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‘The Paths of Lovers Cross in the Line of Duty.’
THE ART MATRIX INTRODUCTION TO FRACTALS.
Copyright (C) 1990 Homer Wilson Smith
All rights reserved.
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‘Mandelbrot Sets and Julia Sets’
120 minute video
Art Matrix, a small company in Ithaca, NY has released a milestone
video on the subject of fractals and chaotic dynamical systems.
The video is 2 hours of true computer animation delving deeply into
the nature of Julia Sets and Mandelbrot Sets. It includes the classic
30 minute segment ‘Nothing But Zooms’, which has been used around the
world in television science shows and award winning PBS documentaries.
Fractals are simple pictures that visualize the behavior of various
mathematical equations. They show what happens to the output of an
equation for every possible input. Every equation has two kinds of
fractal images associated with it, the first is called a Mandelbrot
image and the second is called a Julia image. There is one Mandelbrot
image for each equation but an infinite number of Julia images, and in
fact the Mandelbrot image is a summing statement covering every possible
The video ‘Mandelbrot Sets and Julia Sets’ covers many different
equations including quadratics, cubics, 5th degree polynomials, rational
maps and a host of others. Scenes consist of long deep zooms into the
various mathematical spaces under study. Also included are various
‘promenades’ around the quadratic space showing the resulting Julia
The video is scored with a beautiful music only sound track.
The company says that the video is suitable for all ages, including
the very young. They suggest that providing this experience to young
children will change their view of mathematics, an effect that will last
the rest of their lives.
Those interested in more information about this video or other
educational products related to fractals should contact Art Matrix at PO
880PR Ithaca, NY 14851-0880 or call (607) 277-0959.
HOW HISTORY WILL PERCEIVE US
We can choose two paths.
We can take the stand most acceptable to our present fellows in
which case our future peers will consider us fools.
Or we can take the stand most aggravating to our present fellows,
in which case our future peers will consider us visionaries.
Which is it going to be?
Present acceptance or future acceptance?
Who does History remember?
Those that agreed?
Or those that stood alone?
The Great Ones always stood alone. For a while.
The enclosed material is a short compendium of material on fractals
for the layman. It ranges from the truely simple to the slightly more
complex. I have made an effort not to lessen the subject matter by
writing watered down analogies that would not make any sense to a true
scientist and would give the beginner a false feeling that they had
Thus each of the following pieces invites thought and understanding
by presenting fractals as they really are. The first few pieces do not
assume that the reader has any mathematical back ground at all, and in
fact approach the subject in such a way that a mathematical background
is not needed.
However fractals are eventually mathematical, thus to understand
them one has to foray into the subject of mathematics at least a little
bit. The remaining pieces go into the mathematics of fractals in
greater detail. You should follow this material only as long as you can
continue to understand it and take from it what you need.
Everyone starts off learning about mathematics at the beginning, I
was no exception, therefore you should not feel uncomfortable if you are
at the beginning of this subject. Many people give up mathematics
because it became boring to them, it became useless. Our intention with
fractals is to get people interested in mathematics again even if they
are 90 years old. With fractals it happens every day. People who are
retired and living off their well earned pension are suddenly calling me
up telling me about the crazy new pictures they have just made on their
PC. Just like kids again. And yes their mathematics is rusty.
The Theory behind ‘The Cell and the Womb’.
This is by far the easiest to read and requires no mathematics at
all. It was written long after ‘The Cell and the Womb’ in order to give
a basic understanding of the principles. The idea that mathematics
could be applied to the growth of the human body has met with some
resistance and cat calls. My intention with this piece was to point out
a use of the principles involved that was so simple and so obvious that
no one could refute it. It was hoped that this might then open the door
to the more speculative ideas contained in ‘The Cell and the Womb’.
‘What is a fractal?’
This provides a quick overview of the nature of fractals and what
they might be used for in the biological sciences. It goes over again
in a simpler light the material covered in the first piece and in fact
could be used as a sort of press release about what fractals are and
what they are useful for. This is placed second only because of its
‘Do Fractals Explain Everything?’
Besides trying to answer this question directly, this piece is
directed to a thorough definition of the terms evaluation and iteration,
input and output, periodic and chaotic, and stable and unstable.
