Difference between revisions of "Field:Fractals"

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<gallery caption="" widths="150px" heights="150px" perrow="5">
 
<gallery caption="" widths="150px" heights="150px" perrow="5">
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Image:Fractal_Broccoli.jpg|Romanesco broccoli (''Natural Fractal'')
 
Image:DragonCurve.jpg|Dragon Curve (''IFS'')
 
Image:DragonCurve.jpg|Dragon Curve (''IFS'')
Image:Henon2.jpg|Henon Attractor (''Strange Attractor'')
 
 
Image:LorenzAttractor.png|Lorenz Attractor (''Strange Attractor'')
 
Image:LorenzAttractor.png|Lorenz Attractor (''Strange Attractor'')
 
Image:BrownianTree.png|Brownian Tree(''Random Fractal'')
 
Image:BrownianTree.png|Brownian Tree(''Random Fractal'')
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:'''Iterated function systems (IFS)''' {{Hide|
 
:'''Iterated function systems (IFS)''' {{Hide|
::Fractals created with IFS are always exactly self-similarity, so that these fractals are essential constructed of infinitely many infinitely smaller pieces of itself that overlap. The IFS consists of one of more functions or methods that describe the behavior of the fractal.
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::Fractals created with IFS are always exactly self-similarity, so that these fractals are essential constructed of infinitely many infinitely smaller pieces of itself that overlap. The IFS consists of one of more functions or methods that describe the behavior of the fractal. Many of these images are generated by IFS governed by the ‘’chaos game’’. This is where the IFS consists of a set of functions that each have a fixed probability. A starting point is picked at random, and then one function is picked (according to probability) and applied to the point for an infinite amount of iterations.
::Fractals are often generated by IFS governed by the ‘’chaos game’’. This is where the IFS consists of a set of functions that each have a fixed probability. A starting point is picked at random, and then one function is picked (according to probability) and applied to the point for an infinite amount of iterations.
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::*Examples include: [[Koch’s Snowflake]], [[Harter-Heighway Dragon]], [[Barnsley’s Fern]], and [[Sierpinski's Triangle]].
::*Examples include: [[Koch’s Snowflake]], [[Harter-Heighway Dragon]], [[Barnsley’s Fern]], and [[Sierpinski’s Triangle]].
 
 
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: '''Escape-time (“orbit”) fractals''' {{Hide|
 
: '''Escape-time (“orbit”) fractals''' {{Hide|
::Escape-time fractals are created in the complex plane with a single function, such as <math>f(z) = z^2 + c</math>, where ''z'' is a complex number and ''c'' is any number.  On a computer, each pixel corresponds to a complex number value (click here for more information about [[Complex Numbers]]). Each complex number value is iterated into the function until it reaches infinite or until it is clear that value will converge to zero. The colors assigned to each complex number value or pixel are black if the value converges to zero or a color based on the number of iterations (aka. escape time) it took for the values to reach infinity.
+
::Escape-time fractals are created in the complex plane with a single function, such as <math>f(z) = z^2 + c</math>, where ''z'' is a complex number and ''c'' is any number.  On a computer, each pixel corresponds to a complex number value (click here for more information about [[Complex Numbers]]). Each complex number value is iterated into the function until it reaches infinite or until it is clear that value will converge to zero. The colors assigned to each complex number value or pixel are black if the value converges to zero or a color based on the number of iterations (aka. escape time) it took for the values to reach infinity. The boundary between black and colorful pixels is infinite and increasing complex. And the intermediary numbers that arise from the iterations are referred to as their “orbit”.
::The boundary between black and colorful pixels is infinite and increasing complex. And the intermediary numbers that arise from the iterations are referred to as their “orbit”.
 
 
::*Examples include: [[Mandelbrot Set]], [[Julia Sets]], and [[Lyapunov Fractal]].
 
::*Examples include: [[Mandelbrot Set]], [[Julia Sets]], and [[Lyapunov Fractal]].
 
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Revision as of 14:51, 1 June 2009


Fractals

NorwayCoastline.png

A fractal is often defined as a geometry shape that is self-similarity>, or whose magnified parts look like a smaller copy of the whole. It was coined by Benoit Mandelbolt in 1975 from the latin term ‘’fractus’’ meaning ”fragmented” or “irregular”.

This concept can be explained in a commonly used nature analogy involving the coastline of an island:

 Suppose you wanted to measure the total perimeter of an island. You could begin by roughly estimating
 the perimeter of the island by measuring the border of the island from a high vantage point like an
 airplane and using miles as units. Next, to be more accurate, you could walk along the island's borders
 and measure around its various coves and bays using a measuring tape. Then, if you wanted to be really
 accurate, you could carefully measurement around every single protruding rock and detail of the island
 with a yardstick or even a foot-long ruler.

Clearly, the perimeter of the island would grow as you decrease the size of your measuring device and increase the accuracy of your measurements. Also, the island would more or less like similar (in terms of becoming more and more jagged and complex) as you continued to decrease your measuring device.


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