On 9/19/2012 8:39 AM, Bruno Marchal wrote:


On 18 Sep 2012, at 18:02, meekerdb wrote:

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.

I think Julia worked on the "Mandelbrot's Julia sets", notably. The Mandelbrot set is a classifier of the Julia sets. You can define the Mandelbrot set by the the set of z such that z belongs to its Julia J(z).

The point is that in math and physics such object are hard to miss, even if you need a computer to figure out what they looks like.




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 practice, yes. But if I remember well, the point is that the M sets and alike are discovered, not fictitious human's construction. To see them, we need a computer, but to see a circle you need a compass, or a very massive object, like the sun or the moon, ...




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).

Hmm, the inside mandelbrot set has dimension 2, as you can extrapolate from the big spot, and then the filament ar made of little mandelbrot set. So you can always see something. You are correct, for the filaments: usually we can see them, as the little Mandelbrot sets are too small. The coloring only makes them less thin and more easily observable, but you would see the same basic shape with a pure black and white picture. for example, everywhere on the main (straight) antenna, there is a little mandelbrot set, so even black and white resolution will make a thin line (with always too big pixels, of course). Of course, the line can become thinner and thinner, so with deeper zoom, you will have "to darken the picture", and then light it up, etc. Of course this is true also for a circle, or a straight line, which are too thin to be seen, too, but we don't worry to draw them with chalks or pens, which approximates them quite well.

You can see that phenomenon here:

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


Bruno



Dear Bruno,

Your remarks raise an interesting question: Could it be that both the object and the means to generate (or perceive) it are of equal importance ontologically?

--
Onward!

Stephen

http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html


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