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.