Deciding whether or not a point is in the Mandelbrot set

Recursive Process

Start: Pick a point. Write it as a complex number s.
Create: the algebraic expression x2+s.
  1. Let x=s
  2. Evaluate x2+s
  3. Let x=result (but don't change s)
  4. Go to Step 2
                                          
Repeat this process until you can tell what is happening to the sequence of all these x's: are they increasing and/or decreasing without bound? If so, the point you started with is not in the Mandelbrot set, and should be colored non-black. If not, the point you started with is in the Mandelbrot set,and should be colored black.


Example:

Start: Pick the point (1,0). Written as a complex number, this is the seed s=1+0i=1.
Create: the expression x2+1.
Note: You do not change the ``+1'' part as long as you're working with this seed.

  1. Letx=1
  2. x2+1=12+1=2
  3. Let x=result=2. (Remember, s stays as 1).
  4. x2+1=22+1=5
  5. Let x=result=5.
  6. x2+1=52+1=26
  7. Let x=result=26.
  8. x2+1=262+1=677
  9. Let x=result=677.
                              

The first several terms of the Mandelbrot sequence associated with s=1 are:

{1, 2, 5, 26, 677}
We can see that this sequence will continue to grow without bound.
The point (1,0) is not in the Mandelbrot set, and we will give it a color based on how fast it is escaping.


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