmade spaces and demonstrate how fractal pattern components are associated with positive psychological experiences that can be utilized to promote occupant well-being. These designs are composite fractal patterns consisting of individual fractal ‘tree-seeds’ which combine to create a ‘global fractal forest.’ The local ‘tree-seed’ patterns, global configuration of tree-seed locations, and overall resulting ‘global-forest’ patterns have fractal qualities. These designs span multiple mediums yet are all intended to lower occupant stress without detracting from the function and overall design of the space. In this series of studies, we first establish divergent relationships between various visual attributes, with pattern complexity, preference, and engagement ratings increasing with fractal complexity compared to ratings of refreshment and relaxation which stay the same or decrease with complexity. Subsequently, we determine that the local constituent fractal (‘tree-seed’) patterns contribute to the perception of the overall fractal design, and address how to balance aesthetic and psychological effects (such as individual experiences of perceived engagement and relaxation) in fractal design installations. This set of studies demonstrates that fractal preference is driven by a balance between increased arousal (desire for engagement and complexity) and decreased tension (desire for relaxation or refreshment). Installations of these composite mid-high complexity ‘global-forest’ patterns consisting of ‘tree-seed’ components balance these contrasting needs, and can serve as a practical implementation of biophilic patterns in human-made environments to promote occupant well-being.

{{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>

===Physiological responses===
Humans appear to be especially well-adapted to processing fractal patterns with [[fractal dimension]] between 1.3 and 1.5.<ref>{{cite book |chapter=Fractal Fluency: An Intimate Relationship Between the Brain and Processing of Fractal Stimuli |last=Taylor |first=Richard P. |pages=485–496 |title=The Fractal Geometry of the Brain |editor-last=Di Ieva |editor-first=Antonio |date=2016 |publisher=Springer |series=Springer Series in Computational Neuroscience |isbn=978-1-4939-3995-4}}</ref> When humans view fractal patterns with fractal dimension between 1.3 and 1.5, this tends to reduce physiological stress.<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><ref>For further discussion of this effect, see {{cite journal | last1=Taylor | first1=Richard P. | last2=Spehar | first2=Branka | last3=Donkelaar | first3=Paul Van | last4=Hagerhall | first4=Caroline M. | title=Perceptual and Physiological Responses to Jackson Pollock's Fractals | journal=Frontiers in Human Neuroscience | volume=5 | pages=60 | year=2011 | doi=10.3389/fnhum.2011.00060| pmid=21734876 | pmc=3124832 | doi-access=free }}</ref>

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

{{div col|colwidth=25em}}
* [[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 | pages=e24791 | year=2011 | pmid=21949753 | pmc=3176775|arxiv = 1104.4682 |bibcode = 2011PLoSO...624791C | doi-access=free }}</ref><ref>{{cite web |url=http://library.thinkquest.org/26242/full/ap/ap.html |title=Applications |access-date=October 21, 2007 |url-status=dead |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 | pages=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=http://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=http://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=http://www.google.ch/patents/US20110227921 | access-date=December 27, 2019}}</ref> and indexing of turbulence data.<ref>{{cite web |url=http://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}}