On 19 Sep 2012, at 17:03, Stephen P. King wrote:
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?
Yes. It comes from the embedding of the subject in the objects, that
any monist theory has to do somehow.
In computer science, the "universal" (in the sense of Turing)
association i -> phi_i, transforms N into an applicative algebra. The
numbers are both perceivers and perceived according of their place x
and y in the relation of phi_x(y).
You can define the applicative operation by x # y = phi_x(y). The
combinators are not far away from this, and provides intensional and
extensional models.
I remind you that phi_i represent the ith computable function in some
effective universal enumeration of the partial computable functions.
You can take LISP, or c++ to fix the things.
Bruno
--
Onward!
Stephen
http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html
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