For example:

            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 CELL AND THE WOMB.
                Copyright (C) 1985 by Homer Wilson Smith
     Consider the fertilized cell in the womb.  When it divides, two
identical cells are formed.  When they divide, four identical cells
are formed.  And when each of them divide, eight identical cells are
formed.  The question is, since every cell is an exact duplicate of
the cell before it, how come eventually some cells become skin cells
and some become bone marrow cells?  Or some become blood cells and
some become brain cells?  They are all, every last one of them, merely
future perfect duplicates of the original cell in the womb.
     The answer is, up to the moment when there are eight identical
cells they are all in the same environment.  They form the corners of
a cube.  No cell is in a special or different environment from the
others except maybe for the one that touches the womb wall.  However
with the next division of each cell into two identical cells there are
now 16 cells.  Eight are on the inside and eight are on the outside.
They form two cubes of 8 cells each, one cube on the inside of the
other cube.  (This is a simplified visualization.) The cells on the
outside form a tight well controlled environment for the cells on the
inside.  Each cell knows what environment it is in.  As the cells
divide they change their own environment and the environment of their
fellows by adding the presence of their sister cell.  It is these
different environments that cause cells to develop different traits.
     When a cell divides in two, each half is smaller but identical.
There is a time for growing before the cell is allowed to divide again.
A cell grows by two way interchange of chemical substances across the
cell wall.  The cell grows by taking in MASS from the outside world.  In
different environments it 'eats' different masses so it becomes
something different.  The different insides that result cause
different genes to express themselves, and so the cell begins
to change.

     The cell wall demarks the inside of the cell from the outside of
the cell which is the entire rest of the universe.  How the cell grows
and into what it becomes is determined by the nature and substances of
the insides of the cell and how they react to the nature and substances
of the environment.  Identical cells become very different when allowed
to grow in different environments.  The human body is living proof of
this.  The internal gene pool does not change, but which genes get
expressed does, and so the external manifestation changes dramatically.
     This immediately suggests a simple mathematical model.
     Define a number as any number on the complex plane and define two
different complex planes called the INSIDE PLANE and the OUTSIDE PLANE.
Remember these are two complete and different complex planes, not two
parts of the same plane.  Both the inside and outside complex planes
contain all possible integer, rational, irrational and transcendental
complex numbers from infinity to minus infinity.
     Each number in the inside plane represents the internal state of
one cell at one time.  The entire inside of a cell is represented by one
number.  Likewise each number in the outside plane represents the
environment of that same cell at that same time.  The entire outside of
the cell is represented by one number.  The outside of a cell is the
entire rest of the universe.  The inside of a cell is finite.  The
outside may not be.
     A cell can do two things: grow and divide.  Division results in
two identical but smaller cells each in the new environment of its own
sister.  Division takes place over a very short period of time and the
cell does not eat during the division process.  Growth takes place
over a longer period of time and consists mainly of eating and
incorporating into its body the material which it has absorbed.
Growth results in a bigger but very different cell.  Also important is
that while the cell is taking IN mass, it is also exchanging mass OUT
into its surroundings creating a unique environment for any of its
neighbors.  What the cell puts OUT is of course affected by what it
takes IN, so in different environments the cell will then CREATE even
more different environments from its own effluence.
     Thus as a cell absorbs different masses, and triggers different
genes which themselves produce masses of different kinds, it grows
into a different cell.

     What the cell grows into of course is also a function of what
it's expressing genes produce that remain inside the cell.
     From the inflow that the cell takes in from its immediate
environment, it also creates new and unique outflows that act as
inflows to other cells in its vicinity which then, absorbing this
different environment, become different in their own turn.  They in
turn generate a new outflow which acts as a new environment for the
first cell.
     This might seem to be hopelessly confusing, but it becomes
simpler if we study the grow and divide cycle of just one cell.
