Path: utzoo!utgpu!jarvis.csri.toronto.edu!clyde.concordia.ca!uunet!bywater!scifi!ndla!platt
From: platt@ndla.UUCP (Daniel E. Platt)
Newsgroups: comp.graphics
Subject: Re: Fractals, and Philosophy of Science
Summary: causes...
Message-ID: <266@ndla.UUCP>
Date: 7 Jan 90 22:46:29 GMT
References: <119.256E54C5@uscacm.UUCP> <1247@becker.UUCP> <12707@phoenix.Princeton.EDU>
Lines: 55

In article <12707@phoenix.Princeton.EDU>, markv@gauss.Princeton.EDU (Mark VandeWettering) writes:
> 
> Yes!  Mathematically speaking, the concept of fractal dimension is rigorously
> defined.  Chaos theory has some very specific results, as does complexity 
> theory.  Because of the hype surrounding fractals, one sees grandiose claims
> about the applicability of fractals to describing natural objects.  
> The problem with this is twofold:
> 
> 	a)	rarely are such descriptions analytically compared with
> 		the objects they describe. ... [stuff deleted]
>
> 	b)	descriptions say nothing about the processes that generated
> 		the object in the first place.  Fractal mountains don't
> 		react to erosion or gravity, fractals trees don't grow with
> 		the wind etc....
> 	

Actually, it isn't fair to say that the descriptions say nothing about
the processes that generate the shapes.  While fractality in and of itself
doesn't assert causes to things, that doesn't mean that there aren't
descriptions that define causes quite effectively in terms of fractal
structures.

Two examples come to mind.  First is the formation of blood-vessels, and
the second is the formation of sea-shores.

First, blood vessels, during the time when differentiation of cells and
all that good stuff is going on, will grow in regions where demand is
high; measured by gradients of nutrients and oxygen &c...  This implies
that the formation of blood vessels would grow in a manner that is 
diffusion limited.  In a retina, the growth is restricted to 2-D (sort
of a unique condition).  Measurements of the dimensions of retinal vessel
patterns compare favorably with Diffusion Limited Aggregation clusters.

Second, Sapoval has recently done some simple experiments looking for
exitation modes on fractal gaskets.  He found that 1) there was little
mode coupling between several related regions (where coupled modes could
be expected from symmetry), and 2) that this could be expected to be
due to very low Q values at any and all length scales.  In other words,
fractal resonance chambers are great dampers at all length scales.  This
has also been impirically noticed in acoustical studies where rooms are
designed with points and peaks randomly distributed in size and placement
(essentially making a fractal even though the designers didn't think of
it that way -- they just knew it worked best that way).  So, what does
this have to do with sea-shores?  If a sea-shore starts out with a non-fractal
shape, it will probably have a nice resonance at some frequency (as well
as an associated length at that frequency).  Pounding at that frequency
will distort the shore until that resonance disappears.  The end result
is a shore with very little in the way of a supported resonant frequency --
in other words, fractal.

Thus, there are mechanisms where fractal descriptions actually point to
a description of the cause. 

Dan Platt