Difference between revisions of "Field:Fractals"

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:'''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 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 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.
 
::*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]].
 
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Revision as of 16:00, 30 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|]]



References

Wikipedia, Fractals Page

Cynthia Lanius, Cynthia Lanius' Lessons: A Fractal Lesson

CoolMath.com, Math of Fractals