     One ITERATION is one cycle of grow and divide in the on going
life of a cell.  Cells can live or die.  If they live they continue to
iterate: grow and divide.  If they die they stop iterating.  Usually
they die while trying to grow.  Cells often will not divide unless
they have grown enough, that is attained a large enough mass through
eating.  If they do not have enough proper food they will not grow to
the dividing stage and so die of starvation or poisoning.  If they
make it to the divide stage, they usually have enough food energy to
make it all the way through the division.  Cells do not eat while they
divide. (Again this is a simplification.)
     The purpose of creation is trade in expressions of discovery.
     Cells that do not discover how to iterate by trading stuff
properly with their environment are selected out.  It is hoped
something would be selected in, that would iterate forever.
     (The Mandelbrot set is a statement about all possible Julia sets,
so if you are studying the Mandelbrot Set and don't know what a Julia
Set is, you will become lost.)
     The progress of a cell and what it becomes (its STATE) can be
plotted on the inside (julia) plane as a red dot jumping around.  This
is because the inside plane is a numerical representation of every
possible state the inside of a cell could be in.  As long as the red
dot stays within a finite reasonable arena of operation on the inside
plane, the cell can be considered to be alive and functioning.  (See
'Do Fractals Explain Everything' for a more detailed explanation of
this idea.) 
     If the red dot goes to infinity, the cell dies.

     If the red dot goes to a stasis at one or more finite points, the
exact same points over and over again, it can also be considered to
probably be dead.
     The inside plane is a STATE SPACE of every possible internal state
the cell could be in.  A STATE SPACE is a SPACE of ALL STATES.  As the
cell changes over time, its internal state changes also and its
representative value on the plane of all possible insides also changes.
SURVIVAL is obtained when there is change within reasonable bounds.
Infinite change is death.  You can't change EVERYTHING about you and
still expect to be you.  If you were to change EVERYTHING about you,
chances are you would be a perfume bottle or a turtle or Dust in the
Wind (which might be considered a state of maximum change).

     However those that didn't change state at all, just stayed at one
point forever on the inside plane, would be like a rock, no change, and
no life.
     The JULIA PLANE is the STATE SPACE of the INSIDE of the CELL.
     Cells live forever because of what they are.  But what they are
results from what they were and what their environment was too.  So the
environment plays a determining role in what a cell becomes and if it is
able to live.
     One finds that
     1.) For SOME ENVIRONMENTS,     NO CELLS live forever.
     2.) For NO   ENVIRONMENTS, do ALL CELLS live forever.
     3.) For SOME ENVIRONMENTS,   SOME CELLS live forever.
     First this says that some environments are so DEADLY nothing can
expect to survive.  The inside of a super nova might be an example.  A
nitric acid bath would be another.  The air over Ithaca would be a
third.  Electrical Engineering classes at Cornell would be a fourth.
     Secondly it says that there is no environment that is conducive
to life for every possible kind of cell no matter how malformed or
unsuited for life it may be.  What this means is that if you are going
to survive you must bring a modicum of your own personal survivability
to the situation in which you wish to live.  Then if you should find
an environment amenable to your particular life form, you have a going
     Lastly it says that biological immortality, at least for a
species, is possible as long as there is a correct match between the
nature of the cell and the nature of the environment.
     As for individual BIOLOGICAL immortality, remember that for
entities that survive by dividing and growing, there must be some
mechanism of individual death or else the system will over populate
and THAT is one of the most deadly environments there is.  The
resulting death and disease from over population and excessive numbers
of dead bodies lying around can kill EVERYTHING.
     It is always better to have a famine cut back the population, for
then the few and the strong almost always survive and with them the
species.  In a famine situation the number of dead bodies lying around
is much less than in an overpopulation situation, as the dead bodies
tend to get eaten by other hungry animals who are also starving, and
thus disease has less of a chance to take hold.
