Fractals, Complex Numbers, and Chaos

Date: 01/20/97 at 19:18:18
From: Lua Choon Ngee
Subject: Fractals

I am reading up on fractals. It seems that they use complex numbers 
to plot a mathematical formula (such as the Julia set of fractals). 
Actually, I have not studied complex numbers (such as -2^.5). 
Therefore, I would like to ask about complex numbers and formulas for 
fractals.  I have the formula for the Mandelbrot Set and the Julia Set 
but other that that, I do not have any others. 

Also, is chaos theory related to fractals in any way?

Date: 01/21/97 at 23:18:30
From: Doctor Toby
Subject: Re: Fractals

A lot of questions!

Fractals don't necessarily have anything to do with complex numbers.
But complex numbers are a convenient way to get points on a plane.
A complex number takes the form a + ib, where i^2 = -1.  So you can 
add them: (a + ib) + (c + id) = (a + c) + i(b + d).  And you can 
multiply them: (a + ib) (c + id) = ac + i(ad + bc) + i^2 bd; because 
i^2 = -1, this is equal to (ac - bd) + i(ad + bc).  Just as you can 
plot real numbers on a line, you can plot complex numbers on a plane; 
a + ib has coordinates (a,b).  There's more to complex numbers 
involving taking roots and logarithms and stuff like that, but you 
don't need that for things like Julia sets.

Here's a familiar formula for a new fractal you may not know:  
Compare the formulas for the Mandelbrot set and the Julia sets.  

The Mandelbrot set says start with z = 0 and repeat the process of 
replacing z with z^2 + c. If the iteration is bounded, c is in; 
otherwise not.  

The Julia set associated with the complex number c says start with 
z = k and repeat the process of replacing z with z^2 + c. If this 
iteration is bounded, then k is in the set.  

You can combine these sets into one giant 4 dimensional fractal.  
Start with z = k and iterate by replacing z with z^2 + c.  If the 
iteration is bounded, the pair (c,k) belongs to the set.  The 
Mandelbrot and Julia sets are just cross sections of this 4D fractal.

If you know about Newton's method for solving polynomial equations,
try applying it to an equation such as x^4 = 1. (In the complex 
numbers, this has 4 solutions: 1, -1, i, and -i.)  Starting with the 
complex number c as an initial estimate, color the point c according 
to which solution Newton's method finds. You'll have four patches of 
solid color near the four solutions, but the boundary between them 
will be fractal, like the boundary of the Mandelbrot set.

There are many other fractals that look little like the Mandelbrot 
set, such as the Cantor set, the Sierpinski gasket, and the Koch 
snowflake. (These fractals have nothing to do with complex numbers.)
I don't have the space to describe all these now; if you're 
interested, you should be able to find them in some of the many 
popular books on chaos or fractals.  James Gleick's book _Chaos_ 
(Viking, 1987) is widely cited.

As you might guess from the last paragraph, chaos and fractals 
definitely have a lot to do with each other.  Near the boundary of the 
Mandelbrot set, a small change from one point to the next can make a 
big difference in the point's membership in the set, and there's no 
way to predict this change without going through the entire 
calculation.  That's the hallmark of chaos: sensitive dependence on 
initial conditions.  Basically, any graph of a chaotic process is 
likely to be a fractal, and every fractal is the graph of some chaotic 
process.

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