Although little to no mathematics is required here, this is meant for
serious study and contemplation.
‘The Cell and the Womb’
This is an effort to apply the simplest fractal concepts to a
living system in an environment. It is a semi detailed description of
how the fertilized cell growing in the womb might be described by a set
of equations that give rise to very fractal behavior.
Mandelbrot Sets and Julia Sets.
This was written for teachers and serious students of the subject
who wanted their first introduction to the mathematical foundation of
fractals. It defines in detail the mathematical terms and concepts
surrounding the computation of fractals such as iteration, basins of
attraction, periodic cycles and what a Julia Set is.
There is a definite drum being beaten here for fractals. Namely
that they are important, easy to understand and beautiful. They are
important in that many phenomena of nature not yet susceptible to formal
scientific analysis will soon be. They are easy to understand in that
your child in the 10th grade should be able to program them on his or
her PC. Their beauty is obvious and should need no further explanation.
Even if you yourself feel that fractals are beyond you, you should
consider that fractals are for children for they are the scientific
pioneers of our future. Fractal mathematics will be common curriculum
in 10th grade high schools around the country in a few years if only as
a motivating factor to keep kids interested in math and computers.
Every person that is touched for the first time by a fractal recognizes
in their heart that they have found something new and wonderful,
something that THEY want to be a part of and they will be endlessly
thankful to the person who first turned them onto the subject (you!).
The products that we offer at ART MATRIX are the result of years of
research and millions of dollars worth of computer time and resources.
These funds have been given gladly by the various institutions involved
in the name of research. From this research have come a few token
products that we bring you in an effort to help the rest of the world
see what we have been so fortunate to be buried in for the past few
We are not paid by Cornell or anyone else to do this research. Art
Matrix is our only source of income, thus by helping us get this new
found wonder out to the population at large you are also taking a direct
hand in keeping the research going. At the same time you are inspiring
the little people of this planet to hold onto their hope and give them a
new and wonderful way to contribute to the problems that need to be
solved in the years ahead.
THE THEORY BEHIND ‘THE CELL AND THE WOMB’
Mandelbrot and Julia Survivability Maps.
Consider the Planet Earth. It consists of many climates and
terrains where various things may or may not grow depending on the
hospitality of the environment to the item in question.
Let us consider the case of simple American Corn. Let’s do the
following thought experiment.
Place a grid over every square foot of the Planet Earth so that
every square foot is clearly demarked. This grid covers oceans and
mountains and deserts and fertile ground alike.
Plant one seed of corn in each and every square foot whether it be
under water or on mountain top or on fertile soil. You don’t have to do
this all on the same day, but eventually you will have to study every
square foot of the planet in this fashion.
After the seeds have been planted, come back to each square foot
every day and record if the planted corn is still growing and viable.
If it is, let it be. If however the seed has failed to grow or has died
then mark that square with the number of days the corn managed to stay
alive before it died.
After every square foot of Earth has been mapped in this manner,
color each square foot according to the number marked in it. Color the
numbers according to the simple rainbow starting at Red and continuing
through Yellow, Green, Cyan, Blue and Magenta.
Where the numbers are small, meaning the Corn died quickly assign
the Red end of the spectrum. Where the numbers are large meaning the
Corn lasted for a long time and maybe even went a few generations before
dying assign the Magenta end of the spectrum. For values inbetween the
two extremes assign the colors in the middle of the spectrum in an even
Where the corn is still growing at the end of the experiment color
This is a Mandelbrot Survivability Map. The map represents all the
different possible ENVIRONMENTS the corn could be planted in. The color
represents how long the corn survived before it died. The black areas
represents where the Corn survived best and is indeed still growing.
The reason we are giving DEATH such importance here, by coloring
it, is because death is what distinguishes one square foot from another.
If two different squares both have corn still living in them, they can
not be distinguished from each other. Both are still alive. But if one
or both dies, then they can be distinguished by WHEN they died.
It is the change in state that causes the discrimination to take
place. The corn starts out alive (as a seed). If it ends up still
alive then no change has taken place. But if it dies then that is a
significant change which we record as a color demarking it as different
from its neighbors.