     In this case too much food is much worse for a population than
too little food.  There is almost always enough food for SOME and the
BEST to survive.  But if there is too much food, then animals start to
drown in their own excrement and the bugs that love excrement and dead
bodies, and THIS can infect the entire population forever or wipe it
out over night.  The point being that an endlessly affluent
environment is not always the most conducive to good survival.
Instead an environment that has a measure of roughness and toughness
will far better serve biological immortality.  Biologically speaking,
endless wealth means certain death.  This is true because SPACE is
     One might consider recent experiments wherein mice that were 30
percent underfed vastly outlived their well fed compatriots.  Nature
has learned that endless affluence must be checked against by an early
death rate to avoid the total annihilation consequent to
     The CELL and its ENVIRONMENT.
     As the cell grows and divides it changes its own environment.  It
does this by adding the presence of its sister cell after division and
also by emitting material into the environment for other cells to
absorb which then in turn re-emit new material back out into the
environment for the first cell and others.
     As long as the cell changes its own environment to one that is
supportive of its functioning it will continue to survive and iterate.
If it doesn't it will be selected out (die) in a finite number of
     The outside plane is a STATE SPACE of every possible outside or
environment a cell could be in.
     The OUTSIDE plane is the MANDELBROT PLANE.
     The MANDELBROT plane is the STATE SPACE of the OUTSIDE of the CELL.
     In other words:

     It should be obvious that with the Julia Plane and the Mandelbrot
Plane we have the universe covered.  This is no small point.  If we
become well versed in Mandelbrot Sets and Julia Sets, we will have a
descriptive mechanism to help us deal with, well, everything.
Everything where insides are affected by outsides, and outsides are
affected by insides.  A DESCRIPTIVE MECHANISM mind you, not
necessarily a PREDICTIVE mechanism.
     (See Mandelbrot and Julia Survivability Maps for a further
discussion of this idea.)
     Coloring the MANDELBROT PLANE.
     Assume for a moment that as the cell divides it does NOT change
its own environment.  Then a starting cell can be placed in each and
every possible starting environment represented by each point on the
Mandelbrot plane, and allowed to grow and divide until dead.
     If it dies then that spot on the outside plane is colored
according to the number of divisions the cell made before it choked.
     If the cell never dies in a particular constant environment then
that position on the outside plane is colored black.  Color measures
how long until the cell died.  Black means it never died or took so
long we could not wait to find out.
     Since a cell DOES change its environment when it divides, as the
red dot jumps around on the inside plane, representing the changes
inside one cell over time, a green dot is also jumping around on the
outside plane.  The green dot on the OUTSIDE plane traces the time
evolution of the changes to the ENVIRONMENT of the same cell whose own
INNER evolution is traced by the red dot on the INSIDE plane.
     The position of the red dot on the inside plane specifies the
entire inside state of the cell at that moment; specifically whether
it is a blood, skin, brain or dead cell.  The position of the green
dot on the outside plane specifies the entire outside state of the
cell at that moment; specifically the environment that the cell is
growing in immediately after division.
     The green dot makes one move on the outside plane because the
cell divides creating a new environment for itself.
     The red dot makes one move on the inside plane because the
freshly divided cell grows in its new environment and so becomes a
different cell just before it divides again.
     Thus the two dots move one after the other.  First the cell grows
creating a new inside for itself (red dot moves), then the cell
divides creating a new environment for itself (green dot moves).
     GROWTH is a form of CHANGE.  DIVISION is a form of SURVIVAL.
CHANGE is a form of NON SURVIVAL.  What you were then is not what you
are now.  What you were did not survive.  SURVIVAL is a form of NO
CHANGE.  What you were then is still what you are now.  What you were
did not change.
     The cell CHANGES  because of its OUTSIDES during the GROW   phase.
     The cell SURVIVES because of its INSIDES  during the DIVIDE phase.
     What this means is that two cells that start off identical
(because they both just resulted from a division of a common cell)
will quickly CHANGE into different cells as each feeds in a different
environment.  Hence CHANGE happens during the GROW phase BECAUSE OF
differing OUTSIDES.
