Fractal

A fractal is a geometric object which is rough or irregular on all scales of length, and so which appears to be 'broken up' in a radical way. Some of the best examples can be divided into parts, each of which is similar to the original object. Fractals are said to possess infinite detail, and they may actually have a self-similar structure that occurs at different levels of magnification. In many cases, a fractal can be generated by a repeating pattern, in a typically recursive or iterative process. The term fractal was coined in 1975 by Benoît Mandelbrot, from the Latin fractus or "broken". Before Mandelbrot coined his term, the common name for such structures (the Koch snowflake, for example) was monster curve.

Examples

Some common examples of fractals include the Mandelbrot set, Lyapunov fractal, Cantor set, Sierpinski gasket and carpet, Menger sponge, dragon curve, Peano curve, limit sets of Kleinian groups, and the Koch curve. Fractals can be deterministic or stochastic (i.e. non-deterministic). Chaotic dynamical systems are often (if not always) associated with fractals. The Mandelbrot set contains whole discs, so has dimension 2. This is not surprising. What is truly surprising is that the boundary of the Mandelbrot set also has a Hausdorff dimension of 2.

Related Topics:
Mandelbrot set - Lyapunov fractal - Cantor set - Sierpinski gasket - Carpet - Menger sponge - Dragon curve - Peano curve - Kleinian group - Koch curve - Deterministic - Stochastic - Chaotic dynamical systems - Boundary

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A relatively simple class of examples is the Cantor sets, in which short and then shorter (open) intervals are struck out of the unit interval , leaving a set that might (or might not) actually be self-similar under enlargement, and might (or might not) have dimension d that has 0 < d < 1. A simple recipe, such as excluding the digit 7 from decimal expansions, is self-similar under 10-fold enlargement, and also has dimension log 9/log 10 (this value is the same, no matter what logarithmic base is chosen), showing the connection of the two concepts.

Related Topics:
Cantor set - Unit interval - Digit - Decimal expansion - Enlargement - Logarithm

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