Path: utzoo!attcan!uunet!cs.utexas.edu!hellgate.utah.edu!cs.utah.edu!thomson
From: thomson@cs.utah.edu (Rich Thomson)
Newsgroups: comp.graphics
Subject: Re: fractals as bad science
Keywords: Fractals, ugh!, mathematicians, pure, applied, non-
Message-ID: <1989Nov24.114609.8837@hellgate.utah.edu>
Date: 24 Nov 89 18:46:09 GMT
References: <19544@pasteur.Berkeley.EDU> <1619@crdos1.crd.ge.COM> <3775@celit.fps.com> <5383@orca.WV.TEK.COM> <3842@puff.cs.wisc.edu>
Organization: Oasis Technologies
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In article <5383@orca.WV.TEK.COM> brucec@demiurge.WV.TEK.COM
    (Bruce Cohen) writes:
>I think what you're seeing here is the classic
>antipathy of "pure" mathematicians for "applied" mathematics.

And in article <3842@puff.cs.wisc.edu> zerr@schaefer (Troy Zerr) writes:
[ Long discussion about what applied mathematics is, focusing on the
  concept of rigorous proof as a means of validating research in applied
  mathetmatics -- his example uses differential equations to calculate the
  form of a suspended telephone wire, e.g. a catenary ]

]      Constrast this with the activity of a prototypical "fractal
] geometer", the sort whose books are often seen on bookstore shelves.
] I program a computer to [ create fractal images via typical random
] subdivision method ].

]      Am I to conclude that islands are constructed by some geological 
] process of repeated subdivision?  Of course not!  All I have done is to
] produce a picture which looks (to me) like an island.  

Are we to conclude that at the subatomic level differential equations
represent the structure of the telephone wire?  The example you've used
(random subdivision) may produce a similar structure, at a gross level, to
an island in the same way your differential equation provides a gross level
description of the spatial configuration of a suspended telephone wire.  I
could use the same reasoning to argue that smooth curves (i.e. polynomials)
have absolutely nothing to do with nature because nature isn't smooth!
Then I could argue that fractal descriptions are better because, while they
are not perfect models, are better models than smooth analytic functions.
But of course, this would all be meaningless because a model is only good
until you find a better one and the process evolves an ever-refining
picture (note that the evolutionary development of ideas is sometimes
discontinuous: ex.  relativity).

]      Admittedly, I chose a particularly obnoxious example -- "fractals"
] may indeed be good models of some physical phenomena.

Perhaps you should take a look at "fractals" from another perspective.  Look
at the work done in sting rewriting systems.  First, go read Alvy Ray
Smith's article[1].  Then, after you've familiarized yourself with the
basic notion, take a look at the work that Aristid Lindenmayer has done
with string rewriting systems to model growth.

Another direction one can go with "fractals" is into the embryonic chaos
theory that has been worked out by people like Devaney in his book[2], or
start with Gleick's book _Chaos: The Making of a New Science_.  You may
find out that self-similar curves (in particular, the Cantor Set) have more
relevance to the real world than what you thought.

I agree that there is alot of people crunching on Mandelbrot set
algorithms, etc, without understanding the underlying mathematics and
theories.  I feel that this comes from the fact that the theories are
currently inaccessible to the layman in terms of well-reasoned and
practical explanations.  Go ahead and TRY and read Devaney's book and
you'll see what I mean.  We tried to use it here at the University of
Utah's CS department and found it woefully painful to read and, more
importantly, understand.  Perhaps we're not die-hard mathematicians, but
we're not stupid either; with out little group we had a good coverage of
areas of knowledge that helped us out, but I don't think any of us (there
are about 6) would have gotten much by reading the book alone.
Mandelbrot's work has been more popularized, but he does tend to bathe in
his own "glory" and his books (I found) didn't exhibit a uniform formalism
or theoretic explanation; I found them to be useful introductory surveys.

Things are improving, though currently the trend is to publish books that
show you how to generate all these "fractal" images without understanding
the ideas behind them.  Personally, I feel it is more useful to study the
works of the emerging "chaos theory" and things like Rene' Thom's
catastrophe theory (as outlined in [3]).  If you're looking for real-world
modelling applications using fractal methods, check out the course notes
from this year's SIGGRAPH course taught by Prusinkiwiecz (I hope I spelled
that right) and Hanan[4]

[1] "Plants, Fractals, and Formal Languages", Computer Graphics 18:3,
    1984

[2] _An Introduction to Chaotic Dynamical Systems_, 2nd. ed., Robert L.
    Devaney, Addison-Wesley, 1989

[3] _Structural Stability and Morphogenesis_, Rene' Thom, 197?

[4] _Lindenmayer Systems, Fractals and Plants_, P. Prusinkiwiecz & J.
    Hanan, 1989

] -Troy Zerr
] University of Wisconsin
] Department of Mathematics
] Madison, Wisconsin
] zerr@math.wisc.edu

						-- Rich

Rich Thomson	thomson@cs.utah.edu  {bellcore,hplabs,uunet}!utah-cs!thomson
"Tyranny, like hell, is not easily conquered; yet we have this consolation with
us, that the harder the conflict, the more glorious the triumph. What we obtain
too cheap, we esteem too lightly." Thomas Paine, _The Crisis_, Dec. 23rd, 1776