Chapter 12:
Mandelbrot Sequences
The Mandelbrot Set is described mathematically
using computations of complex numbers.....
b
Recall that complex
numbers can be graphed
in a Cartesian coordinate
system...
a
a + bi
3 + 2i
-4 + 5i
The Mandelbrot Set is constructed by plotting points
from a Mandelbrot Sequence.
The Mandelbrot Sequence is a sequence of
complex numbers that begins with a seed (s) and
then each successive term is obtained recursively
by adding the seed to the previous term squared.
seed = s
s1 = s 2 + s
s2 = s12 + s
s3 = s22 + s
So, each point on the Cartesian plane is a complex number,
and thus a seed of some Mandelbrot Sequence.
The pattern of growth for the seed determines where the
seed is located, and the color it is assigned.
Mandelbrot Sequences can be:
*Escaping
*Periodic
*Attracted
Escaping Sequences:
(the terms get larger and larger very quickly)
Example: Let the seed be 1
*The seeds of escaping Mandelbrot sequences are not in the
Mandelbrot set and must be assigned a color other than black.
"Hot" colors are used for seeds that escape slowly.
"Cool" colors are used for seeds that escape quickly.
Periodic Sequences:
(the terms repeat themselves in a cycle)
Example: Let the seed be -1
Example: Let the seed be i
When the Mandelbrot Sequence is periodic, the seed
is a point in the Mandelbrot set and colored black.
Attracted Sequences:
(the terms get closer and closer to a fixed complex number)
Example: Let the seed be -0.75
When the Mandelbrot Sequence is attracted, the seed
is a point in the Mandelbrot set and colored black.
More practice with complex numbers:
Let the seed be 1 + 3i