Fractals
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 selfsimilarity
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:
z_{n+1} = z_{n}^{2} + 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.
This page was created in September 1999 by Attila Narin <attila@narin.com>
and was last updated on October 6, 1999.
This page and the Fractal Generator are Copyright © 1999 Attila Narin. All Rights Reserved.
