The Images Of Chaos!

About Mandelbrot and Julia Sets

The Mandelbrot and Julia Sets are among the most well known of all fractals. Countless breathtaking shapes and curves hide within their depths, beautifully displaying the concept of self-similarity that can be found in many fractals. Zooming into the edges of the "apple man" will expose many more "apple men" with very similar details as the outer one. Julia Sets can be generated for points on the Mandelbrot Set. Interesting Julia Sets can be found at places where the Mandelbrot Set exhibits interesting details as well. The pattern of the Julia Set for a particular point of the Mandelbrot Set is typically quite similar to an enlargement of the Mandelbrot Set at the given point and adds a different kind of beauty to the pattern.

Interactive Explorer

The following is a generator for Mandelbrot and Julia Sets that allows for interactive exploration of the depth and beauty of both sets. The program is written as a Java Applet and should work with Microsoft's Internet Explorer and Netscape's Navigator. Simply press the button below to start. Enjoy your journey through the fractal world of Mandelbrot and Julia Sets!

Mathematical Background

The following iterated equation produces both the Mandelbrot and Julia Sets:

zn+1 = zn2 + c
where both z and c are complex numbers. For the Mandelbrot Set, the first value for z is zero, and different values for c are taken over a selected range (e.g. the real part corresponding to a point on the horizontal axis of the display, and the imaginary part relating to the vertical axis). For the Julia Set, a particular value for c is chosen, and the starting value for z is taken over a selected range corresponding to the area that is to be displayed. The interesting question now is whether the value for z will grow towards infinity or approach zero after a certain number of iterations. When z approaches zero, the point c is part of the Mandelbrot Set and is traditionally colored in black. When z grows towards infinity, the point c is outside the Mandelbrot Set, and its color depends on how many iterations it took for z to grow over a certain threshold. Something very similar is true for Julia Sets. For the Julia Set at c, the point corresponding to the initial value for z is colored in black if the function approaches zero after continuously iterating; otherwise the number of iterations until the function grows beyond a certain threshold will determine the color.

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This page was created in September 1999 by Attila Narin <> and was last updated on October 6, 1999.
This page and the Fractal Generator are Copyright © 1999 Attila Narin. All Rights Reserved.