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### Fractals, Complex Numbers, and Chaos

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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?
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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/
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Associated Topics:
High School Fractals
High School Imaginary/Complex Numbers

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