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ART MATRIX  PO 880  Ithaca, NY  14851-0880  USA
(607) 277-0959, Fax (607) 277-8913

'The Paths of Lovers Cross in the Line of Duty.'

THE THEORY BEHIND 'THE CELL AND THE WOMB'
Part 1

Copyright (C) 1988 by Homer Wilson Smith

The following interpretation has received much criticism and some
praise.  It is not presented here as TRUE, only as food for thought.

Some people seem to take immediate offense at the thought that
fractal math might explain something.  I don't really understand why,
they seem so defensive by saying things like 'we don't need fractals to
explain this.'

My position on this is as follows.  Every equation that is non-
linear and iterated (not merely evaluated) will show fractal
manifestation.  These manifestations fall into three categories.

1.)  Stability/Unstability.  This describes how small or
infinitesimal changes to the input affect the output.

2.)  Periodicity/Chaos.  This describes the behavior of the output.

3.)  Fractal Dimension.  This describe the convolutedness of
various boundaries or shapes involved with input spaces and output
spaces.

One example where I have been unceremoniously attacked for
suggesting that fractals might apply is to planetary motion.  Planets
clearly do not show the rich and varied behavior that most quickly
associate with fractals.  Their motion is like a pendulum, very boring
and uninteresting.  Thus on the surface it might seem that fractals
have nothing to do with planetary motion.

However their very 'boring' behavior immediately comes under the
classification of PERIODIC.

Furthermore perturbations to their orbits do not especially change
in wild disarray what they were originally doing.  This comes under the

Next, planetary motion is known to be the result of equations
containing 1/R**2 terms which is highly NON LINEAR.

Lastly, a planet's position can be thought of as being a function
of it's just previous postion, so clearly this comes under the heading
of ITERATION.

All that is missing is the complex swirls and convoluted
boundaries that people normally associate with fractals.  Thus they
claim that fractals don't apply here.

STABLE, PERIODIC behavior can be one type of FRACTAL BEHAVIOR!

It obliges us therefore to look where the planetary equations might
start acting in an UNSTABLE fashion producing strange PERIODS or even
CHAOTIC behavior.

Fractals do not EXPLAIN anything at all.  Fractals are not CAUSE,
they are EFFECT.  Fractal behavior is merely a description of what
some equations do given some inputs.

Thus any system modeled on non linear iterated equations, should
be considered to be showing fractal behavior even if it is STABLE and
PERIODIC.  If one looks further one should be able to show how the
input could be changed to create chaotic output results and to map the
input areas of greatest instability, for example where the output
changes without warning from periodic to chaotic and visa versa.

Voila, pretty pictures!

Thus people who claim that 'Fractals do not apply here' are
almost uniformly wrong except in the very few cases where the model
uses a linear equation or is a non iterated system.

If the model is linear and non iterated and works, one can not
argue about that.  However very few things can truely be modeled on a
straight line.  And almost everything in existance is a function of
what it was just before.  Thus it behooves us to look at non iterated
models to see if they can't be rewritten in an iterated form.

I expect to be attacked for this view.  It's like wearing a sign
that says 'Kick me'.

Homer
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