For example:

            ART MATRIX  PO 880  Ithaca, NY  14851-0880  USA
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            'The Paths of Lovers Cross in the Line of Duty.'
                Copyright (C) 1985 by Homer Wilson Smith
     Consider the equation Y equals Z squared (Y = Z*Z) on the positive
real number line from 0 to infinity. If Z starts off as a number bigger
than 1, Y becomes a number even bigger. For example 2 squared is 4, 4
squared is 16, 16 squared is 256. This sequence soon approaches
infinity. This process of using the OUTPUT as the next INPUT is called
FORWARD ITERATION of the equation Y = Z*Z. For Y = Z*Z, infinity is
ATTRACTIVE for all numbers greater than 1.  Infinity is also a FIXED
POINT because infinity squared is infinity (sort of).  If Z starts off
as less than 1, Y becomes smaller and smaller approaching 0.  For
example, .5 squared is .25, .25 squared is .0625 etc.  For Y = Z*Z, 0 is
ATTRACTIVE for all numbers less than 1.  0 is also a FIXED POINT because
0 squared is 0. The number 1 is a very important number because it forms
a dividing line between those numbers that are REPELLED from it towards
0 and those that are REPELLED from it towards infinity.  Yet 1 squared
is 1, so 1 is also a FIXED POINT. 1 is a REPULSIVE FIXED POINT where 0
and infinity are ATTRACTIVE FIXED POINTS.  1 is UNSTABLE because any
number even infinitesimally different from 1 will decay rapidly inward
towards the center or outward towards infinity.
     Expanding this concept to include the entire COMPLEX PLANE, the
number 1 becomes a CIRCLE of radius 1, called a JULIA SET, named after
Gaston Julia, a French mathematician.  This is shown in Fig. J1 on the
other side.  Using standard rules for COMPLEX MULTIPLICATION, any number
chosen INSIDE the circle will FORWARD ITERATE inward towards 0 at the
center, and any number chosen OUTSIDE the circle will FORWARD ITERATE
outward towards infinity, and any number chosen ON the circle will
FORWARD ITERATE to some other number ON the circle.  This is shown by
the green dots and red dots in Fig J1.  The green dot outside the circle
iterates to the red dot further outside the circle, etc.  Thus there are
three distinct regions.  INSIDE the JULIA SET, ON the JULIA SET, and
OUTSIDE the JULIA SET.  Each region has its own FIXED POINT. 0 is the
JULIA SET and infinity is the FIXED POINT for OUTSIDE the JULIA SET.
Notice that although 1 is a REPULSIVE FIXED POINT it does ATTRACT points
ON the JULIA set just as it ATTRACTS itself. The Julia set is an
extension of 1, and so acts similar to 1.
     Fig. J1 is called a Z-SPACE picture because it is in the space of
all possible Z's. In this case Z ranges from -2 to 2 on both axes with 0
in the center.  J1 is actually computed for the equation Y = Z*Z + C
where C is 0. The question arises, what happens if C is NOT 0?  The
pretty color picture in the upper left corner is the MANDELBROT SET,
named after Benoit Mandelbrot of IBM.  It is a C-SPACE picture because
it is the space of all possible C's. C = (0,0) is marked clearly by the
+ next to the letters J1.  If C is chosen on the MANDELBROT SET from the
point marked J2, then the JULIA SET of Fig. J2 becomes evident.
Infinity is an ATTRACTIVE FIXED POINT for all JULIA SETS, but the two
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yellow FIXED POINTS on J2 have moved off their original values of 1 and
0 to become some other numbers.  Since both are now ON the JULIA SET,
the are both REPULSIVE.  Any point starting INSIDE this JULIA SET
FORWARD ITERATES towards the two red points.  The ITERATION ALTERNATES
first near one, then near the other. Every TWO ITERATIONS the point is
back near where it was.  In Fig. J3 there is an ATTRACTIVE CYCLE of
THREE POINTS.  Notice how they surround the original yellow FIXED POINT
that is ON the JULIA SET. J2 is taken from the TWO-BALL of the
MANDELBROT SET, so called because all JULIA SETS taken from within this
region have an ATTRACTIVE CYCLE OF TWO, and J3 is taken from the THREE-
BALL.  J32 is taken from the TWO-BALL off the THREE-BALL. This produces
an ATTRACTIVE CYCLE of SIX POINTS. J0 is taken from the COLORED area of
the MANDELBROT SET picture.  Notice its JULIA set is not a CLOSED CURVE.
Thus points INSIDE the JULIA SET escape to infinity as well as points
OUTSIDE. This kind of JULIA SET is called a CANTOR SET and is composed
of DUST. No point ON this JULIA SET is CONNECTED to any other point.
Thus there is no CLOSED CURVE to fence in BASINS of ATTRACTION with
     Which brings us to how the MANDELBROT SET was computed.  For Y =
Z*Z + C there is a theorem which says that if the JULIA SET is CLOSED
and therefore there does exist a CYCLIC BASIN of ATTRACTION then 0 will
fall into it.  That is if you start with 0 and FORWARD ITERATE Y = Z*Z +
C, then the points will be attracted to whatever ATTRACTIVE BASIN
exists.  If there is NO ATTRACTIVE BASIN because the JULIA SET is DUST,
then 0 will go towards infinity very quickly as in Fig.  J0. The
MANDELBROT SET picture is colored according to how fast 0 escapes to
infinity for all C's where the JULIA SET is DUST.  If the JULIA SET is
CLOSED and 0 finds an ATTRACTIVE BASIN, then the point will not have
escaped even after 1000 iterations, and so there the MANDELBROT SET is
colored BLACK.  Each BALL on the MANDELBROT SET is surrounded by other
BALLS which are surrounded by other BALLS ad infinitum. Each BALL
relates to how many CYCLE POINTS the BASINS of ATTRACTION contain in the
JULIA SET for that value of C.
     During FORWARD ITERATION the JULIA SET REPELS all points not ON it.
equation for the BACKWARD ITERATION of Z1 = Z0*Z0 + C is Z0 = plus or
minus the SQRT(Z1 - C). SQRT is the SQUARE ROOT function.  This is shown
in Fig. J3 where the red starting point BACKWARD ITERATES into two green
same red FORWARD IMAGE (starting point).  Note that the BACKWARD IMAGES
are closer to the JULIA set than the starting point. Repeated
application of BACKWARD ITERATION to each of these points would create
more points closer to the JULIA SET.  This works from the INSIDE of the
JULIA SET as well.  INSIDE, points that would normally fall inward under
FORWARD ITERATION will move outward towards the JULIA SET under BACKWARD
ITERATION.  The JULIA SETS created for this demonstration were generated
by taking the repeated BACKWARD IMAGES of a point already ON the JULIA
SET, namely the right hand yellow FIXED POINT.  The BACKWARD IMAGES of
this FIXED POINT are itself and another point.  It is from this other
point that the further BACKWARD IMAGES are taken.
     The application of all this is to systems which are a function of
themselves and something else.  That is, any system of Z's where Z =
f(Z,C).  Where Z at the next moment of time is equal to some function of
Z at the previous moment of time plus all other influencing factors.
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For example the number of moths in a forest is clearly a function of the
number of moths in the forest just prior and all the other factors that
affect how they breed, eat and get eaten, and grow and die.  For a
further discussion of this please read "The Cell and the Womb",
obtainable from ART MATRIX. -(C) 1985 HWS
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