Difference between revisions of "Field:Fractals"

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: '''Escape-time (“orbit”) fractals''' {{Hide|
 
: '''Escape-time (“orbit”) fractals''' {{Hide|
::Escape-time fractals are created in the complex plane with a single function, such as <math>f(z) = z^2 + c</math>, where ''z'' is a complex number and ''c'' is any real number.  On a computer, each pixel corresponds to a [[Complex Numbers | complex number]] value. Each complex number value is applied recursively to the function until it reaches infinity or until it is clear that value will converge to zero. A color is assigned to each complex number value or pixel: the pixel is either colored black if the value converges to zero or the pixel is given a color based on the number of iterations (aka. escape time) it took for the value to reach infinity.  The intermediary numbers that arise from the iterations are referred to as their “orbit”. The boundary between black and color pixels is infinite and increasingly complex.
+
::Escape-time fractals are created in the complex plane with a single function, some <math>f(z)</math>, where ''z'' is a [[Complex Numbers|complex number]].  On a computer, each pixel corresponds to a [[Complex Numbers | complex number]] value. Each complex number value is applied recursively to the function until it reaches infinity or until it is clear that value will converge to zero. A color is assigned to each complex number value or pixel: the pixel is either colored black if the value converges to zero or the pixel is given a color based on the number of iterations (aka. escape time) it took for the value to reach infinity.  The intermediary numbers that arise from the iterations are referred to as their “orbit”. The boundary between black and color pixels is infinite and increasingly complex.
 
::*Examples include: [[Mandelbrot Set]], [[Julia Sets]], and [[Lyapunov Fractal]].
 
::*Examples include: [[Mandelbrot Set]], [[Julia Sets]], and [[Lyapunov Fractal]].
 
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Revision as of 14:58, 4 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