Fraktal: Perbedaan antara revisi

Konsensus di kalangan ahli matematika adalah bahwa fraktal teoretis adalah konstruksi matematika [[Iterasi|yang berulang]] dan terperinci dengan kemiripan yang tak terhingga, yang banyak [[Daftar fraktal menurut dimensi Hausdorff|contohnya]] telah dirumuskan dan dipelajari.<ref name="Mandelbrot19833">{{Cite book|last=Mandelbrot|first=Benoît B.|year=1983|url=https://books.google.com/books?id=0R2LkE3N7-oC|title=The fractal geometry of nature|publisher=Macmillan|isbn=978-0-7167-1186-5}}</ref><ref name="Falconer">{{Cite book|last=Falconer|first=Kenneth|year=2003|title=Fractal Geometry: Mathematical Foundations and Applications|publisher=John Wiley & Sons|isbn=978-0-470-84862-3|pages=xxv|nopp=true}}</ref><ref name="patterns">{{Cite book|last=Briggs|first=John|year=1992|title=Fractals:The Patterns of Chaos|url=https://archive.org/details/fractalspatterns0000brig|location=London|publisher=Thames and Hudson|isbn=978-0-500-27693-8|page=[https://archive.org/details/fractalspatterns0000brig/page/148 148]}}</ref> Fraktal tidak terbatas pada pola geometris, tetapi juga dapat menggambarkan proses dalam waktu.<ref name="Gouyet">{{Cite book|last=Gouyet|first=Jean-François|year=1996|title=Physics and fractal structures|location=Paris/New York|publisher=Masson Springer|isbn=978-0-387-94153-0}}</ref><ref name="vicsek">{{Cite book|last=Vicsek|first=Tamás|year=1992|title=Fractal growth phenomena|url=https://archive.org/details/fractalgrowthphe0000vics_2edi|location=Singapore/New Jersey|publisher=World Scientific|isbn=978-981-02-0668-0|pages=31; 139–146}}</ref><ref name="time series">{{Cite book|last=Peters|first=Edgar|year=1996|title=Chaos and order in the capital markets : a new view of cycles, prices, and market volatility|location=New York|publisher=Wiley|isbn=978-0-471-13938-6}}</ref><ref>{{Cite journal|last=Krapivsky|first=P. L.|last2=Ben-Naim|first2=E.|year=1994|title=Multiscaling in Stochastic Fractals|journal=Physics Letters A|volume=196|issue=3–4|page=168|bibcode=1994PhLA..196..168K|doi=10.1016/0375-9601(94)91220-3}}</ref><ref>{{Cite journal|last=Hassan|first=M. K.|last2=Rodgers|first2=G. J.|year=1995|title=Models of fragmentation and stochastic fractals|journal=Physics Letters A|volume=208|issue=1–2|page=95|bibcode=1995PhLA..208...95H|doi=10.1016/0375-9601(95)00727-k}}</ref><ref>{{Cite journal|last=Hassan|first=M. K.|last2=Pavel|first2=N. I.|last3=Pandit|first3=R. K.|last4=Kurths|first4=J.|year=2014|title=Dyadic Cantor set and its kinetic and stochastic counterpart|journal=Chaos, Solitons & Fractals|volume=60|pages=31–39|arxiv=1401.0249|bibcode=2014CSF....60...31H|doi=10.1016/j.chaos.2013.12.010}}</ref> Pola fraktal dengan berbagai tingkat kemiripan diri telah dirender atau dipelajari dalam media visual, fisik, dan aural<ref name="music">{{Cite journal|last=Brothers|first=Harlan J.|year=2007|title=Structural Scaling in Bach's Cello Suite No. 3|journal=Fractals|volume=15|issue=1|pages=89–95|doi=10.1142/S0218348X0700337X}}</ref> dan ditemukan di [[Fraktal|alam]], <ref name="cerebellum">{{Cite journal|last=Liu|first=Jing Z.|last2=Zhang|first2=Lu D.|last3=Yue|first3=Guang H.|year=2003|title=Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging|journal=Biophysical Journal|volume=85|issue=6|pages=4041–4046|bibcode=2003BpJ....85.4041L|doi=10.1016/S0006-3495(03)74817-6|pmc=1303704|pmid=14645092}}</ref><ref name="neuroscience">{{Cite journal|last=Karperien|first=Audrey L.|last2=Jelinek|first2=Herbert F.|last3=Buchan|first3=Alastair M.|year=2008|title=Box-Counting Analysis of Microglia Form in Schizophrenia, Alzheimer's Disease and Affective Disorder|journal=Fractals|volume=16|issue=2|pages=103|doi=10.1142/S0218348X08003880}}</ref><ref name="branching">{{Cite book|last=Jelinek|first=Herbert F.|last2=Karperien|first2=Audrey|last3=Cornforth|first3=David|last4=Cesar|first4=Roberto|last5=Leandro|first5=Jorge de Jesus Gomes|year=2002|url=https://books.google.com/books?id=FFSUGQAACAAJ|title=Workshop proceedings: the Sixth Australia-Japan Joint Workshop on Intelligent and Evolutionary Systems, University House, ANU|publisher=University of New South Wales|isbn=978-0-7317-0505-4|editor-last=Sarker, Ruhul|chapter=MicroMod-an L-systems approach to neural modelling|oclc=224846454|quote=Event location: Canberra, Australia|access-date=February 3, 2012}}</ref> [[Fraktal|teknologi]], <ref name="soil">{{Cite journal|last=Hu|first=Shougeng|last2=Cheng|first2=Qiuming|last3=Wang|first3=Le|last4=Xie|first4=Shuyun|year=2012|title=Multifractal characterization of urban residential land price in space and time|journal=Applied Geography|volume=34|pages=161–170|bibcode=2012AppGe..34..161H|doi=10.1016/j.apgeog.2011.10.016}}</ref><ref name="diagnostic imaging">{{Cite journal|last=Karperien|first=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|year=2008|title=Automated detection of proliferative retinopathy in clinical practice|journal=Clinical Ophthalmology|volume=2|issue=1|pages=109–122|doi=10.2147/OPTH.S1579|pmc=2698675|pmid=19668394}}</ref><ref name="medicine">{{Cite book|last=Losa|first=Gabriele A.|last2=Nonnenmacher|first2=Theo F.|year=2005|url=https://books.google.com/books?id=t9l9GdAt95gC|title=Fractals in biology and medicine|publisher=Springer|isbn=978-3-7643-7172-2}}</ref> [[Fraktal|seni]],<ref name="novel">{{Cite web|last=Wallace|first=David Foster|date=August 4, 2006|title=Bookworm on KCRW|url=http://www.kcrw.com/etc/programs/bw/bw960411david_foster_wallace|publisher=Kcrw.com|archive-url=https://web.archive.org/web/20101111033857/http://www.kcrw.com/etc/programs/bw/bw960411david_foster_wallace|archive-date=November 11, 2010|access-date=October 17, 2010|url-status=dead}}</ref><ref name="African art">{{Cite web|last=Eglash|first=Ron|year=1999|title=African Fractals: Modern Computing and Indigenous Design|url=http://www.rpi.edu/~eglash/eglash.dir/afractal/afractal.htm|publisher=Rutgers University Press|location=New Brunswick|archive-url=https://web.archive.org/web/20180103005701/http://homepages.rpi.edu/~eglash/eglash.dir/afractal/afbook.htm|archive-date=January 3, 2018|access-date=October 17, 2010|url-status=dead}}</ref> dan [[arsitektur]].<ref name="springer.com 9783319324241">Ostwald, Michael J., and Vaughan, Josephine (2016) ''[[Dimensi Fraktal Arsitektur|The Fractal Dimension of Architecture]]'' Birhauser, Basel. {{Doi|10.1007/978-3-319-32426-5}}.</ref> Fraktal memiliki relevansi khusus dalam bidang [[Teori kekacauan|teori chaos]] karena mereka muncul dalam penggambaran geometris dari sebagian besar proses chaos (biasanya sebagai penarik atau sebagai batas antara cekungan tarikan).<ref>{{Cite web|last=Baranger|first=Michael|title=Chaos, Complexity, and Entropy: A physics talk for non-physicists|url=http://necsi.edu/projects/baranger/cce.pdf}}</ref>