     Division however is the sign of SURVIVAL, a sign that the cell
made it.  After division there are two of them after all and surely
that means it survived.  But it made it because of the correctness of
its INSIDES in their ability to function properly in the environment
given.  Division is sort of a reward for having successfully made it
through the growth phase to maturity.  Thus division is a sign of
survival.  Thus SURVIVAL happens during the DIVIDE phase BECAUSE OF
viable INTERNAL CONSTRUCTION (INSIDES).  However the result is two
IDENTICAL cells.  No CHANGE takes place (except in size) during the
division process in the internal nature of the cell.  Thus GROWTH is
associated with CHANGE which is a form of NON SURVIVAL, and DIVISION
is associated with SURVIVAL which is a form of NO CHANGE.
     When CHANGE takes place during DIVISION a mutation occurs.  What
was is no longer, although it might fare better.  When SURVIVAL takes
place during GROWTH, the cell has failed to differentiate properly due
to its surroundings.  This isn't always bad, we want mature skin cells
to remain skin cells, but for a zygote failure to differentiate at the
right time can be catestophic.
     In this sense CHANGE and SURVIVAL are dicoms, DIchotomies of
Comparable and Opposite Magnitude.  Change is a form of non survival,
you are no longer what you just were.  What you just were did not
survive because it became what you are now.  Survival on the other
hand is a form of no change.  It means you persisted as you were
without change across a span of time.
     Of course in biological systems, the overall cycle of change and
survive should SURVIVE as this is the process of life going through
time.  But notice that during GROWTH the cell SHOULD CHANGE, if it
doesn't then something is very very wrong.  At the same time during
DIVISION the cell had BETTER NOT CHANGE, because the purpose of
division is to exactly replicate the DNA structure within.  If the DNA
changes during a division then a mutation has occurred, which means
the original blue print definitely did not survive.  In general this
can mean the end of the cell.  If you don't believe me, try eating
some radium some time.  Sometimes the non survival of one chain of DNA
and the continued survival of the mutant is good for the ongoing cycle
of life as a whole, but the original cell that did not divide properly
definitely did not survive even if it improved the chances for its
     Thus the iteration and movement of the red dot on the inside
plane happens during the growth phase and tracks the changes in the
cell brought on by the environment.
     The iteration and movement of the green dot on the outside plane
happens during the divide phase and tracks the changes in the
environment brought on by the new daughter cell.  
     This again is a simplification, because in truth as the new cell
grows, it will continue to change the enivronment in different way for
its bretheren right up until the moment it divides again.
     The red dot is allowed to go anywhere but infinity stasis.
Infinity means too much change in the cell and this means death.
Stasis means no change at all, which again is death.  Life is
reasonable change within finite arenas of operation.
     The green dot is allowed to stay in the black forever or wander
in the colored areas for a while but not so long as to cause the
cell's fatality.  The green dot staying in the black areas of the
outside plane MEANS the red dot DID NOT go to infinity and so
survived.  This is true by the definition of how we color the outside
plane in the first place according to whether or not the cell lives or
dies.  But the coloring of the outside plane represents a CAUSAL
quality of the environment namely how it affects the longevity of an
initial cell.  The red dot goes to infinity BECAUSE the green dot
stayed in a colored area too long.  The green dot in a colored area
means that the red dot WILL GO TO INFINITY after N number of
iterations and so WILL die if the green dot does not get back into a
     It is possible that if the green dot stays in a colored area for
too long, the red dot will go infinity even if the green dot gets back
into a black area before the red dot does go to infinity.  Thus there
is a point of no return, and point of no RECAPTURE.
     In more lay terms, it is OK to smoke a cigarette every once in a
while (bad environment) but don't chain smoke.  Likewise it is OK to
visit Los Angeles (or downtown Ithaca for that matter) but don't move
in for the long haul.  (Please see 'The Theory Behind The Cell and the
Womb' for a more detailed explanation of RECAPTURE.)
