Fractals:
According to Benoit Mandelbrot; 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"a property that is called 'self-similarity'.
The term was coined by Benoit Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured."
History:The mathematics behind fractals began as early as the 17th century when mathematician and philosopher Leibniz considered recursive self-similarity.In 1872, Karl Weierstrass gave an example of a function with the property of being everywhere continuous but nowhere differentiable (These are calculus terms. If you haven't taken calculus, don't worry about it). In 1904, Helge von Koch, dissatisfied with Weierstrass's very abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch Snowflake (see: Koch Snowflake). In 1915, Waclaw Sierpinski constructed his famous triangle fractal (called "The Sierpinski Gasket" [see: Sierpinski Gasket)] ). In 1883, Georg Cantor also gave examples of subsets of the real line with unusual properties. these 'Cantor sets' are also now recognized as fractals (see: Cantor Sets). 
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Iterated functions, using imaginary numbers were investigated in the late 19th and early 20th centuries by Henri Poincare, Felix Klein, Pierre Fatou and Gaston Julia. However, without the aid of modern computer graphics, they lacked the means to visualize the beauty of many of the objects that they had discovered.
In the 1960s, Benoit Mandelbrot started investigating self-similarity in papers such as "How Long Is the Coast of Britain?: Statistical Self-Similarity and Fractional Dimension", in which Mandelbrot discusses the idea that the coast of Britian (and other natural formations) are infinitely long. When you consider that the coastline is actually made of jagged cliffs that can be measured more accurately by sticking rulers and whatnot INTO the jagged edges. Using this method, you would end up with a much longer measurement. Likewise, the jagged cliffs themselves are made of jagged rocks which are made of jagged dust which are made of jagged molecules etc... In other words, there is no perfect way to obtain a perfect measurement of the coast.
So to answer the question: "How Long Is the Coast of Britain?The length will be measured differently according to the accuracy of the measuring device. And since no device will accurately measure to the finest detail (and since there IS no "finest detail" when considering that the coastline is made of molecules, which are made of atoms, which are made of quarks etc...), the coastline is INFINITELY LONG!!!!
Finally, in 1975 Mandelbrot coined the word "fractal" to refer to these new ideas and disoveries. Mandelbrot is most famous for his discovery of a particular fractal which has come to be known as the "Mandelbrot Set". This set has been plotted using computer graphics, and could never have been created or viewed without the aid of modern technology. (see: Mandelbrot Set)
Koch Snowflake:
A Koch curve has an infinitely repeating self-similarity when it is magnified, as shown below:
Fun Fact: With every iteration, the perimeter of this shape grows by 1/3rd. The Koch snowflake is the result of an infinite number of these iterations, and has an infinite length, while its area remains finite.
Sierpinski Gasket:
The Sierpinski triangle, also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal named after Waclaw Sierpinski who described it in 1915.The Sierpinski triangle is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction.
Note that this infinite process is not dependent upon the starting shape being a triangle. The first few steps starting, for example, from a square also tend towards a Sierpinski gasket.Michael Barnsley used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals."[3]
Fun Fact: The area of a Sierpinski triangle is zero!! The Sierpinski triangle has NO area! Since pieces of the triangle are constantly removed, the area gets smaller and smaller after each iteration. Since the Sierpinki gasket examples shown on this page have only had a finite amount of triangles removed, there is still some area left to be seen. However...after infinite iterations, the area will be completely removed, as illustrated below:
Cantor Sets:
One example of a Cantor set is created by repeatedly deleting the open middle thirds of a set of line segments. One starts by deleting the open middle third from the initial line segment, leaving two line segments remaining. Next, the open middle third of each of these remaining segments is deleted, leaving four line segments.This process is continued ad infinitum.
The Cantor ternary set contains all points in the interval [0, 1] that are not deleted at any step in this infinite process.
The first six steps of this process are illustrated below:
The following equation is a means of determining how much of the initial line segment is removed after infinite iterations.
The answer is "1". Therefore, after infinite iterations, the ENTIRE line will be gone and there will be nothing left, since the sum of the lengths of the removed intervals is equal to the length of the original interval.
(ie. The proportion left is 1 - 1 = 0)
However, a closer look at the process reveals that there MUST be something left, since removing the "middle third" of each interval will always leave 2/3 remaining.
Fun Fact: The Cantor set described above has NO length left after infinite iterations:
on the other hand:
The Cantor set has INFINITE points left after infinite iterations.
This is completely awesome.
Hilbert Curve:A Hilbert curve (also known as a Hilbert space-filling curve) is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891. |
Apollonian Gasket:An Apollonian gasket or Apollonian net is a fractal generated from triples of circles, where any circle is tangent to two others. It is named after Greek mathematician Apollonius of Perga. |
Mandelbrot Set:
The Mandelbrot set is an infinitely complex fractal, composed by using the following seemingly simple equation:
When various numbers are used in place of the variable C, after several iterations, they will either tend to lead toward infinity, or they will lead toward zero. When constructing the Mandelbrot set, not only are Real numbers used, but also Imaginary numbers (such as i, 2i, 3i etc... where 'i' is defined as the square root of negative 1). Any number that leads toward infinity is discarded, and all other numbers remaining are considered elements of the Mandelbrot set.
For example, c = 1 gives the sequence 0, 1, 2, 5, 26; which leads to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set.
On the other hand, c = i gives the sequence 0, i, (-1 + i), -i, (-1 + i), -i, which is bounded, and so it belongs to the Mandelbrot set.
Fun Fact: Only after infinite numbers have been placed into the equation Z = Z2 + C, will the full set be truly created. Since it is impossible for humans (or even computers for that matter) to input INFINITE numbers into the equation, we will never know (and CAN never know) the FULL mandelbrot set!!
However...
When computed and graphed on the complex plane using several hundred, thousand, million etc... numbers, the Mandelbrot Set is seen to have an elaborate boundary, which does not simplify at any given magnification. This qualifies the boundary as a fractal.
The infinite complexity of the Mandelbrot set is illustrated below:
 
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The video above shows a magnification of the Mandelbrot set. The song playing in the video is Jonathan Coulton's "Mandelbrot Set". If you listen to the lyrics, he explains how to construct a Mandelbrot set. He also mentions other fractals such as the Koch curve, Seirpinski gasket, and Cantor Ternary Set. ENJOY!! |
External links:
http://en.wikipedia.org/wiki/Fractal
http://boojum.as.arizona.edu/~jill/NS102_2006/FRACTALS/xaos1.mpg
http://boojum.as.arizona.edu/~jill/NS102_2006/chaos.html
http://kluge.in-chemnitz.de/documents/fractal/node2.html
http://classes.yale.edu/fractals/Panorama/Nature/NatFracGallery/NatFracGallery.html
http://www.ba.infn.it/~zito/project/nfractals.html
http://tiger.towson.edu/~gstiff1/fractalpage.htm
http://members.aol.com/SpinChaos/PageFract.html#Spins%20Fractal%20Gallery
http://www.coolmath.com/fractals/fractals_lesson.html
http://people.cs.uchicago.edu/~kaharris/15100/lab7/lab7.html
http://mthwww.uwc.edu/wwwmahes/fractals/index.htm