On 9/18/2012 8:13 AM, Bruno Marchal wrote:


On 17 Sep 2012, at 22:25, meekerdb wrote:

But did anybody think z' = z^2 + c was interesting before that?


Yes. This was known by people like Fatou and Julia, in the early 1900.

I knew they considered what are now called fractal sets, but not that 
particular one.

Iterating analytical complex functions leads to the Mandelbrot fractal sets, or 
similar.

The computer has made those objects famous, but the mathematicians know them both from logic (counterexamples to theorem in analysis, like finding a continuous function nowhere derivable), or from dynamic system and iteration.

If you iterate the trigonometric cosec function on the Gauss plane C, you can't miss the Mandelbrot set.

But this iteration is a tedious and impractical *construction* which in practice depends on computers.


In nature too as the following video does not illustrate too much seriously :)

http://www.youtube.com/watch?v=JGxbhdr3w2I

In such beautiful imagery it is generally overlooked that it is not the Mandelbrot set you are looking at, but rather regions colored according how close they are to the set (which cannot be seen at all).

Brent

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