The Mandelbrot Set is probably one of the most well known fractals, and probably one of the most widely implemented fractal in fractal plotting programs. It was originally discovered by Benoit B. Mandelbrot, hence the name. It is produced by the formula

z_{n+1} = z_{n}^{2}
+ c

where z and c are complex numbers, z_{0} = 0, and c is
a point on the plain. The formula is iterated until
|z_{n}| (the magnitude of z) is greater than or equal to
the bailout value 2. Then the pixel that c corresponds to is
coloured according to the number of iterations that occured before
the process bailed out. The uninteresting black area of the image
is the actual Mandelbrot Set. It consists of all the values for c
where |z_{n}| never got larger than 2. Of course this
area is impossible to comute accurately, so this program decides
to colour black all pixels for which |z_{n}| never gets
larger than 2 for a given number of iterations (256).

It is amazing that such a simple formula can produce such an interesting picture. If you zoom into the image (click zoom and drag out an area on the image) you will find sections that look the same as portions of the image at other zooms. This infinite level of detail is common to all fractals and makes them so facinating. If you find that the image is too blocky for your liking, you can change the resolution by choosing a different number from the list box on the left of the controls (The smaller the number, the smaller the pixels will be) and press the redraw button.

By now you are probably getting bored of this fractal, but
don't change to a different one yet; there is still an infinite
number of Julia Sets to look at. Julia Sets are produced with the
same formula as the Mandelbrot set, but the starting values are
different. That is c is some constant, and z_{0} is the
starting point on the plane. From this definition, you can see
that there is an infinite number of Julia Sets; one for each value
of c. In fact, there is a Julia Set that corresponds to each
point on complex plane. There is an interesting relationship
between the Mandelbrot Set and the Julia Sets. In a way, you can
think of the Mandelbrot Set as an index for the Julia Sets. For
values of c that are inside the Mandelbrot Set, you will get
connected Julia Sets. That is all the black regions are
connected. Conversely, those values of c outside the Mandelbrot
Set, you get unconnected sets.

Because of this relationship, I have written the applet so you can switch over to a Julia Set by just clicking on the "Switch (M<->J)" button and clicking on part of the set. You will notice that the coordinates of the pointer are displayed on the status line during selection. To get back to the Mandelbrot Set, just click on the button again.

By thinking of the Mandelbrot Set as an index for Julia Sets, the question comes up about what happens if a point on the edge of the Mandelbrot Set is chosen as the value of c. The answer is you get a null set. That is there should be no black section. If you have trouble finding a point on the edge of the set, try the point (0, 1). This is one of the few points on the edge of the Mandelbrot Set with rational coordinates which is why I name it. Have fun exploring!

Copyright © 2005 — James Henstridge <james@jamesh.id.au>