     Life functions in the high iteration areas bounding the black and
colored areas of the outside plane.
     The Mandelbrot plane does not determine how a cell will evolve
but demarks how a cell can evolve and still be viable.  The evolution
is determined by what a cell does with its present environment to make
itself a new environment through division.
     If the cell creates environments in the colored areas of the
outside plane it will cause its own demise.  If it creates
environments in the black areas of the outside plane it will survive
forever.  Since survival forever is equivalent to death forever
through over population, the ideal survival for the SPECIES is
obtained by the cell creating environments for itself that wander
around the chaotic boundaries of the outside plane where color and
black, death and survivability intermix and swirl around each other in
and endless array of beauty, confusion and amazement.  This guarantees
the death of the individual cell but the survival of the ongoing
     Individual death of old age is the result of intentional failed
recapture.  The offspring are injected into the system where recapture
is relatively secure.
                     Z = Z*Z + C  and  C = C/2 + Z
                           Please see Fig 1.
     If Z stands for ZYGOTE which is the cell in the womb, and if C
stands for (external) CONDITIONS, then the equation Z = Z*Z + C says
that what the zygote becomes is what the zygote was squared plus the
number representing its environment.  This iteration represents the
GROW phase of the zygote as it changes and prepares itself for its
next divide phase.  We know this because this equation represents the
iteration of the zygote (Z).  Z appears both on the left and the right
of the equal sign, thus it is the zygote that is being iterated.  It
shows that the zygote changes from what it was before, to what it is
one iteration later.  The zygote CHANGES while it GROWS not while it
divides.  Thus Z = Z*Z + C models the growth phase starting just after
division and ending just before its next division.  During this
process the environment (C) does not change, only Z changes.  (This is
a simplification.)
     The second equation, C = C/2 + Z says that the environment
(external Conditions) becomes what the environment was divided by 2 with
the new sister zygote added in.  This iteration represents the DIVIDE
phase of the zygote as it changes its environment by adding the presence
of its sister cell during division.  
     Do not confuse the C/2 with the zygote dividing in two, we simply
picked a simple function here for C to change by.

     We know this second equation represents the the divide phase of
the zygote because C is on both the left and the right of the equal
sign, therefore it is the environment (C) that is being iterated.
This equation says that C changes from what it was before division to
what it is after division.  The zygote does not change during this
phase (even though it is the zygote that is dividing!).  C (the
environment) changes.
     Z is the red dot jumping around on the Julia plane.  C is the
green dot jumping around on the Mandelbrot plane.  Assume an initial
zygote called Z0 in an initial environment called C0.  Then during the
growth phase Z0 grows into a new and different zygote called Z1 but it
does this growing in the original environment called C0.  Then as the
new and fully matured zygote Z1 divides, it produces two smaller but
identical versions of itself, both still called Z1, but now the
environment C0 that the first Z1 was in now includes the presence of
the second Z1, so becomes C1.
     Recognizing that any equation may be used to model the growth and
division of a cell, we may write this in general mathematical terms as
follows.  Let Z1 = Z0*Z0 + C0 be generalized to Z1 = f(Z0,C0) and let
C1 = C0/2 + Z1 be generalized to C1 = g(C0,Z1).  The iteration of Z1 =
f(Z0,C0) relates to the GROWTH of the cell in environment C0 from the
smaller Z0 to the bigger but different Z1.  The iteration of C1 =
g(C0,Z1) relates to the change in environment from C0 to C1 caused by
the DIVISION of cell Z1 from the larger Z1 to two smaller but
identical Z1's.  The sister copy of Z1 becomes part of the new
environment C1 of the first copy of Z1.
                   The MOTH and the FOREST.
