Fractal curve

Mathematical curve whose shape is a fractal

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, that is, its graph takes the form of a fractal.[1] In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite length — and every subarc longer than a single point has infinite length.[2]

A famous example is the boundary of the Mandelbrot set.

## Fractal curves in nature

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Fractal curves and fractal patterns are widespread, in nature, found in such places as broccoli, snowflakes, feet of geckos, frost crystals, and lightning bolts.[3][4][5][6]

See also Romanesco broccoli, dendrite crystal, trees, fractals, Hofstadter's butterfly, Lichtenberg figure, and self-organized criticality.

## Dimensions of a fractal curve

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Most of us are used to mathematical curves having dimension one, but as a general rule, fractal curves have different dimensions,[7] also see fractal dimension and list of fractals by Hausdorff dimension.

Zooming in on the Mandelbrot set

## Relationships of fractal curves to other fields

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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.[1]

## Examples

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* 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

## See also

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* The Beauty of Fractals
* Fractal antenna
* Fractal expressionism
* Fractal landscape
* Hexaflake
* Mosely snowflake
* Newton fractal
* Orbit trap
* Quasicircle
* The Fractal Geometry of Nature

## References

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1. ^ a b "Geometric and topological recreations".
2. ^ Ritzenthaler, Chella. "Fractal Curves" (PDF).
3. ^ McNally, Jess. "Earth's Most Stunning Natural Fractal Patterns". Wired. wired.com. Retrieved 17 May 2020.
4. ^ Tennenhouse, Erica (July 5, 2016). "8 Stunning Fractals Found in Nature".
5. ^ LaMonica, Martin (March 30, 2017). "Fractal patterns in nature and art are aesthetically pleasing and stress-reducing".
6. ^ Gunther, Shea (April 24, 2013). "14 amazing fractals found in nature". Retrieved 2020-05-17.
7. ^ Bogomolny, Alexander. "Fractal Curves and Dimension". cut-the-knot.

## External links

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* Wolfram math on fractal curves
* The Fractal Foundation's homepage
* fractalcurves.com
* Making a Kock Snowflake, from Khan Academy
* Area of a Koch Snowflake, from Khan Academy
* Youtube on space-filling curves
* Youtube on the Dragon Curve

* v
* t
* e

Fractals

Characteristics|

* Fractal dimensions
* Assouad
* Box-counting
* Higuchi
* Correlation
* Hausdorff
* Packing
* Topological
* Recursion
* Self-similarity

Iterated function
system|

* Barnsley fern
* Cantor set
* Koch snowflake
* Menger sponge
* Sierpiński carpet
* Sierpiński triangle
* Apollonian gasket
* Fibonacci word
* Space-filling curve
* Blancmange curve
* De Rham curve
* Minkowski
* Dragon curve
* Hilbert curve
* Koch curve
* Lévy C curve
* Moore curve
* Peano curve
* Sierpiński curve
* Z-order curve
* String
* T-square
* n-flake
* Vicsek fractal
* Gosper curve
* Pythagoras tree
* Weierstrass function

Strange attractor|

* Multifractal system

L-system|

* Fractal canopy
* Space-filling curve
* H tree

Escape-time
fractals|

* Burning Ship fractal
* Julia set
* Filled
* Newton fractal
* Douady rabbit
* Lyapunov fractal
* Mandelbrot set
* Misiurewicz point
* Multibrot set
* Newton fractal
* Tricorn
* Mandelbox
* Mandelbulb

Rendering techniques|

* Buddhabrot
* Orbit trap
* Pickover stalk

Random fractals|

* Brownian motion
* Brownian tree
* Brownian motor
* Fractal landscape
* Lévy flight
* Percolation theory
* Self-avoiding walk

People|

* Michael Barnsley
* Georg Cantor
* Bill Gosper
* Felix Hausdorff
* Desmond Paul Henry
* Gaston Julia
* Niels Fabian Helge von Koch
* Paul Lévy
* Aleksandr Lyapunov
* Benoit Mandelbrot
* Hamid Naderi Yeganeh
* Lewis Fry Richardson
* Wacław Sierpiński

Other|

* Coastline paradox
* Fractal art
* List of fractals by Hausdorff dimension
* The Fractal Geometry of Nature (1982 book)
* The Beauty of Fractals (1986 book)
* Chaos: Making a New Science (1987 book)
* Kaleidoscope
* Chaos theory

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