Difference between revisions of "Field:Fractals"

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|BasicDesc=[[Image: NorwayCoastline.png|left|225px]]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".
 
|BasicDesc=[[Image: NorwayCoastline.png|left|225px]]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 nature analogy involving the coastline of an <balloon title="Actually the image is a picture of the border of Norway, not an island!" style="color:green"> island </balloon>:
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This concept can be explained in a commonly used analogy involving the coastline of an <balloon title="Actually the image is a picture of the border of Norway, not an island!" style="color:green"> island </balloon>:
 
   Suppose you wanted to measure the total perimeter of an island. You could begin by roughly estimating
 
   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
 
   the perimeter of the island by measuring the border of the island from a high vantage point like an
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   a yardstick or even a foot-long ruler.
 
   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.
+
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 self-similar (in terms of becoming more and more jagged and complex) as you continued to decrease your measuring device.
 
|FurtherInfo=In addition to self-similarity, there are other traits exhibited by fractals:
 
|FurtherInfo=In addition to self-similarity, there are other traits exhibited by fractals:
 
:* Fine or complex structure at small scales
 
:* Fine or complex structure at small scales
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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. 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.
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::Fractals created with IFS are always exactly self-similar, 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 that are generated by IFS are also 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 its probability and applied to the point for an infinite amount of iterations.
 
::*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]].
 
}}
 
}}
  
 
:'''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. These points are actually not completely random and are in fact evolving towards a structure that can eventually be seen as the form of the attractor.
+
::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 can eventually be seen as the form of the attractor.
 
::*Examples include: [[Lorenz Attractor]], [[Henon Attractor]], [[Cantor Set]] , and [[Rossler Attractor]].
 
::*Examples include: [[Lorenz Attractor]], [[Henon Attractor]], [[Cantor Set]] , and [[Rossler Attractor]].
 
}}
 
}}
  
 
:'''Random fractals''' {{Hide|
 
:'''Random fractals''' {{Hide|
::These fractals are created through stochastic methods, meaning there are no set functions that determine the behavior of the fractals. Although the fractals still must have rules that they follow, they are unlike chaotic IFS fractals because chaotic fractals are governed by a set of functions that are picked according to probability and these follow some sort of a pattern (and grow exponentially), while random fractals are simply random (and grow randomly).
+
::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.
 
::*Examples include: [[Levy Flights]], [[Brownian Motion]], [[Brownian Tree]],  and fractal landscapes.
 
::*Examples include: [[Levy Flights]], [[Brownian Motion]], [[Brownian Tree]],  and fractal landscapes.
 
}}
 
}}
  
 
: '''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.  The intermediary numbers that arise from the iterations are referred to as their “orbit”. The boundary between black and colorful pixels is infinite and increasingly complex.
+
::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.
 
::*Examples include: [[Mandelbrot Set]], [[Julia Sets]], and [[Lyapunov Fractal]].
 
::*Examples include: [[Mandelbrot Set]], [[Julia Sets]], and [[Lyapunov Fractal]].
 
}}
 
}}
  
 
}}
 
}}

Revision as of 13:40, 3 June 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. Then, if you wanted to be really
 accurate, you could carefully measure 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 self-similar (in terms of becoming more and more jagged and complex) as you continued to decrease your measuring device.


Contents

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