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Fractal

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Generating fractals

Four common techniques for generating fractals are:

* Escape-time fractals – (also known as "orbits" fractals) These are defined by a formula or recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal, the Nova fractal and the Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly.

* Iterated function systems – These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Highway dragon curve, T-Square, Menger sponge, are some examples of such fractals.
* Random fractals – Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Lévy flight, fractal landscapes and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters.
* Strange attractors – Generated by iteration of a map or the solution of a system of initial-value differential equations that exhibit chaos.

Classification

Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals:

* Exact self-similarity – This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity. For example, the Sierpinski triangle and Koch snowflake exhibit exact self-similarity.