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,

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 takes the form of a fractal.

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. 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.

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 (see list of fractals by Hausdorff dimension).

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.

Examples

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