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
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 fractal dimension and list of fractals by Hausdorff dimension .
Zooming in on the Mandelbrot set
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
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