Dimulai pada abad ke-17 dengan gagasan [[rekursi]], fraktal telah beralih melalui perlakuan matematis yang semakin ketat hingga mempelajari fungsi [[Fungsi kontinu|kontinu]] tetapi tidak [[Fungsi terdiferensialkan|terdiferensiasi]] pada abad ke-19 oleh karya penting [[Bernard Bolzano]], [[Georg Friedrich Bernhard Riemann|Bernhard Riemann]], dan [[Karl Weierstrass]],<ref>{{Cite journal|last=Segal|first=S. L.|date=June 1978|title=Riemann's example of a continuous 'nondifferentiable' function continued|journal=The Mathematical Intelligencer|volume=1|issue=2|pages=81–82|doi=10.1007/BF03023065}}</ref> dan hingga munculnya kata ''[[wiktionary:fractal|fraktal]]'' pada abad ke-20 yang kemudian diikuti dengan berkembangnya minat terhadap fraktal dan pemodelan berbasis komputer pada abad ke-20.<ref name="classics">{{Cite book|last=Edgar|first=Gerald|year=2004|title=Classics on Fractals|url=https://archive.org/details/classicsonfracta0000unse|location=Boulder, CO|publisher=Westview Press|isbn=978-0-8133-4153-8}}</ref><ref name="MacTutor">{{Cite web|last=Trochet|first=Holly|year=2009|title=A History of Fractal Geometry|url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/fractals.html|website=MacTutor History of Mathematics|archive-url=https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html|archive-date=March 12, 2012|url-status=dead}}</ref>

:* [[Fraktal acak]] — Dihasilkan melalui [[proses stokastik]], misalnya [[landskap fraktal]] dan [[penerbangan Lévy]].

Benda-benda yang mendekati fraktal bisa ditemukan dengan mudah di alam. Benda-benda tesebut menunjukkan struktur frakral yang kompleks pada skala tertentu. Contohnya adalah awan, gunung, jaringan sungai, dan sistem [[pembuluh darah]].

Fraktal biasanya digambar oleh komputer dengan [[perangkat lunak]] fraktal. Lihat daftarnya di bawah.