Difference between revisions of "Field:Fractals"

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*General description
*General description
**history, self-similarity, iterating
**history, self-similarity, iterating
|FurtherInfo='''Generating fractals'''
Three common techniques for generating fractals are:
:* Escape-time fractals — (also known as "orbits" fractals) These are defined by a 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-Heighway 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. --[www.wikipedia.org| Wikipedia]
**Mandelbrot 1975
**complex, <math>z = z^2 + c\,</math>
***to zero = black
***infinity - color, how fast is what color
*Fractal dimension
**Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, space-filling curve, Koch curve, Lyapunov fractal, and Kleinian groups.
*[http://support1.mathforum.org/~swatimage/Interactive/Puzzle.htm Link to image 1]
*[http://support1.mathforum.org/~swatimage/Dynamic_Systems/7.htm Link to image 2]
*[http://mathworld.wolfram.com/Fractal.html Fractal] from Wolfram MathWorld
*[http://en.wikipedia.org/wiki/Fractal Fractal] from Wikipedia

Revision as of 09:12, 1 June 2009

{{Field Page |Field=Fractals |BasicDesc=A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,"[1] a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured."

  • General description
    • history, self-similarity, iterating