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Concept# Fractal curve

Summary

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. 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.
A famous example is the boundary of the Mandelbrot set.
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.
Most of us are used to mathematical curves having dimension one, but as a general rule, fractal curves have different dimensions, also see also fractal dimension and list of fractals by Hausdorff dimension.
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.
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
Trees
Natural fractals
Weierstrass function
Wolfram math on fractal curves
The Fractal Foundation's homepage
fractalcurves.

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Menger sponge

In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension. The construction of a Menger sponge can be described as follows: Begin with a cube. Divide every face of the cube into nine squares, like Rubik's Cube.

Fractal dimension

In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured. It is also a measure of the space-filling capacity of a pattern, and it tells how a fractal scales differently, in a fractal (non-integer) dimension. The main idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions.

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