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
Brent
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