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### Mandelbrot and Julia Sets

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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
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/
```
Associated Topics:
High School Fractals

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