How do you generate a fractal?
Fractals are generated using formulas which contain a complex number.  These formulas are iterated, then they are "graphed", usually on a complex plane.  The formulas are shown below.

How do you "graph" the results from a iterated formula?
After each iteration in a formula, you are given two main numbers to act as your "x and y" coordinates.  Since a complex number is used, a will act as the x-coordinate while b acts as the y-coordinate.  As you continue to iterate the formula, you have new coordinates to plot.  After iterating various times, you produce a series of lines.  This is where a rather sophisticated computer is needed.  Since it is impossible to graph an imaginary number, the computer that is used must be able to graph these coordinates and interpret these lines. 

Basic iteration formula
You must define f(x) to apply this formula.

          x0 = 0
x1 = f(x0)
x2 = f(x1) = f(f(x0))
x3 = f(x2) = f(f(x1)) = f(f(f(x0)))
... and so forth...
In this case, the next number in the series is on the left side of the equation.  On the right side, however, the function of the number (s) on the left side of the previous equation is used.


Most commonly used iteration formula (derived from the basic iteration formula):
You must define c to apply this formula. c will always be a complex number.  There are limitations for a and b:  

f(zn) = zn2+c = zn+1 OR  zn+1 = zn2+c

This is what the latter formula means:
z0 = 0
z1 = (z0)2 + c 
z2 = (z1)2 + c
z3 = (z2)2 + c
z4 = (z3)2 + c
...and so forth...
The next number in the series is always on the left side of the equation.  On the right side of the equation, the left side of the previous equation is squared and c is added.  This is a recursive definition.



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This webpage is copyrighted by Freda Auyeung and Jane Panamaneechot (since 2000)