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| Guest |
 Posted: Fri Dec 10, 2004 7:33 pm |
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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
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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| Guest |
| Posted: Fri Dec 10, 2004 7:41 pm |
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Sorry I posted this twice. It's a pet peeve of mine
when other people do so.
My connection/google server were not responding when
I tried to post... very annoying.
-Eric |
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| Tony Roberts |
| Posted: Mon Dec 13, 2004 5:50 pm |
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Guest
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boo@fractalfreak.com wrote:
[quote:16b3f6ece4]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'.
That is good because the set is NOT chaotic. The set is just a map,[/quote:16b3f6ece4]
whereas chaos is about dynamics and things evolving in time.
[quote:16b3f6ece4]
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.
Powers of 2 arise because the Mandelbrot set is a map of a dynamical[/quote:16b3f6ece4]
process (not necessarily chaotic), and powers or 2 arise as the map of
the period doubling route from order to chaos. Read James Geick's book
and Feigenbaum's contribution.
[quote:16b3f6ece4]
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.
Nope. Mathematics has fundamental things to say. The Mandelbrot set is[/quote:16b3f6ece4]
just one tiny corner of the greatest human enterprise ever constructed.
Tony |
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| Roger Bagula |
| Posted: Tue Dec 14, 2004 8:02 am |
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Guest
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Dear Tony Roberts,
The boundary of the Mandelbrot set is d=2.
That is when you look at it very close up very patterned and very
chaotic at the same time.
The 2 dimensional flow of the sort:
dz/dt=-z+z^2+c
can be broken into the map:
z'-z=-z+z^2+c
When the flow is looked at it has chaotic results.
As to the "universality" of the Mandelbrot set:
the square law is the basis of the "field" theory
that governs most of ordinary physics. The
square measure:
m(x,y)=Sqrt[x^2+y^2]
is imbedded in the law of curvature used
in most calculations.
You can't have right triangles without a square law and
trigonometry ( sine and cosine) assumes a square law as well.
It is one reason that fractals haven't become exclusively mathematics.
Dr. Mandelbrot says that fractals are a "cross" discipline science.
He has refused to make it a solely mathematical study.
The point here is that the Mandelbrot set has become symbolic for
science at the "edge" of human knowledge.
Tony Roberts wrote:
[quote:db5445ebc1]boo@fractalfreak.com wrote:
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'.
That is good because the set is NOT chaotic. The set is just a map,
whereas chaos is about dynamics and things evolving in time.
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.
Powers of 2 arise because the Mandelbrot set is a map of a dynamical
process (not necessarily chaotic), and powers or 2 arise as the map of
the period doubling route from order to chaos. Read James Geick's
book and Feigenbaum's contribution.
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.
Nope. Mathematics has fundamental things to say. The Mandelbrot set
is just one tiny corner of the greatest human enterprise ever constructed.
Tony
[/quote:db5445ebc1]
--
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
alternative email: rlbtftn@netscape.net
URL : http://home.earthlink.net/~tftn |
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| Guest |
| Posted: Tue Dec 14, 2004 8:41 pm |
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Roger Bagula wrote:
[quote:a945fecec1]Been thinking a bit about the Mandelbrot set lately.
I haven't looked into fractals, yet. Could you give me a good link to
get a basic understanding?
(...)[/quote:a945fecec1]
I really don't have any interesting links to offer you.
There's a considerable amount of stuff on the Mandelbrot
set on-line, if you do a search...
(...)
[quote:a945fecec1]
Now this really captures my attention. I found the same types of
patterns in my research as well. I'd like to learn more about these
patterns and when they occur.
Dave
[/quote:a945fecec1]
Again, don't have much to offer you. My initial 'attraction' to
the Mandelbrot set was the imagery - and it still is. Sorry
I don't have more to offer. I was interested more in any
feelings/implications people may have regarding the set.
-Eric B |
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| Guest |
| Posted: Tue Dec 14, 2004 8:49 pm |
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Tony Roberts wrote:
[quote:105e6728b4]boo@fractalfreak.com wrote:
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'.
That is good because the set is NOT chaotic. The set is just a map,
whereas chaos is about dynamics and things evolving in time.
[/quote:105e6728b4]
Right. I knew someone would bring this up. I was just trying to
clarify this fact, because many people somehow consider the
Mandelbrot set an example of chaos. It's not - and this includes
the underlying dynamics of the feedback loop.
[quote:105e6728b4]
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.
Powers of 2 arise because the Mandelbrot set is a map of a dynamical
process (not necessarily chaotic), and powers or 2 arise as the map
of
the period doubling route from order to chaos. Read James Geick's
book
and Feigenbaum's contribution.
Right.[/quote:105e6728b4]
[quote:105e6728b4]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.
Nope. Mathematics has fundamental things to say. The Mandelbrot set
is
just one tiny corner of the greatest human enterprise ever
constructed.
Tony
[/quote:105e6728b4]
Nope, as in you don't think the Mandelbrot set itself may be
significant, beyond just interesting images? And I never
excluded mathematics as being fundamental, either. Sheesh.
-Eric B |
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| Tony Roberts |
| Posted: Wed Dec 15, 2004 6:38 pm |
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Guest
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Roger Bagula wrote:
[quote:2ead2f2ff2]Dear Tony Roberts,
The boundary of the Mandelbrot set is d=2.
That is when you look at it very close up very patterned
Very true.
and very
chaotic at the same time.
Very false, at least in the mathematical sense that I assume we are[/quote:2ead2f2ff2]
using. I reiterate, mathematical chaos is a dynamical process that
happens to have (usually) intricate geometric facets. The boundary of
the Mandelbrot set is not chaotic in the sense you should be using.
[quote:2ead2f2ff2]The 2 dimensional flow of the sort:
dz/dt=-z+z^2+c
can be broken into the map:
z'-z=-z+z^2+c
When the flow is looked at it has chaotic results.
No need to teach me basic dynamics, I have been researching them for 20+[/quote:2ead2f2ff2]
years.
[quote:2ead2f2ff2]
As to the "universality" of the Mandelbrot set:
the square law is the basis of the "field" theory
that governs most of ordinary physics. The
square measure:
m(x,y)=Sqrt[x^2+y^2]
is imbedded in the law of curvature used
in most calculations.
You can't have right triangles without a square law and
trigonometry ( sine and cosine) assumes a square law as well.
Irrelevant to the issues under discussion.
It is one reason that fractals haven't become exclusively mathematics.
Dr. Mandelbrot says that fractals are a "cross" discipline science.
He has refused to make it a solely mathematical study.
The point here is that the Mandelbrot set has become symbolic for
science at the "edge" of human knowledge.
Irrelevant.[/quote:2ead2f2ff2]
Tony |
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