Difference between revisions of "Field:Fractals"

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{{Field Page
 
{{Field Page
 
|Field=Fractals
 
|Field=Fractals
|BasicDesc=A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,"[1] a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured."
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|BasicDesc=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”.
  
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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.
  
*General description
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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.
**history, self-similarity, iterating
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|FurtherInfo=In addition to self-similarity, there are other traits exhibited by fractals:
|FurtherInfo='''Generating fractals'''
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* Fine or complex structure at small scales
Three common techniques for generating fractals are:
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*Too irregular to be described by traditional geometric dimension
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*Defined by a recursive statement
  
:* Escape-time fractals — (also known as "orbits" fractals) These are defined by a recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal, the Nova fractal and the Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly.
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==Self-Similarity==
:* Iterated function systems — These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals.
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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.
:* Random fractals — Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Lévy flight, fractal landscapes and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters. --[www.wikipedia.org Wikipedia]
 
  
*History, Mandelbrot 1975
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==Irregular Dimension==
*Self-similarity
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[[Image: NorwayCoastline.jpg]]
**Iterating
 
**complex, <math>z = z^2 + c\,</math>
 
***to zero = black
 
***infinity - color, how fast is what color
 
  
*Fractal dimension
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Fractals are too irregular to be defined by tradition or Euclidean geometry language. Objects that can be described by Euclidean geometric dimensions include a line (1 dimension), a square (2 dimension), and a cube (3 dimension). Fractals are instead described by a specific term 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.
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Click here to learn more about [[Fractal Dimension]] and how it is calculated.
  
*Examples: Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, space-filling curve, Koch curve, Lyapunov fractal, and Kleinian groups.
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==Recursive==
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Fractals are defined by recursive or iterating statements that can be equations or geometric curves. Basically, a recursive statement is a rule that defines the shape or behavior of a fractal and is applied over and over again, using the output calculated from the previous statement as the input for the next statement. This can be seen as a kind of positive feedback loop, where the same definition or statement is applied infinitely by using the results from the previous iteration to start the next iteration.
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Click here to learn more about [[Iterated Functions]] and its mathematical implications.
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==Examples of fractals==
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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, including lightening, broccoli, blood vessels, and in landscapes.
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*Iterated function systems (IFS)
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{{Hide|
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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.
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::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]].
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}}
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*Strange attractors
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{{Hide|
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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. 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.
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::Examples include: [[Lorenzo Attractor]], [[Henon Attractor]], [[Cantor Dust]] , and [[Rossler Attractor]].
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}}
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*Random fractals
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{{Hide|
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::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).
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::Examples include: [[Levy Flights]], [[Brownian Motion]], [[Brownian Tree]], and fractal landscapes.
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}}
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* Escape-time (“orbit”) fractals
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{{Hide|
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::Escape-time fractals are created in the <balloon title = “For more information, click here [[Complex Numbers]]” style = “green”> complex plane <\balloon> with a single <balloon title = “such as <math>f(z) = z^2 + c<\math>, where z is a complex number and c is any number” > function<\balloon>, where each pixel corresponds to a complex number. 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.
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::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”.
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::Examples include: [[Mandelbrot Set]], [[Julia Sets]], and [[Lyapunov Fractal]].
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}}
  
*Nature
 
 
|Links=http://en.wikipedia.org/wiki/Fractals
 
|Links=http://en.wikipedia.org/wiki/Fractals
 
http://mathworld.wolfram.com/Fractal.html
 
 
}}
 
}}

Revision as of 14:01, 1 June 2009


Fractals

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.


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

Links to Other Information

Browse Fractals Images

http://en.wikipedia.org/wiki/Fractals