Difference between revisions of "Field:Fractals"

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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!"> 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. 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.</div>''
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   ''<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>''
 
<!-- 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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   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 self-similar (in terms of becoming more and more jagged and complex) as you continued to decrease your measuring device.
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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.
 
|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
 
:*Too irregular to be described by traditional geometric dimension
 
:*Too irregular to be described by traditional geometric dimension
 
:*Defined recursively
 
:*Defined recursively
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===Self-Similarity===
 
===Self-Similarity===
Although all fractals exhibit self-similarity, they do not necessarily have to possess exact self-similarity. The coastline fractal explained above does not have exact self-similarity, but its parts are very similar to the whole, while fractals made by iterated function systems (explained later) have exact-similarity.
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[[Image:Sierp-zoom.gif|200px|thumb|right|Self-Similiarity of Sierpinkisi's Triangle]]
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Although all fractals exhibit self-similarity, they do not necessarily have to possess exact self-similarity, which would mean that the parts look exactly like the whole. The coastline fractal explained above does not have exact self-similarity, but its parts are very similar to the whole, while fractals made by iterated function systems (such as [[Sierpinski's Triangle]], shown at the right) have exact-similarity.
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===Fractal (Non Integer) Dimension===
 
===Fractal (Non Integer) Dimension===
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 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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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.
  
Click here to learn more about [[Fractal Dimension]] and how it is calculated.
 
  
 
===Recursive===
 
===Recursive===
Fractals are defined by a recursive equation(s) or process that governs the behavior of a fractal. If the fractal is defined by a single equation or by a system of equations, the fractal is created by taking an initial starting value and applying the recursive equation(s) to that value over and over again. 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 segment and then continuously iterated to the segments resulting from the previous iteration. This 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.
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Fractals are defined by a recursive equation(s) or process that governs the behavior of a fractal. If the fractal is defined by a single equation or by a system of equations, the fractal is created by taking an initial starting value and applying the recursive equation(s) to that value over and over again. 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. This 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.
 
Click here to learn more about [[Iterated Functions]] and its mathematical implications.
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===Examples of Fractals===
 
===Examples of Fractals===
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</gallery>
 
</gallery>
  
There are four main types of fractals that are categorized by how they are generated. In addition, numerous fractals occur naturally, including lightening, broccoli, blood vessels, and in landscapes.
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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|
 
:'''Iterated function systems (IFS)''' {{Hide|
::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|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.
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::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.
 
::*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. 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.
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::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).
 
::*Examples include: [[Lorenz Attractor]], [[Henon Attractor]], [[Cantor Set]] , and [[Rossler Attractor]].
 
::*Examples include: [[Lorenz Attractor]], [[Henon Attractor]], [[Cantor Set]] , and [[Rossler Attractor]].
 
}}
 
}}

Revision as of 10:58, 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