Re: Why does the Mandelbrot set work?
- From: "T.H. Ray" <thray123@xxxxxxx>
- Date: Sat, 21 Apr 2007 10:32:51 EDT
This question has puzzled me forever.
The Mandelbrot set has always fascinated me for one
ne thing:
An incredibly simple iterative formula like "z = z^2
+ c"
produces an astonishingly wide variety of organic
shapes.
I have collected some of my findings in this page:
http://warp.povusers.org/snaps/fract/
It's just incredible that all those images have
ve been created
by simply iterating that simple formula and then
coloring the
pixels according to how many iterations it takes to
bailout.
Using a clever color palette produces these stunning
images.
If someone (who is not a math guru and has never
er heard of the
Mandelbrot set) was given the problem of iterating a
formula like
"z = z + c", where c is a complex number in the
complex plane and
then coloring the equivalent pixel according to the
number of
iterations it takes for z to go outside a circle of a
given radius,
and then this person is asked what would the result
be like, he
would probably think a moment and guess, correctly,
that the result
would be a series of colored concentric circles. As
the formula is
iterated z simply grows in the direction of c until
it goes outside
the circle and that's it.
Now, tell this person "how about z = z^2 + c
c instead?". His first
guess will still most probably be "concentric
circles, the radii
probably not being arranged linearly but in a
quadratic way" or
something similar.
Even if you point out that "z^2" doesn't actually
ly just go in
the same direction as the previous iteration due to
how complex
number multiplication works, he would still most
probably just think
that the result is some simple pattern. If he thinks
about that for
a bit he might perhaps guess that it maybe forms a
spiral or something
like that.
When this person is then shown what it actually
ly produces, the
result is most astonishing and unexpected. For a
complete layman
like me, who doesn't understand anything about
complex number
dynamics, the result is most astonishing and
unexpected.
I just can't understand *why* that formula produces
es those results.
I just can't even begin to comprehend how it is even
possible.
If I had no experience whatsoever about the
he Mandelbrot set and
I was given any of the images on that webpage of mine
I mentioned
above and was asked to guess the mathematical formula
it was
produced with, and after having no idea the correct
answer of
"iterating z = z^2 + c and coloring the result
according to the
number of iterations until bailout" (more detailedly
explained,
of course), I would not believe it. There's no way I
could believe
it without actually trying it for myself by making a
program which
tests the claim.
The formula is just so astonishingly small for what
at it actually
produces. It's possible to write an executable (DOS)
binary which
draws the Mandelbrot set in less than 100 bytes (I
have actually
done that).
I have tried to find the answer to this question in
in the internet.
I have failed miserably. There are tons of websites
which explain
*how* to calculate the Mandelbrot set. That's not
what I'm looking
for. There are also some sites which present some
geometrical and
mathematical properties of the set (eg. related to
the bulbs and
numbers of "antenna" branches in each, etc), and
while interesting,
that still doesn't explain *why* it happens, so it
doesn't answer
my question.
Does anyone know the answer to this question, or
or any website which
explains it?
And mind you, I'm not a mathematician. Most complex
ex dynamics theory
papers would probably go well over my head. I would
like a simpler
explanation to this phenomenon.
Suppose one were to ask, why is 2 the sum of
1 + 1? Eventually, one would would probably get
to the convention, that 1 + 1 = 2 is a definition.
Definitions don't explain anything; however, the
elemental term of induction also leads to an
astonishingly wide variety of results, wouldn't
you agree?
What arithmetic iteration and Mandelbrot's complex
plane iteration share, are facts of proportion and
recursion. These characteristics are native to
the one-dimensional real number line; Mandelbrot's
formula is in the two-dimensional complex plane,
where not every recursion is proportionate to the
line and so does not belong to the set. Boundaries
of shape define themselves thus, by maintaining a
Hausdorff dimension that exceeds the topological
dimension. The Hausdorff dimension is associated
with a non-negative real number in the closed
interval [0,inf].
But the deeper facts of self-similarity and self-
limitation elude us.
Tom
.
- References:
- Why does the Mandelbrot set work?
- From: Juha Nieminen
- Why does the Mandelbrot set work?
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