Originally Posted by Abra
So I guess there could be discreet levels, then. They'd be indistinguishable from any other level. Right?
Depends on what kind of fractal we're talking about. All fractals have selfsimilarity but only a subset of those have exact copies of themselves.
Originally Posted by Abra
In that case, in this entire thread I mean to talk only about dynamical systems like Mandelbrot (that are fractal and chaoticthough what are examples of chaotic dynamic systems that are not fractals? Is that like how emergent properties of chemistry, as chemicals grow larger and more complicated, give rise to properties of biology?). Ones that are chaotic and do not produce exact replicas.
The Mandelbrot set is a set of points in the complex plane defined by whether or not a given point escapes to infinity when the complex quadratic function is applied to it repeatedly. It's a set, not a dynamical system. The dynamical system is the quadratic function.
Fractals are sets. Dynamical systems are functions. You may as well be asking whether there's such thing as a grapefruit that can fly. Now, fractals can be generated by drawing the results of dynamical systems in certain ways, but you would need to define that. I don't think anyone can tell you definitively whether or not there's such thing as a chaotic dynamical system that can't be iterated in such a way as to produce a fractal set. Aside from maybe the trivial systems.
Originally Posted by Abra
In a dynamical system, there are technically no discreet levels, as there would be for a sierpinski. Is this true?
See above.
Originally Posted by Abra
Also, I repeat my question about program B a few posts above, this time only as it applies to dynamical system fractals.
Ok, so you want to know if there's a program that takes as input a set of points and determines whether or not it's a fractal in a finite time. My instinct says the answer is No, because a fractal can be arbitrarily complicated and memory space is finite. With infinite memory, maybe. That's a deceptively difficult thing to answer, because it involves varying degrees of "infinity". You would probably have to ask a PhD to get a straight answer.
Originally Posted by Abra
On the side, what do you mean by applying an escape algorithm? What is that?
Pick a point in C (the complex plane). Apply the dynamical system z > z^2 + c (where c is some constant to determine the location of the set you end up drawing), and do it X number of times until z goes to infinity ("escapes" to infinity, hence the name) or you're satisfied that it won't go to infinity. If it escapes, it's not in the Mandelbrot set. Otherwise, it is.
Example:
z = 1 + i, c = 0.
(1+i)^2 = 2 + 2i, so z = sqrt(8)
(2+2i)^2 = 8 + 8i, so z = sqrt(128)
etc...
You can see that z grows very quickly. Obviously, (1+i) is not in the Mandelbrot set. You can also perhaps see why the Mandelbrot wasn't discovered until the 1970s, when computers became common.


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