Now let’s change the experiment. This time we collect together one
seed of every known edible vegetable on Earth. This of course would
include the corn used above. We pick just one particular square foot of
Earth from the above Mandelbrot Survivability Map. It does not matter
whether that square foot is a colored one or not, but for our first
experiment let’s pick a black square foot. This guarantees that at
least something will grow there, namely our corn seed.
We then plant each one of our seed collection in that one square
foot of Earth. We do not plant them all at the same time, but rather we
plant each in turn and let it grow and see what happens. Then we repeat
the experiment with the next seed in line and so on until all seeds have
been planted and been allowed to grow and die (or live).
For each seed we keep a record of how long it survived in that
particular square foot of Earth. If a seed seems to grow forever though
we cut the experiment short at some reasonable arbitrary time so that we
have time to finish the rest of the experiment with the other seeds.
After all of our seeds have been allowed to grow and die in our
single plot of land, we arrange them in a square grid on a table (a very
large table!). We mark on the table the number of days the seed managed
to survive before it died or was cut short by the need to get on with
the other seeds. We then color the table with the same method used in
the Mandelbrot case.
Those positions on the table having small numbers in them get the
Red end of the spectrum. This means they died quickly in this one plot
of land. Those having large numbers get the Magenta end of the
spectrum, and those inbetween get all the other colors between Red and
Magenta. Where the seeds survived so long they had to be cut short,
color the table black.
Thus you have a Julia Survivability Map. A Julia Map represents
every possible item that could grow in ONE PARTICULAR environment and
how long it was able to survive there.
Clearly a Julia Survivability Map could be made for every square
foot on Earth planting all the seeds for all the vegetables in each
square foot just like we did for the first square foot.
Thus each and every point in the Mandelbrot Survivability Map has a
complete Julia Survivability Map associated with it.
In terms of knowledge what is gained from these two types of maps
is a thorough knowledge of what grows where. Pretty important don’t you
Defining some terms here, let’s call the starting seeds ENTITIES
and the square of land they are planted in the ENVIRONMENT. If the
entities can be called INSIDES then the environments can be called
A Mandelbrot Map is a picture of every possible ENVIRONMENT one
seed or entity could be planted in and it is colored according to how
how well that one entity survived in each different environment. It
records how well one starting INSIDE does in every possible OUTSIDE.
A Mandelbrot Survivability Map can be made for any INSIDE of interest
and shows how well that one INSIDE survived in all possible OUTSIDES
A Julia Survivability Map is a picture of every possible SEED or
ENTITY that could be planted in just one chosen ENVIRONMENT, and each
different SEED is colored according to how well it survived in that one
chosen environment. It records how well every possible INSIDE survives
in one possible OUTSIDE. A Julia Survivability Map can be made for any
single OUTSIDE of interest and shows how well all possible INSIDES in
existance survived in that one chosen OUTSIDE.
With a complete set of Mandelbrot and Julia Survivability Maps for
all the food on Earth and all the places you could plant it, you would
have a complete directory of what grew where and how well.
Nothing new and no big deal. Its just a way of looking at it.
WHAT IS A FRACTAL?
What is a fractal? A fractal is a picture. A fractal is a picture
demonstrating in color what the output of an equation does for any given
input. The color picture represents the space of all possible inputs,
or some zoomed-in blow up of such a space, and each point in the picture
is colored in a way that represents what the output of the equation does
for that particular input.
Often equations are used to represent the population of living
systems such as a colony of Moths in a Forest. The output of the
equation is the population of Moths at any moment of interest. The
input to the equation is the number of Moths just prior to the moment of
interest and another number representing the total environment the Moths
live in. The Forest in other words.
Thus the equation takes in two numbers, one being the number of
Moths that exist right now, and the other a number representing the
living conditions of the Forest, and the equation puts out just one
number representing the number of Moths at the next moment of time down
Clearly by taking this new number of Moths and plugging it right
back into the equation (along with the second number representing the
Forest), you will get yet another number of Moths even further down the
road. The unit of time being cycled through here can be seconds, days,
weeks, months, years, or anything at all. The purpose of this is to
describe what will happen to the Moth population tomorrow according to
what the population is now and the living conditions they find
The most interesting kind of picture that can be made from this
little game is called a Mandelbrot Map. A Mandelbrot Map is a fractal
and it is a colored picture just like we said, and the colors describe
how long it takes, how many cycles it takes, until the Moth population
dies. Where the Mandelbrot Map is black, the Moths live forever in
happy harmony. Please see the upper left color image of the Mandelbrot
Set on the sheet ‘Mandelbrot Sets and Julia Sets’. Black and white
images can be made many different ways. Sometimes they try to emphasize
the contouring on the outside leaving the black inside area white. The
images supplied with this compendium are of this nature. You might
compare the black and white image of the Mandelbrot Set with the color
image of the same.
