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 Posted: Fri Dec 10, 2004 7:37 pm |
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Sorry if this gets posted twice)
Been thinking a bit about the Mandelbrot set lately.
One thing, after spending years looking at, and studying,
the damn thing, is I no longer can consider it 'chaotic'.
It's absolutely, perfectly, ordered. Like a crystal. Just
because this order may be a bit beyond the grasp of the
human mind doesn't make it any less so.
Tis a strange place, where two reaches for infinity. A
bizarre uber-fractal, limit fractal. A corrupted radial
symmetry.
One of the things so amazing about it, at least at first,
is all the *apparent* complexity which arises from such a
simple feedback loop invloving x and y themselves. Humans have
learned to use numeric quantites to model physical reality.
Coming from my background using computers, I'm used to
creating simple, regular, shapes. Circles, spheres, polygons,
elipses, etc. So all the strangeness found in the Mandelbrot
set came as a surprise to me, and many other people
apparently.
There are many way to look at the Mandelbrot set. My understanding
is Mandelbrot himself was interested in a map of all
the connected Julia sets. This lead him to the Mandelbrot set.
I think that he downplays the role of computers in its
discovery. I recall he was asked why it wasn't discovered sooner,
and he responded to the effect that no one bothered to look.
I feel it would have been too tedious a structure to likely
have been discovered before computers. Too many computations
involved even to render a small image.
Coming at it from my point a view, which was writing a program
to view the set, I was struck how many time times 2 and powers
of two came up. All over the place. This gemoemtric explosion
turned into an implosion and curled ever tighter into smaller
regions of space. How when progressing to Mandelbrot 'islands'
the chains of features are always powers of two. Just goes to
show the touch of the infinite can make something as simple
as this interesting.
What is to be learned from the Mandelbrot set? What are
its 'practical' uses? I feel there are many things, but most
of them have yet to be realized. The Mandelbrot set is
fundamental, and I feel it may have fundamental things to
say about many things.
A few of the obvious one:
The futility of reductionism. Much of science takes a
reductionistic approach to things. Especially physical
sciences. It is hoped that though an understanding of
the basic parts of matter (for example) that a greater
undertanding can be reached. But reductionism can rarely
be used to understand the emergent behaviour which arises
in physical systems (life itself being one of them). If
even something as simple as the feedback loop invloved
in creating the Mandelbrot set cannot be entirely predicted,
this should give one pause when trying to evaluate more
complex systems in a reductistic manner.
Lessons in connectedness. I feel this is a big one. My
feeling/intuition is there may be something big here
being overlooked. The exact nature of how the set is connected
is still not known. It may have implications for other
types of organized processes.
If anyone has any thoughts they'd like to add regarding
their 'philosophy' of the Mandelbot set, please do so.
-Eric B
http://www.fractalfreak.com
{note, korrect email address boofus {ta) fractalfreak.com {{
PS Haven't done any work on the 'fractalfreak' site lately.
It's still up though. I pawned my computer this summer. 
I'm using a friend's computer to post this
I'm interested in implementing my integer code on an Athlon-64.
The 64x64->128 (bit) multiply would double the speed of my
program. Ideally I'd like to have 2-4 machines working in
parallel to cut the computational time down. Anyone have
any hardware they want to send me?  |
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