## The Beauty of Fractals

I'm a reasonably intelligent
galoot, but what I can't understand is this: That beauty such as this
exists neatly tucked away in common, garden-variety complex numbers. Take a
small area along the periphery of the region known as the

Mandlebrot Set.
For each complex number

**c** in that area, iterate the
following:

**z**(0) = **c**

**z**(*n+1*) = **z**(*n*)^{2} + **c**

Now, the magnitude of **z**(*n+1*) will either remain finite or
not. If it's going to go infinite, it will exceed 2. If that happens,
assign a color to that point based on the value of *n* when it
happens. Otherwise, continue iterating until the maximum number of
iterations is reached, and assign black to that point. Then plot out all of
the points. I used to do this in FORTRAN on a VAX, plotting on an old
low-res paintjet printer. It's a lot easier
now.

lower lefthand corner = 0.25932155869498 +
0.00145611369495**i**

width = 5.459818e-008

maximum iterations = 2048

lower lefthand corner = -0.75411018025570 +
0.05536016198561**i**

width = 6.146660e-010

maximum iterations = 2048

lower lefthand corner = -0.75411017990214 +
0.05536016227582**i**

width = 8.817630e-011

maximum iterations = 2048

lower lefthand corner = -0.75411017990214 +
0.05536016227582**i**

width = 8.817630e-011

maximum iterations = 10000

Copyright © Peter L Berglund, 2006. All rights
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