The picture itself is the space of all possible different FORESTS
the Moths could be in, so it represents the input number that represents
the living conditions the Moths find themselves in during this time.
Forests come in all kinds of shapes and sizes and states of well being.
Thus the Mandelbrot Map shows at a glance which Forests are
conducive to life and which are not. Those for example that are soaked
in Acid Rain, presumably would have a harmful effect on the
survivability of the Moth population. Those Forests therefore that were
deadly to a Moth population would show up as vibrant colors in the
Mandelbrot Map of all possible Forest types. If we were using the
rainbow as our color scheme, then red would be the most deadly with the
Moths dying off in the shortest number of cycles, and violet would be
the least deadly bordering on a good healthy environment. Those Forests
that were truely good for the little beasts would show up as black.
It is a large leap of imagination to go from real Forests to
numbers representing Forests, especially when those numbers are numbers
in the complex number plane. However rather than break your mind with
the details of such things let me assure you that scientists have been
modeling physical things with numbers and equations for a very long
time. In fact they have done quite well in this field for the simpler
phenomena of nature. With this new concept of the Mandelbrot Map the
door has been opened to applying strict scientific scrutiny to things
like weather, and chaotic turbulence which have stumped the best minds
Quickly, things like weather systems always exist in a larger
system from which the smaller system in question takes its life. By
understanding how things survive in their environments, and by having a
tool to describe, compute and predict such relationships, it becomes
possible to study these things at a formal level.
It is pretty obvious that if the whole rest of the atmosphere
around a hurricane were removed, the hurricane would dissipate
forthwith. Thus any hurricane depends for its survival on the calm and
sunny afternoons that day on the other side of the planet.
The fractalness of these pictures comes about because of certain
characteristics that are common to such equations. The primary
characteristic is the tendency of the output of the equation to change
drastically with the slightest change to the input. This is called
INSTABILITY or SENSITIVITY to INITIAL CONDITIONS. Thus the slightest
change in the acidity of a Forest could dramatically alter it from a
living forest to a dead one. And the slightest change in the atmosphere
could precipitate a global ice age.
Fortunately there are large areas in these pictures where things
are relatively stable and NOT sensitive to initial conditions. The
broad areas of common color show this. But where the colors intermingle
in a chaotic frenzy, you know that the slightest change to the input
conditions means a great change in the output result. By making such a
picture we can see easily for the first time where such a system is VERY
unstable and sensitive to initial conditions. Presumably we could check
our present environments to see if they approach these unstable areas
and take heed if caution were indicated.
DO FRACTALS EXPLAIN EVERYTHING?
To answer this question the following must be considered:
Do equations explain everything?
Are these equations non-linear?
Are these equations merely evaluated or are they iterated?
Fractals are not cause, they are effect. Fractal behavior is a
manifestation of non-linear equations when iterated; that is repeatedly
evaluated using the output as the next input.
If the physical phenomena under study is modelable with
mathematical equations, and if these equations are non-linear, and if
these equations are iterated rather than merely evaluated, then the
physical phenomena will manifest fractal behavior.
What is EVALUATION?
If you take 1000 bees and put them in a closed room and start to
lower the temperature you will notice that as the temperature goes down,
more and more bees cool it and sit on the floor. After a certain point
all the bees are no longer flying around. As you warm up the room, more
and more bees take off and start buzzing around in intense activity. If
you make a graph of the number of bees that are airborne at each
temperature between room temperature and 32 degrees fahrenheit, you will
notice a definite curve. Later if you want to know how many bees would
be flying around at a given temperature, all you have to do is plug the
temperature you are interested in into your equation (curve) and your
output would be the percentage of bees still air born. This is simple
evaluation of an equation. One input gives one output.