     The question naturally arises, is it reasonable to represent the
entire inside state of a cell by one number?  Or even more ridiculous
the entire rest of the universe by one number?  Consider a population
of moths in a forest.  Here the inside is the system of moths trying
to survive in the forest environment which is the outside.  The number
of moths in the forest at any time is a function of the number of
moths just prior plus the environment.
     Clearly the population of moths in the forest at any one time can
be represented by one number.  But can the entire rest of the forest
be represented in this same way?  The forest is a large system of
interacting subsystems, like the number of trees, the number of birds,
the number of oxygen molecules in the air.  It would seem that if you
broke the forest into its parts you might be able to represent the
forest as a system of things each of which can be represented by one
number.  Hence representing the entire forest at any one time as a
function of many single numbers resulting in one overall number is not
so wild.  Just so with the inside and outside of a cell.  Of course
the arrangement of things can be important too.  But in large systems,
the DENSITIES of things can be more important that exact positioning.
And where positioning becomes important, it would be taken into
account by the number representing the environment containing the
particular object whose position was important.
     (Finding a single complex number to represent an actual
environment is not as simple as it might seem, and in fact might be a
daunting task.)
                         The DNA and the SOUP.
     Consider the primordial sea.  This is an all pervasive
environment that contains all the parts for a DNA molecule to start
building itself.  DNA molecules survive by perfectly duplicating
themselves.  They can not see ahead and so do not 'plan' their own
changes.  A DNA molecule that is different after it has duplicated has
NOT survived.  The environment of the DNA molecule is constantly
trying to destroy it and scatter its well collected parts back into
the soup.
     DNA molecules also tend to eat each other and eat each other's
parts.  The ones that survive are the ones that can continue to
perfectly duplicate in spite of an environment that is trying to
destroy them.  Thus survival is always measured by no change in inside
state.  The outside environment is directed towards changing the
inside state.
     The environment is not trying to intelligently build a better DNA
molecule.  But if a better DNA molecule should happen to form via
environmental influences then it will begin to out survive the
environmental destruction.  Thus is obtained a classic case of insides
surviving in the presence of outsides.  Clearly all of life has
evolved because of the ability of insides to out survive the changes
brought on by the outsides.
     Every living thing that continues to exist today, does so
because it survived when everything else didn't.
     What is a FRACTAL and why is there one in every PAW?
     Every equation of the form M = f(M,F) has two questions that can
be asked of it.  If M stands for Moths, and F stands for Forest,
clearly the number of Moths in the Forest depends on the number of
Moths just prior and also on the Forest.  One would want to know
therefore what happened to the number of Moths for every possible
starting number of Moths given a constant Forest, and also what
happened to the number of Moths, for every possible Forest given a
constant starting number of Moths.
     The first question is for each and every possible starting M and
a constant F, what happens to M?  This is the Julia plane.  The second
question is for each and every possible F and a constant starting M,
what happens to M?  This is the Mandelbrot plane.  A fractal is thus
the pictorial representation of either one of these questions.  There
are Julia fractals and Mandelbrot fractals.  And of course there are
hybrid fractals like the Tarantula resulting from iterating equations
in both M and F.
     Fractal math is a way of looking at equations and physical
phenomena.  Just like calculus is a way of looking at equations and
physical phenomena.  Calculus deals with related rates.  Fractals deal
with iteration, in this case applied to insides and outsides.  Fractal
math has been called the most important discovery since calculus and
has been rated with Relativity and Quantum Mechanics as one of the
three great discoveries of the 20th century.  Calculus is important to
all of life.  So are fractals.
     What in existence does not have to do with insides and outsides?
     Fractal math has to do with any system of insides trying to
survive in a system of outsides.  Even a hurricane depends on and
feeds upon the surrounding atmosphere where there is no hurricane.  If
you were to vanish all the rest of the CALM air on the planet
surrounding the hurricane, the hurricane would vanish too.
     Fractals ARE a description of whether or not insides survive in
various outsides of interest.  From that point of view they underlie
every operating system in existance.