Re: What has fractal theory achieved?


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.

Tom


Sierpinski is deterministic ? Hmmmm....

I know that there are different ways to generate the
Sierpinski
triangle, but there is only one that I know really
well. It goes like
this:


[1] You have a triangle ABC.

[2] Pick any point inside the triangle ABC to be the
starting point,
call it X1

[3] Pick a corner at random from A,B,C.

[4] Find the halfway point from X1 to that corner,
call this point X2

[5] Rename X2 as X1

[6] Goto [3]



This algorithm makes nice gaskets, regardless of
whether you use
triangles, squares, hexagons, whatever. And it is
based on a random
selection of which corner.

Can you really say this is deterministic ? I dont
know - been thinking
about that one for 20 years now.


"Deterministic" means that the outcome is determined
by the initial conditions input. Nothing mysterious
going on here.

Tom


.

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