{{Short description|Mathematical curve whose shape is a fractal}}
[[File:Gosper 6.gif|thumb|right|upright=2.25|Construction of the Gosper curve]]
A '''fractal curve''' is loosely a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified or scaled, that is, its graph is either space-filling or takes the form of a fractal.<ref name="Geometric and topological recreations">{{Cite web |title=Geometric and topological recreations |url=https://www.britannica.com/topic/number-game/Geometric-and-topological-recreations#ref396163}}</ref>
In general, fractal curves are nowhere rectifiable — that is, they do not have finite length — and every subarc longer than a single point has infinite length.<ref name="Fractal Curves">{{Cite web |title=Fractal Curves |url=https://www.whitman.edu/Documents/Academics/Mathematics/ritzenmc.pdf |last=Ritzenthaler |first=Chella}}</ref> A famous example is the boundary of the Mandelbrot set.
==In nature==
Fractal curves and fractal patterns are widespread in nature, found in such places as broccoli, snowflakes, feet of geckos, frost crystals, and lightning bolts.<ref name="Earth's Most Stunning Natural Fractal Patterns">{{cite magazine |title=Earth's Most Stunning Natural Fractal Patterns| magazine=Wired | url=https://www.wired.com/2010/09/fractal-patterns-in-nature/ | publisher=wired.com | accessdate=17 May 2020| last1=McNally | first1=Jess }}</ref><ref name="8 Stunning Fractals Found in Nature">{{Cite web |title=8 Stunning Fractals Found in Nature |url=http://thescienceexplorer.com/nature/8-stunning-fractals-found-nature |last=Tennenhouse |first=Erica |date=July 5, 2016}}</ref><ref name="Fractal patterns in nature and art are aesthetically pleasing and stress-reducing">{{Cite web |title=Fractal patterns in nature and art are aesthetically pleasing and stress-reducing |url=https://theconversation.com/fractal-patterns-in-nature-and-art-are-aesthetically-pleasing-and-stress-reducing-73255 |last=LaMonica |first=Martin |editor-first1=Maggie |editor-last1=Villiger |date=March 30, 2017 |doi=10.64628/AAI.v7dkdpjs4 }}</ref><ref name="14 amazing fractals found in nature">{{Cite web |title=14 amazing fractals found in nature |url=https://www.mnn.com/earth-matters/wilderness-resources/blogs/14-amazing-fractals-found-in-nature |last=Gunther |first=Shea |date=April 24, 2013 |access-date=2020-05-17}}</ref>
See also Romanesco broccoli, dendrite crystal, trees, fractals, Hofstadter's butterfly, Lichtenberg figure, and self-organized criticality.
==Dimension==
Mathematical curves are one dimensional spaces. However, fractal curves have different fractal dimension or Hausdorff dimension<ref name="Fractal Curves and Dimension">{{Cite web |title=Fractal Curves and Dimension |url=https://www.cut-the-knot.org/do_you_know/dimension.shtml |last=Bogomolny |first=Alexander |website=cut-the-knot}}</ref> (see list of fractals by Hausdorff dimension).
[[Image:Mandelbrot sequence new.gif|thumb|250px|left|Zooming in on the Mandelbrot set]]
==Relationship to other fields==
Starting in the 1950s, Benoit Mandelbrot and others have studied self-similarity of fractal curves, and have applied theory of fractals to modelling natural phenomena. Self-similarity occurs, and analysis of these patterns has found fractal curves in such diverse fields as economics, fluid mechanics, geomorphology, human physiology and linguistics.
As examples, "landscapes" revealed by microscopic views of surfaces in connection with Brownian motion, vascular networks, and shapes of polymer molecules all relate to fractal curves.<ref name="Geometric and topological recreations" />
==Examples== {{div col}} * Blancmange curve * Coastline paradox * De Rham curve * Dragon curve * Fibonacci word fractal * Koch snowflake * Boundary of the Mandelbrot set * Menger sponge * Peano curve * Sierpiński triangle * Weierstrass function {{div col end}}
==See also== {{div col}} * ''The Beauty of Fractals'' * Fractal antenna * Fractal expressionism * Fractal landscape * Hexaflake * Mosely snowflake * Newton fractal * Orbit trap * Quasicircle * ''The Fractal Geometry of Nature'' {{div col end}}
==References==
{{Reflist}}
==External links==
* [https://demonstrations.wolfram.com/FractalCurves/ Wolfram math on fractal curves] * [https://fractalfoundation.org/ The Fractal Foundation's homepage] * [http://www.fractalcurves.com/ fractalcurves.com] * [https://www.khanacademy.org/math/geometry-home/geometry-volume-surface-area/koch-snowflake/v/koch-snowflake-fractal Making a Koch Snowflake, from Khan Academy] * [https://www.khanacademy.org/math/geometry-home/geometry-volume-surface-area/koch-snowflake/v/area-of-koch-snowflake-part-1-advanced Area of a Koch Snowflake, from Khan Academy] * [https://www.youtube.com/watch?v=RU0wScIj36o Youtube on space-filling curves] * [https://www.youtube.com/watch?v=UBuPWdSbyf8 Youtube on the Dragon Curve]
{{Fractals}}
Category:Fractal curves Category:Types of functions