What is ITERATION?
Iteration on the other hand is a bit stranger. In this case the
output you just received from your first input becomes your next input.
For the case of the bees above this does not make a whole lot of sense;
the output is in units of bees and the input is in units of temperature.
But consider instead the population of moths in a forest. Clearly the
number of moths in the forest at any time T is a function of how many
moths there where just a moment before, plus all the things that affect
how moths grow and die, eat and get eaten. It is not unfeasible to
postulate an equation that specifies the probable number of moths in the
forest at any time T as a function of the number of moths at time T – 1
plus all the other factors. Then to trace the population of moths over
a year, you would start the equation with the starting number of moths,
and get your number of moths for the next day. You would then stick
that number back into the equation to get your number of moths for the
day after that, etc.
This is iteration. And this produces fractal behavior in non-
linear equations (equations of degree 2 or higher).
INPUTS and OUTPUTS.
The behavior of equations under simple evaluation is relatively
straight forward. But with iteration their behavior can be nothing
short of amazing. Every equation has an INPUT and an OUTPUT. The
output is always just one variable, but the input can be as many
variables as you want INCLUDING the OUTPUT variable. For example in the
equation Z = Z*Z + C, the variable Z is in the output and the variables
C AND Z are in the input. Because Z is both in the output and the
input, this equation is iterable. What this equation says is that Z is
a function of itself plus something else (C). Or a wider interpretation
is that Z is a function of itself and EVERYTHING ELSE IN THE WORLD THAT
IS NOT Z. That is what C represents. Any equation where the output
variable is also in the input, can be iterated. Just take the output
and put it back into the input and do it again. In a sense Z is trying
to SURVIVE, that is why it is going into the equation and coming out of
the equation slightly changed but still Z. But Z is also changed by its
environment or external influences and that is represented by C.
Obviously when it comes to biological growth or evolution of most
systems of any kind, the state of the system is usually equal to some
function of its state just prior and plus all other determining factors.
Thus you would expect that iteration would play an important role in the
mathematical description of the various systems of existance.
WHAT is FRACTAL BEHAVIOR?
Fractal behavior can be a number of different things.
1. STABILITY and INSTABILITY.
First it can be sensitivity to input (initial) conditions. This
means that tiny changes in the value of the input can cause wildly
different changes in the behavior of the output as it is iterated.
Technically this is called the STABILITY/INSTABILITY dichotomy of
fractal behavior. In a stable input area, quite large changes of input
values will cause little to no change in output result. In an unstable
area, the tiniest possible change of input value can cause totally
different output results. All fractal equations have input areas of
stability and instability, hence any physical phenomenon modeled with
these equations may manifest either.
Stability and instability refer to the INPUT and describe the
effect small changes to the input have on the output.
2. PERIODICITY and CHAOS.
Second, fractal behavior can relate to the behavior of the output
given one particular input. In general there are two possible behaviors
of an output result. The first is when the output settles down to a
fixed boring routine. A kettle of hot water left to cool on a table
manifests this as it looses temperature to the atmosphere and settles
down to one temperature, room temperature. When left alone that is all
it does, stay at one final temperature, room temperature.
A similar situation is our yearly weather cycles, which instead of
settling down to 1 fixed end condition, settle down to 4 fixed end
conditions called winter, spring, summer and fall. The season always
goes from one condition which is winter, to another condition which is
spring, which goes to summer and fall. Eventually however the season
goes back to winter again and repeats the cycle all over again ad
This sort of behavior is called periodicity and is part of the
PERIODICITY/CHAOS dichotomy of fractal equations. A cycle of
periodicity can be one cycle long as in the kettle of hot water cooling
to room temperature, or it can be 4 cycles long as in the seasons of the
Earth or it can be 50,000,000 cycles long. But it is always finite and
eventually returns to its starting point at which moment it begins to
repeat its past history over and over again with out change.
Chaotic behavior on the other hand is similar to infinite
periodicity. In this case the system never returns to the same value
twice and never repeats itself. Chaos in this sense does not refer to
random, wild, undetermined, uncontrolled or totally unpredictable
behavior. It refers to a lack of simple periodicity in the behavior of
the output. Usually chaotic systems are well behaved and their values
stay within a reasonable range. They just never settle down to some
boring routine. Instead they are forever landing on new values
contained within a finite and reasonable arena of operation. They can
however change abruptly and without apparent warning from one arena to
another as in the famous Lorenz attractor.
