Difference between revisions of "Field:Fractals"

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{{Field Page
 
{{Field Page
 
|Field=Fractals
 
|Field=Fractals
|BasicDesc=[[Image: NorwayCoastline.png|left|160px]]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".
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|BasicDesc=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".
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<div style="float:left; width: 50%;"> __TOC__</div>
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<div style="float:right;width:50%;">[[Image:Anna1.jpg]]</div>
  
 
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!"> island</balloon>:
 
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!"> island</balloon>:
  
   ''<div style="color:#545454; position:relative; left:15px">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.</div>''
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   ''<div style="position:relative; left:15px">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.</div>''
 
<!-- I changed it to the above formatting for aesthetic purposes... Can be changed back, though.
 
<!-- I changed it to the above formatting for aesthetic purposes... Can be changed back, though.
 
   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
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===Fractal (Non Integer) Dimension===
 
===Fractal (Non Integer) Dimension===
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Fractals are too irregular to be defined by traditional or Euclidean geometry language. Objects that can be described by Euclidean geometric dimensions include a line (1 dimension), an square (2 dimension), and a cube (3 dimension). Fractals are instead described by what is called Hausdorff or [[Fractal Dimension|fractal dimension]] that measures how fully a fractal seems to fill space. For example, going back to the coastline example above, the coastline of Norway has an estimated fractal dimension of about 1.52 so it is not quite a line, but not quite an area either.
 
Fractals are too irregular to be defined by traditional or Euclidean geometry language. Objects that can be described by Euclidean geometric dimensions include a line (1 dimension), an square (2 dimension), and a cube (3 dimension). Fractals are instead described by what is called Hausdorff or [[Fractal Dimension|fractal dimension]] that measures how fully a fractal seems to fill space. For example, going back to the coastline example above, the coastline of Norway has an estimated fractal dimension of about 1.52 so it is not quite a line, but not quite an area either.
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===Recursive===
 
===Recursive===
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A fractal must have a recursive definition, meaning that the fractal is defined in terms of itself. Fractals can be described by a single equation or by a system of equations, and created by taking an initial starting value and applying the recursive equation(s) to that value over and over again (a process called '''iteration'''). This iteration takes the output calculated from the previous iteration as the input for the next statement. Similarly, if the recursive definition of a fractal is a process, that process is first applied to the starting geometric shape and then continuously iterated to the segments resulting from the previous iteration. Recursive can be seen as a kind of positive feedback loop, where the same definition is applied infinitely by using the results from the previous iteration to start the next iteration.
A fractal must have a recursive definition, meaning that the fractal is defined in terms of itself. Fractals can be described by a single equation or by a system of equations, and created by taking an initial starting value and applying the recursive equation(s) to that value over and over again (a process called iteration). This iteration takes the output calculated from the previous iteration as the input for the next statement. Similarly, if the recursive definition of a fractal is a process, that process is first applied to the starting geometric shape and then continuously iterated to the segments resulting from the previous iteration. Recursive can be seen as a kind of positive feedback loop, where the same definition is applied infinitely by using the results from the previous iteration to start the next iteration.
 
 
 
Click here to learn more about [[Iterated Functions]] and its mathematical implications.
 
}}
 
 
 
 
 
===Examples of Fractals===
 
  
<gallery caption="" widths="150px" heights="150px" perrow="5">
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Click here to learn more about [[Iterated Functions]].
Image:Fractal_Broccoli.jpg|Romanesco broccoli (''Natural Fractal'')
 
Image:DragonCurve.jpg|Dragon Curve (''IFS'')
 
Image:LorenzAttractor.png|Lorenz Attractor (''Strange Attractor'')
 
Image:BrownianTree.png|Brownian Tree (''Random Fractal'')
 
Image:Anna1.jpg|Julia Set (''Escape-Time Fractal'')
 
</gallery>
 
{{hide|1=
 
  
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== Types of Fractals ==
 
There are four main types of fractals that are categorized by how they are generated. In addition, numerous fractals occur naturally in lightening, broccoli, blood vessels, landscapes, and other phenomena.
 
There are four main types of fractals that are categorized by how they are generated. In addition, numerous fractals occur naturally in lightening, broccoli, blood vessels, landscapes, and other phenomena.
  
:'''Iterated function systems (IFS)''' {{Hide|
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===Iterated function systems (IFS)===
 
::A IFS fractal consists of one of more recursive equations or processes that describe the behavior of the fractal and are iterated (or applied continually). 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.
 
::A IFS fractal consists of one of more recursive equations or processes that describe the behavior of the fractal and are iterated (or applied continually). 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]].
}}
 
  
:'''Strange attractors'''  {{Hide|
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===Strange attractors===
::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).
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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 points in 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]].
}}
 
  
:'''Random fractals''' {{Hide|
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===Random fractals===
 
::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.
 
::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.
}}
 
  
: '''Escape-time (“orbit”) fractals''' {{Hide|
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===Escape-time (orbit) fractals===
::Escape-time fractals are created in the complex plane with a single function, some <math>f(z)</math>, where ''z'' is a [[Complex Numbers|complex 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.
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::Escape-time fractals are created in the complex plane with a single function, some <math>f(z)</math>, where ''z'' is a [[Complex Numbers|complex number]].  On a computer, each pixel corresponds to a 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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==Examples of Fractals==
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<gallery caption="" widths="150px" heights="150px" perrow="5">
 +
Image:Fractal_Broccoli.jpg|Romanesco broccoli (''Natural Fractal'')
 +
Image:DragonCurve.jpg|Dragon Curve (''IFS'')
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Image:LorenzAttractor.png|Lorenz Attractor (''Strange Attractor'')
 +
Image:BrownianTree.png|Brownian Tree (''Random Fractal'')
 +
Image:Anna1.jpg|Julia Set (''Escape-Time Fractal'')
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</gallery>
 
}}
 
}}
  

Revision as of 13:16, 25 July 2011


Fractals

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".

Anna1.jpg

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