Mandelbrot and Julia Sets

Date: 06/29/98 at 18:10:34
From: Paulo Jorge Matos
Subject: Fractal equations (Mandelbrot and Julia)

Hi Dr. Math,

I started studying fractals on my own because I will only learn complex 
numbers in Math next year. I'm studying to be a software engineer and I 
know that Mathematics is a very important subject to be a good software 
programmer but I also know that it is not easy. If anybody asks you 
about a good book to start understanding fractals, I'd recommend the 
following. I've read it. It's my first about fractals: _The Magic 
Machine: A Handbook for Computer Sorcery_ by A. K. Dewdney. 

I've made a program to build the Mandelbrot set, but I'm having some 
difficulty with the Julia set. I know that both equations are 
z = z^2+c, where z and c are complex numbers. Computers just don't get 
complex numbers, so I after some equations I got to the following to 
the Mandelbrot set if z is composed by x + yi and c is composed by 
a + bi:

   x = x^2 - y^2+a 
   y = 2xy + b

But for the Julia set I have reached wrong equations. I've reached the 
following:

   y = b - 2ab
   x = a - (a^2 - b^2)

But it is not possible because I have to iterate x and y, and they 
need to appear also on the right part of the equation. What's wrong? 
Is it too confusing? Hope you can help.

Regards,
Paulo Jorge Matos

Date: 07/01/98 at 22:56:11
From: Doctor Pete
Subject: Re: Fractal equations (Mandelbrot and Julia)

Hi,

The difference between the Mandelbrot set (M-set) and Julia set 
(J-set) is not in the formula, but in the type of iteration involved.  
The complex number c is called a parameter. The M-set is plotted in 
the *parameter space* of c, whereas the J-set is not. It is instead 
plotted in the orbit space z.  That is, for the M-set, you calculate:

   0, c, c^2 + c, (c^2 + c)^2 + c, ...

for a given complex number c, and see if the sequence converges. 
If it does, that point c in the complex plane is in the M-set. So what 
you're plotting is a picture of all complex c such that the mapping 
z |-> z^2 + c is bounded, where the initial condition is z = 0.  
However, the J-set is plotted for a *single* value of c, but you are 
now changing the initial condition:

   z, z^2 + c, (z^2 + c)^2 + c, ....

The value of c is fixed for all points z. So in a J-set you're plotting 
in the space of starting values of z.

In short, you plot the M-set by iterating over many different values 
of c, whereas in the J-set you fix c and iterate over many different 
values of z. In both cases, the formula for iteration is the same; 
only two things change:

   M-set:  starting condition is z = 0
           pick a point in the plane, c
           iterate and see if bounded; if so then c is in the M-set

   J-set:  fix a number c -- this never changes
           pick a point in the plane, z
           iterate and see if bounded, if so then z is in the J-set.

So there are many J-sets, one for each c, but only one M-set. If you 
think about it, the M-set is a "guide" to all the J-sets. What I mean 
is that if a point c is in the M-set, then the J-set with starting 
value c is connected. This is because there is a theorem which says 
the point z = 0 is in a J-set if and only if that J-set is connected.

For an online introduction to fractals, please see Paul Bourke's page:

  http://astronomy.swin.edu.au/~pbourke/fractals/fracintro/   

- Doctor Pete, The Math Forum
Check out our web site! http://mathforum.org/dr.math/