Fractal info

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  1. Information about Above Graphic
    1. Original Images - click here to see the original sequence of eight images that I created using the Fractal Microscope.  I then used Macromedia Flash to create the above graphic showing the source of each image. 
    2. Introduction to Fractal Microscope - an interactive tool designed by the Education Group at the National Center for Supercomputing Applications (NCSA) for exploring the Mandelbrot set and other fractal patterns.
           Direct Link to Fractal Microscope - 
  2. Madelbrot Set
    1. Definition

      1. Mathematical Definition - The Mandelbrot set is defined as the set of points c in the complex plane for which the iteratively defined sequence 
               zn+1 = zn2 + c 
        with z0 = 0 does not tend to infinity.

      2. Approximation - It can be shown that once the modulus of zn is larger than 2, then the sequence will tend to infinity, and c is therefore outside the Mandelbrot set. This value, known as the bail-out value, allows the calculation to be terminated for points outside the Mandelbrot set. If the modulus of zn remains less than 2 after a given number of iterations, then there is frequently no way to know whether it would diverge if additional iterations were performed. No matter how many iterations are performed, any visualization of the Mandelbrot set will inevitably include some points that would have diverged if still more iterations were performed. Therefore, all visualizations of the Mandelbrot set are approximations that overestimate the size of the set. As the number of maximum iterations is increased, the image of the Mandelbrot set gradually shrinks toward a more accurate shape.

      3. Color - Mathematically speaking, the picture of the Mandlebrot set is black and white. Either a point is in the set (usually colored black) or it is not (colored white).  Most fractal rendering programs display points outside of the Mandelbrot set in different colors depending on the number of iterations before it bailed out.  This is what creates the multi-colored images.
        Source for the above is

    2. Mandelbrot Biography - - short biography
    3. Benoit Mandelbrot - 
    4. The (MIS)behavior of Markets - - preview of Mandelbrot's popular book, some pages are omitted.
           Reviews of The (Mis)behavior of Markets - compiled at Yale University.
           Reviews on Amazon -
    5. Mandelbrot - - video of Mandelbrot lecturing at MIT
    6. PeoplesArchive - - video interviews with Mandelbrot
  - dedicated to collecting the stories of the great thinkers of our time,
                   whose work has changed our world.

  3. Search Engine Directories about Fractals
    1. Google Fractals - 
    2. Yahoo Fractals - 
  4. University Links about Fractals
    1. Drexel University - 
    2. University of Tennessee, Knoxville - 
    3. Rice University - Cynthia Lanius - - introduction to fractals for elementary and middle-school children.
    4. Boston University - 
  5. Other Links about Fractals
    1. Wikipedia -
    2. FAQ (Frequently Asked Questions) Fractals - 
    3. Canadian National Lab for Particle and Nuclear Physics - - maintained by Noel Giffin.
    4. Encyclopedia of the Mandelbrot Set - - maintained by Robert P. Munafo.
    5. Fractalus - - has detailed information about fractal techniques, fractal generating software, and more.  Primary webmaster is Damien M. Jones.
    6. Infinite Fractal Loop - - links to many fractal websites
    7. Fractint - 
    8. Academic Info - 

                        (This page was last edited on April 28, 2009 .)