Editing Fractal (section)

==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=http://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]].