This question has puzzled me forever.
The Mandelbrot set has always fascinated me for one 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 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 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 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 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 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 those results.
I just can't even begin to comprehend how it is even possible.
If I had no experience whatsoever about the 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 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 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 any website which
explains it?
And mind you, I'm not a mathematician. Most complex dynamics theory
papers would probably go well over my head. I would like a simpler
explanation to this phenomenon.