Another thing the output can do is be chaotic within a cycle of
periodicity. This is still chaotic behavior but there will be clearly
periodic areas the value keeps going to. For example although the
seasons are always winter, spring, summer and fall, which is clearly
periodic of period 4, each winter is always different from every other
winter. No two winters are the same, as is true for the other seasons,
so indeed weather has a chaotic cycle in four parts.
The weather on Earth is also an example of how a chaotic cycle can
change abruptly from one arena of operation to another. Scientists have
long wondered about what brings on the ice ages and why they last. Well
there is a very interesting and frightening explanation to this
phenomenon. To start with it has been suggested that Earth has two
stable world wide climates. By stable is meant general arena of
operation different from the other but none the less chaotic and always
changing. (What stable really means here is that changes to the input
caused by the output going back into the input will not kick the system
over from one arena to the other very easily.)
The first climate is the one we have now. The other is a global
ice age. Ice is inherently unstable stuff. It melts. If you covered a
large section of North America with ice, you would find that within a
while it would melt away probably flooding the place with water but
certainly no ice age would result. But ice reflects sun light and in
fact it is the sunlight absorbed by the land AROUND the ice that warms
the land under the edge of the ice causing it to melt.
This means that if you covered ENOUGH of the Earth with ice, then
most of the sunlight would be reflected back into space and the ice
would never melt. A permanent ice age would result. But if you then
melted a big enough hole in the ice, enough warmth would be absorbed by
the exposed earth to melt the rest.
So you see there are two stable states to Earth’s climate and one
is a global ice age. The other is this rotten weather we have in
Ithaca. It is possible that the equations that run our weather may
periodically switch over from one arena of operation to the alternate
arena to stay there for a while before switching back to the present one
again. If it is in the math to do this, then no other explanation for
ice ages need be found and the predictability of the switch over may or
may not be out of reach as will be explained later.
In summary therefore, periodicity and chaos refer to the OUTPUT of
an equation and describe whether or not the output ever repeats itself,
or is forever new. In this sense chaos means ‘without simple repeating
pattern’. It does not mean a lack of order, determinism or proper
progression of events. In this sense chaos is not anarchy.
SURVIVAL, DEATH and BIOLOGICAL IMMORTALITY.
When applying iteration to the various operating systems of
existance the concept of a STATE SPACE comes in handy. The output
variable which is destined to be iterated lives in the space of all the
possible values it can ever take on. If Z represents a biological or
physical entity then every value in the state space represents the state
of that entity when Z has that particular value.
Every object in existance has a state. This state is represented
somewhere in the state space of values for Z. Thus if Z lands on that
value, Z has become that object. A live human being and a dead human
being both have values in the state space. Since all objects are
changing constantly from moment to moment, the value that represents
their state is also ever changing in the state space.
For biological systems, or any system for that matter, the iterated
variable refers to the subject of interest under study and how it is
affected by itself and its environment (not self.) The first thing to
note is that too much change means death. Thus if Z goes off to
infinity (in the state space) under iteration then the system can be
considered to have died, as nothing can change infinitly and still be
considered to be what it was. Thus if one is studying biological
populations, infinities showing up in the output usually mean non
Another form of non survival would be a low periodicity of say one
or even more. In this case the subject has become one thing that is
absolutely unchanging for ever more. This is akin to attaining
immortality through being a rock or a statue. This is not life.
Another form of non survival would be to change to something that
is still functional but not at all like what the subject originally was.
A moth turning into a tire or a perfume bottle or even a turtle can not
be said to have survived even if the turtle it turned into is surviving
just fine. Turtles, tires and perfume bottles all have their position
in the state space of life. Thus if your Z values happen to land on
such a thing, you become a perfume bottle. Not ridiculous.
Thus survival is measured by the output value of the equation
staying in a finite arena of operation, not becoming heavily periodic,
and not changing so much as to become something else entirely.
CAUSE and EFFECT.
