Some general questions about the Mandelbrot Set (Fractals)

[Kyriakos]

Some very general questions:

1) Why was this particular polynomial set as the one iterated?


z_{n+1}=z_n^2+c

(http://en.wikipedia.org/wiki/Mandelbrot_set)

2) I read that the Set, with the characteristic shape ( http://en.wikipedia.org/wiki/File:Ma...elbrot_set.jpg ) is the collection of all the points for which it still holds true that the series created by the iteration of X (ie x, F(x), F(F(x), F(F(F(x) and so on) will remain bounded, which means that it will not escape to infinity. I can see why the series (which involves complex numbers) remaining bounded is significant, but what is the special math significance of an iterated series regardless of complex numbers or not?

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Thanks in advance for any help with the two questions. I suppose i will read more of this, but the first question is by far the one i mostly would like a specific answer to

[Sphere]

The answer is computers.

If the idea is to just to get some sort of set of bounded complex numbers, the simplest way to get there is just a squaring function. Mandelbrot was an IBM guy working with rather primitive computers so when he went about trying to plot them, he kept the function simple so it wouldn't take 1,000 years to compute.

Now with advanced computing people have taken the idea and run with it, trying out far more complicated functions.

[Kyriakos]

That is a pretty big let-down for me

I sort of hoped that the specific set (Mandelbrot) had some particular reason to be bounded, for example that plant life or other formations in the cosmos appeared to have a similar pattern.

So, is there any more significant (in the above manner) iteration and resulting fractal which i could look into?

[Sphere]

There seem to be some interesting Julia sets which use the golden ratio.

A Julia set is similar, except for instead of asking what set of "c = ?" complex values are bounded when you start with z0 = 0. You instead pick a single value for "c" and ask what set of "z0=?" complex values would converge.



According to wiki this is the julia set for...



c=(φ−2)+(φ−1)i =-0.4+0.6i

φ is the golden ratio