# Difference between revisions of "Field:Fractals"

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|Field=Fractals | |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." | |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." | ||

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+ | ===Basic Description of Fractals=== | ||

+ | |||

+ | *General description | ||

+ | **history, self-similarity, iterating | ||

+ | |||

+ | |||

+ | ===Further Information=== | ||

+ | |||

+ | *History | ||

+ | **Mandelbrot 1975 | ||

+ | |||

+ | *Self-similarity | ||

+ | |||

+ | *Iterating | ||

+ | **complex, <math>z = z^2 + c\,</math> | ||

+ | ***to zero = black | ||

+ | ***infinity - color, how fast is what color | ||

+ | |||

+ | *Fractal dimension | ||

+ | |||

+ | |||

+ | |||

+ | *Examples | ||

+ | **Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, space-filling curve, Koch curve, Lyapunov fractal, and Kleinian groups. | ||

+ | |||

+ | *Nature | ||

+ | |||

+ | ===Links to Other Information=== | ||

+ | *[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 | ||

+ | |||

+ | |||

+ | |||

|FurtherInfo='''Generating fractals''' | |FurtherInfo='''Generating fractals''' | ||

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:* 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. | :* 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] | :* 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] | ||

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|Links=http://en.wikipedia.org/wiki/Fractals | |Links=http://en.wikipedia.org/wiki/Fractals | ||

}} | }} |

## Revision as of 15:01, 29 May 2009

# Fractals

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."

### Basic Description of Fractals

- General description
- history, self-similarity, iterating

### Further Information

- History
- Mandelbrot 1975

- Self-similarity

- Iterating
- complex,
- to zero = black
- infinity - color, how fast is what color

- complex,

- Fractal dimension

- Examples
- Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, space-filling curve, Koch curve, Lyapunov fractal, and Kleinian groups.

- Nature

### Links to Other Information

- Link to image 1
- Link to image 2
- Fractal from Wolfram MathWorld
- Fractal from Wikipedia

[[Image:|300px|thumb|right|]] |

## Links to Other Information

http://en.wikipedia.org/wiki/Fractals