Difference between revisions of "Field:Fractals"

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:'''Iterated function systems (IFS)''' {{Hide|
 
:'''Iterated function systems (IFS)''' {{Hide|
::Fractals created with a IFS are always exactly self-similar, so that you can forever zoom into one of the these fractals and see the same structure. A IFS consists of one of more equations or processes that describe the behavior of the fractal and are recursively applied. Many of these images that are generated by a IFS are also governed by the "[[Chaos|chaos]] game". This is where a 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 its probability and applied to the point for an infinite amount of iterations.
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::A IFS fractal consists of one of more equations or processes that describe the behavior of the fractal and are recursively applied. These fractals are always exactly self-similar and are made up of an infinite number of self-copies that are transformed by a function or set of functions.
 
::*Examples include: [[Koch Snowflake]], [[Harter-Heighway Dragon]], Barnsley’s Fern ([[Blue Fern]]), and [[Sierpinski's Triangle]].
 
::*Examples include: [[Koch Snowflake]], [[Harter-Heighway Dragon]], Barnsley’s Fern ([[Blue Fern]]), and [[Sierpinski's Triangle]].
 
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:'''Strange attractors'''  {{Hide|
 
:'''Strange attractors'''  {{Hide|
::Fractals that are considered strange attractors are generated from a set of functions called attractor maps or systems. These systems are chaotic, because the functions map points in a seemingly random order. However, these points are not actually completely random and are in fact evolving towards a structure that is called the attractor (because it attracts the points into a certain shape).
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::Fractals that are considered strange [Attractors|[attractors]] are generated from a set of functions called attractor maps or systems. These systems are [[Chaos|chaotic]] and [[Dynamical Systems|dynamic]]. Initially, the functions appear to map pointsin a seemingly random order, but the points are in fact over time evolving towards a recognizable structure called an attractor (because it "attracts" the points into a certain shape).
 
::*Examples include: [[Lorenz Attractor]], [[Henon Attractor]], [[Cantor Set]] , and [[Rossler Attractor]].
 
::*Examples include: [[Lorenz Attractor]], [[Henon Attractor]], [[Cantor Set]] , and [[Rossler Attractor]].
 
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:'''Random fractals''' {{Hide|
 
:'''Random fractals''' {{Hide|
::These fractals are created through stochastic methods, meaning that there are no set functions that determine the behavior of the fractals. Although the fractals must still obey rules, they are unlike chaotic IFS fractals because chaotic fractals are governed by a set of functions that are picked according to probability and follow some sort of a pattern. Also, chaotic fractals have errors (the difference between one plotted value to the next) that grow exponentially, while random fractal errors are simply random.
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::These fractals are created through stochastic methods, meaning that the behavior of these fractals depend on a random factor and usually probability restraints. One way to differentiate between chaotic and random fractals is to observe that chaotic fractals have errors (the difference between one plotted value to the next) that grow exponentially, while random fractal errors are simply random.
 
::*Examples include: [[Levy Flights]], [[Brownian Motion]], [[Brownian Tree]],  and fractal landscapes.
 
::*Examples include: [[Levy Flights]], [[Brownian Motion]], [[Brownian Tree]],  and fractal landscapes.
 
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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 real number.  On a computer, each pixel corresponds to a [[Complex Numbers | complex number]] value. 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 color assigned to each complex number value or pixel is either black if the value converges to zero or a color based on the number of iterations (aka. escape time) it took for the value to reach infinity.  The intermediary numbers that arise from the iterations are referred to as their “orbit”. The boundary between black and color pixels is infinite and increasingly complex.
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::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 real number.  On a computer, each pixel corresponds to a [[Complex Numbers | complex number]] value. Each complex number value is applied recursively to the function until it reaches infinity or until it is clear that value will converge to zero. A color is assigned to each complex number value or pixel: the pixel is either colored black if the value converges to zero or the pixel is given a color based on the number of iterations (aka. escape time) it took for the value to reach infinity.  The intermediary numbers that arise from the iterations are referred to as their “orbit”. The boundary between black and color pixels is infinite and increasingly complex.
 
::*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 11:47, 1 July 2009


Fractals

NorwayCoastline.png

A fractal is often defined as a geometric shape that is self-similar, that is, whose magnified parts look like a smaller copy of the whole. The term "fractal" 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 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 and foot as units. Then, if you wanted to be really accurate, you could carefully measure around every single protruding rock and detail of the island with foot-long ruler and use inches as a measuring unit.

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 self-similar (in terms of becoming more and more jagged and complex) as you continued to shorten your measuring device.


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



References

Wikipedia, Fractals Page

Cynthia Lanius, Cynthia Lanius' Lessons: A Fractal Lesson

CoolMath.com, Math of Fractals