Fractal 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]
An extremely famous example is the boundary of the Mandelbrot set.
Fractal curves 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.[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[]
Most of us are used to mathematical curves having dimension one, but as a general rule, fractal curves have different dimensions,[7] also see also fractal dimension and list of fractals by Hausdorff dimension.
Relationships of fractal curves 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.[1]
Examples[]
See also[]
References[]
- ^ a b "Geometric and topological recreations".
- ^ Ritzenthaler, Chella. "Fractal Curves" (PDF).
- ^ "Earth's Most Stunning Natural Fractal Patterns". Earth's Most Stunning Natural Fractal Patterns. wired.com. Retrieved 17 May 2020.
- ^ Tennenhouse, Erica (July 5, 2016). "8 Stunning Fractals Found in Nature".
- ^ LaMonica, Martin (March 30, 2017). "Fractal patterns in nature and art are aesthetically pleasing and stress-reducing".
- ^ Gunther, Shea (April 24, 2013). "14 amazing fractals found in nature". Retrieved 2020-05-17.
- ^ Bogomolny, Alexander. "Fractal Curves and Dimension". cut-the-knot.
External links and references[]
- 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
- Fractal curves
- Types of functions
- Geometry stubs