Re: What has fractal theory achieved?

On 23 Jun, 17:51, "T.H. Ray" <thray...@xxxxxxx>
wrote:
Dirk Van de moortel wrote:

Tomoko Kanazawa dom arigato
<huangxienc...@xxxxxxxxx> wrote in message


09400d1b-c1fb-47a0-a3ac-4d0dd844a...@xxxxxxxxxxxxxxxx
legroups.com

Harold Weissman wrote:

We are all aware that fractals result in
very
nice pictures,
useful compression schemes and some
cosmological
models.
My question, what has the impact of fractals
and
fractal
theory been on mathematics and physics as a
whole? Have
they opened any new avenues of exploration?

In my opinion -

The study of fractals points toward a very
curious
problem. You have a
very simple algorithm which is based on
randomness
which creates all
of this incredible struture and beauty.

So...how is it that you get all of these
beautiful
structures from a
process which is based on randomness ?

It seems to me that we are mixing order with
disorder. It is a
chemistry experiment, and there is no known
mathematical apparatus
available to properly explain how to do such a
thing purposefully. We
dont have the nomenclature, or the philosophy
in
place to properly
begin mixing order with disorder.

So - in my opinion - fractals are like a very
loud
HINT that a very
big chunk of math is out there somewhere just
waiting to be invented.

Where is the randonmess in a simple algorithm
like
for instance the one
that produces the Mandelbrot set?
As far as I can tell, it is just a simple but
completely
determined transformation with nothing random
about
it.
I admit that it's a while since I read some of
these books,
but ... did I miss something?

Dirk Vdm

Yes, you missed something. _Other_ simple
algorithms
may produce e.g.
snowflake patterns, which look rather superficial
though. And _these_
"rigid" fractals are made more natural by adding
a
pinch of randomness
to them. That's the way e.g. clouds are done in
computer animations.
Disclaimer: iff I'm still informed well ..

That's something else. The subject was the
Mandelbrot
set. The only possibly random element is the
choice of
initial condition. Other self-similar
constructions,
e.g., Julia set, Koch snowflake, Sierpinski
triangle,
are also deterministically generated.


This is not correct. The Mandelbrot set is simply
recursive, not
recursively enumerable. Namely, you know if an
element is NOT in the
set (it escapes the unit circle in the sequence), but
you do not know
if an element is in the set (you just cut off after a
certain amount
of iterations, just like you cannot solve the halting
problem).

That's true. The program is, however, deterministic
in that its output is determined by its input.
Sensitive dependence on initial conditions does not
imply random output, even if uncomputable. An example
of true uncomputability is Chaitin's number, Omega,
outputting the probability of halting, which is
"maximally unknowable."

Tom


On the applications of fractal theory, I have the two
books from
Kenneth Falconer on Fractal Geometry. The whole
matter is way too
technical for my understanding, so here is just
something I have
picked up from the table of contents, on the
applications:

-- Iterated function systems (self-similar and
self-affine sets);

-- Examples from number theory (continued fractions
and diophantine
approximation, etc.);

-- Graphs of functions (dimension and
autocorrelation);

-- Examples from pure mathematics (groups and rings
of fractal
dimension, etc.);

-- Dynamical systems (continuous dynamical systems,
etc.);

-- Iteration of complex functions (Julia sets,
Newton's method);

-- Random fractals (random Cantor set, fractal
percolation);

-- Brownian motion and Brownian surfaces;

-- Multifractal measures;

-- Physical applications:

Fractal growth.
Singularities of electrostatic and gravitational
potentials.
Fluid dynamics and turbulence.
Fractal antennas.
Fractals in finance.

And from the sequel:

-- Cookie-cutters and bounded distortion
-- The thermodynamic formalism
-- The ergodic theorem and fractals
-- The renewal theorem and fractals
-- Martingales and fractals
-- Tangent measures
-- Dimension of measures
-- Some multifractal analysis
-- Fractals and differential equations

-LV


Tom

Han de Bruijn
.

Relevant Pages

  • Re: Newbie Programming Language Question
    ... > After reading several books on fractals, I want to write programs to display ... If you have used Turbo Pascal maybe FreePascal whith it's OpenGL ... the books I know deal with progamming OpenGL with C. ... exists a lot of language bindings to diverse languages for OpenGL. ...
    (comp.programming)
  • Re: Eagle in the Snow, by Wallace Breem
    ... > the usual popular science books. ... If you like fractals, you might want to check out: ... those cases that are bounded and exhibit sensitive dependence ...
    (rec.arts.books)
  • Re: What has fractal theory achieved?
    ... snowflake patterns, which look rather superficial ... On the applications of fractal theory, I have the two books from ... -- Dynamical systems ... -- Random fractals ...
    (sci.math)
  • Re: What has fractal theory achieved?
    ... these books, ... Mandelbrot ... And chaos and fractals are very ... -- The ergodic theorem and fractals ...
    (sci.math)
  • [Fwd: Definition of fractals]
    ... Mandelbrot defines fractals, tentatively he says, to be sets for which the Hausdorff dimension strictly exceeds the topological covering dimension. ...
    (sci.fractals)