Difference between revisions of "Field:Fractals"

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:* 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 by a recursive statement
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:*Defined recursively
  
 
===Self-Similarity===
 
===Self-Similarity===
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===Recursive===
 
===Recursive===
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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Fractals are defined by recursive equation(s) or processes that governs the shape or 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 starting value or segment and continuously iterated to the results of the previous iteration.
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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.

Revision as of 14:57, 25 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.


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



Additional Resources

Reference used - Wikipedia, Fractals Page
Reference used - Cynthia Lanius, Cynthia Lanius' Lessons: A Fractal Lesson
Reference used - CoolMath.com, Math of Fractals