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            'The Paths of Lovers Cross in the Line of Duty.'
 
               THE THEORY BEHIND 'THE CELL AND THE WOMB'
                  An example of a NON Fractal system.
 
                Copyright (C) 1990 by Homer Wilson Smith
                          All Rights Reserved
 
 
     We have learned in early school that the equations relating
distance to velocity and acceleration is:
 
     D = 1/2*A*T**2 + V*T + D
     V = A*T + V
 
     Easily enough what this says is that after time T your distance
away from your starting point will be your original distance D away from
D = 0, plus distance gained by virtue of your original velocity V at T =
0 plus distance gained by virtue of added velocity caused by
acceleration.
 
     No problem.  Further a graph of D vs T will show a non linear plot
basically like Y = X*X.
 
     Being non linear one might immediately wonder if there is potential
for fractal behavior in this system.
 
     The answer is no.
 
     To start with, the system as modeled here is not iterated, it is
merely evaluated.  We can however turn it into an iterated system by
choosing a unit of time to match the unit of iteration.  Let's choose
our unit of time to be 1.
 
     Thus the preceding equation can be remodeled and iterated as
follows.
 
     D = 1/2A + V + D
     V = A + V
 
     D is the iterating variable in the first equation, and V is the
iterating variable in the second equation.
 
     If D and V both start off with value of 0, and A is a constant
acceleration, then each iteration will give us new values for D and V
for each second down the road.
 
     This is an iterated system and not an evaluated system because you
can't just plug in the number 10 and get the final distance 10 seconds
down the road.  You have to operate the pair of equations 10 separate
times to get the final answer.
 
     However you will notice that both equations are now linear.  Thus
there is no fractal behavior evident.
 
     The equations have to be non linear in the ITERATING VARIABLE in
order for fractal behavior to be manifest.  Our original equation was
non linear in T but iterated in D.  That is why it is non fractal.
 
     However consider the situation where the acceleration is no longer
constant but is a function of D itself such as in a spring system or a
gravity field.  If A is a non linear function of D, then indeed the
equation in D is non linear and will show fractal behavior.  If we
consider relativistic effects, it is possible that the acceleration will
also be a non linear function of V too.  Then BOTH equations have non
linear terms in the iterating variables D and V and will show dualistic
fractal effects.

     Homer