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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'
Recapture

Copyright (C) 1990 by Homer Wilson Smith

A Julia Set acts as a fenced in boundary to Z's starting off inside
the Julia Set.  Those starting off outside quickly go to infinity, and
those inside quickly attain their stable periodic or chaotic orbits.

Thus if a Z starts off inside the Julia Set it will still be inside
after one iteration and closer 'in' so to speak.  If a Z starts off
outside the Julia Set, even by an infinitesimal amount, it will be
further outside after one iteration.

The question arises, what happens if C changes between iterations?
its Julia Set and a Z inside that same Julia Set.  After one iteration
of Z = Z*Z + C, Z will have moved to some other point still inside.  If
we now change C, perhaps by the equation C = C/2 + Z, then a new Julia
Set will form in place of the first one.

If this new Julia Set contains the Z point we just iterated, then
the next iteration of Z using the NEW C will move Z to another point
still inside the new Julia Set.  If however the new Julia Set does not
contain the Z, then the next iteration of Z will cause it to move to
another point further away from the new Julia Set.

Clearly with each change in C, the new Julia Sets either will or
will not contain the previous Z.  If they do contain the Z, the Z value
will stay CAPTURED in the ever changing Julia Sets.  Even if a few Julia
Sets do NOT contain the Z, the Z will move away towards infinity but may
still be RECAPTURED by further Julia Sets before it reaches a point of
no return.  If however a Z moves far enough away (point of no return),
no possible Julia Set can ever recapture it and it will go to infinity
and get colored.

The Tarantula Rose, a movie on the video tape 'Mandelbrot Sets and
Julia Sets' was made using just such an iteration.  C is made to change
after each iteration of Z making it very hard for Z to guarantee that it
will stay inside the sequences of Julia Sets.  In fact it is not obvious
that any Z would ever stay captured at all.  Only by looking at the
computer pictures does it become obvious that life is indeed possible in
a changing environment.  Z stays 'alive' by NOT going to infinity which
means it stays captured with in the reasonable bounds of the ever
changing Julia Sets.

Z = Z*Z + C
C = C/2 + Z

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