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Mandelbrot Set explanations
Name: david e maxwell
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995
Question:
Could someone define the mandelbrot set for me in layman's terms?
Replies:
The layman will have to know how to add and multiply complex numbers.
Let C be some complex number. Use it to generate a sequence of complex
numbers, by applying the following rules:
Let the first member of the sequence be 0 + 0i. (2) If Z is the latest
member of the sequence, the next member is Z^2 + C (where Z^2 means "Z
squared" i.e. Z*Z). Thus, for example, the second member of the sequence is
C, the third is C^2 + C. If Z = A + Bi, where A and B are real and
i = SQRT(-1), a way to measure the "size" of Z is SQRT(A^2 + B^2). (If you
plot Z in the complex plane, then this size is in fact the distance between
Z and the origin.) If you keep computing more and more terms of the se-
quence, one of two things will happen: either the size of the terms will
become very large (with each succeeding term larger than its predecessor) or
else the size of all of the terms will remain bounded (in fact, will remain
less than 2; that is, they will all be less than 2 units from the origin).
The bounded sequences do not converge (settle down on one number); they jump
all over, but all of the members stay within 2 units of the origin. If your
choice of C gives a sequence that stays bounded, then C is in the Mandelbrot
set (the "gingerbread man", usually colored black); otherwise it is not.
The coloring of the other points (the "pretty" parts of the images) is meant
to indicate how rapidly the terms of their sequences get large.
That is the "theoretical" definition. Now suppose you are writing a
program to check various values of C to test whether they are in the
Mandelbrot set or not. You cannot compute an infinite number of terms for a
sequence; you have to decide on a maximum number of terms to compute. (Of
course, if any term of the sequence has size of 2 or larger, there is no
problem; you can quit with that sequence, and the C value you used is not in
the set.) But some C values that are not in the set take many iterations to
exceed the 2-limit. These "slow grow" C values tend to be around the "edge"
of the Mandelbrot set. So no matter how many sequence terms you compute,
there will almost always be some C values that appear to be M-set members
but really are not. For graphical purposes, that really does not matter; a
modest maximum number of iterations (say, 100) will produce a pretty good
approximation.
rcwinther
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