Difference between revisions of "Field:Fractals"

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{{Field Page
 
{{Field Page
 
|Field=Fractals
 
|Field=Fractals
|BasicDesc=[[Image: NorwayCoastline.png|left|225px]]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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|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>:
 
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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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.
 
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.
  
===Irregular Dimension===
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===Fractal (Non Integer) Dimension===
 
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.
 
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.
  

Revision as of 11:32, 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 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 measurearound 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.


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