Whether or not the output of a system is periodic or chaotic
depends on the initial input conditions. Some input values will cause
periodic behavior in the output result, while other input values will
produce ‘chaotic’ behavior in the output result.
This brings us back to the stable/unstable aspect of fractal
equations. Periodicity and chaos refer to the behavior of the output of
the system which of course is dependant on the input to the system.
Stable and unstable refer to the input of the system and how large and
small changes in input can cause large and small changes in output. The
basic change that can be caused in the output of a system is to change
the output from periodic behavior to chaotic behavior or visa versa.
(Another kind of change that can be caused to an output is to change the
period from one cycle to another, for example from a period of 4 to a
period of 5. The third kind of change that can happen to an output is
to change the actual value of the period point drastically from some
finite number, let’s say, to infinity.)
For example, if the output for a given starting input is behaving
in a periodic manner, and significan’t changes in the input cause the
output to continue to act in a periodic manner, then the input area can
be considered stable.
Or if the output is behaving in a chaotic manner and continues to
behave in a chaotic manner even under significan’t changes in the input,
then the input area would still be considered stable.
If however small changes in the input cause the output to switch
over from periodic behavior to chaotic behavior or visa versa, then that
input area can be considered unstable.
An example of this is the picture of the Mandelbrot set which is an
input area of C’s to the iterated equation Z = Z*Z + C. If you pick a C
inside the main cardiod of the Mandelbrot Set and follow the forward
iterates of Z = 0 for Z = Z*Z + C, you will find the forward images
(iterates) of Z tend towards a one cycle fixed point near 0. This
behavior is periodic, with period of one. If C is near the center of
the cardiod, considerable changes can be made to the input value of C
and still the forward iterates of Z will tend toward a period one cycle
in the same general area. Thus the inside of the Mandelbrot set is a
stable input area, and results in a periodic output of constant period
(one) and similar value (somewhere near 0).
In a likewise fashion, if C is chosen outside of the Mandelbrot set
entirely, then the forward iterates of Z = 0 go to infinity, again a
single point of period one. Thus the entire outside of the Mandelbrot
set can be considered a stable input area. Notice however that infinity
is a wildly different value for the period point than the one approached
when C is chosen inside the Mandelbrot Set. Somewhere between the
inside and outside of the Mandelbrot Set there is an area of input C’s
with great change-over and instability.
If C is chosen from the very edge of the cardiod then the forward
iterates of Z = 0 form a never ending circular disk called a Siegel
disk. Z never returns to the same point twice yet always stays in a
finite and reasonable arena of activity. This is the mark of chaotic
This output behavior though comes from a VERY unstable input area
because even the tiniest change in C can cause C to lie inside the
Mandelbrot set or outside the set where in both cases the output
behavior becomes immediately periodic again.
3. PRETTY PICTURES.
The boundaries between input areas that give rise to periodic
output behavior and input areas that give rise to chaotic output
behavior can be infinitly convoluted and intricate thus giving rise to
the third type of fractal manifestation: the gorgeous and complex swirls
that most people recognize as the hallmark of a fractal.
It is also these areas that give birth to the idea of UNPREDICTABLE
but DETERMINISTIC CHAOS. This needs to be clarified in order to rid it
of its romance and associations. How can something be UNPREDICTABLE and
DETERMINISTIC at the same time? And does this have anything to do with
THREE LEVELS of PREDICTABILITY.
In the face of all this what is the significance of UNPREDICTABLE
but DETERMINISTIC CHAOS? Well in the first place it is not just chaos,
but unpredictable periodicity OR chaos. There are three levels of
predictability pertinent here.
The first level is the simplest one where a person has observed a
phenomenon so many times that it is obvious to him what is going to
happen next. It doesn’t take much to know that spring will soon follow
winter because it has happened so many times. There is no need to know
the equations that govern weather, or even if anything governs weather
at all; the periodicity of the seasons is so absolute that predicting
them is not much trouble. In fact the first level of predictability
derives directly from the simple and OBSERVABLE periodicity of the
2. KNOWING the EQUATIONS.
The next level of predictability comes from knowing the actual
equations that govern the system under observation. From these
equations and postulated initial conditions (starting input values) you
can tell what will happen for the rest of the life of the system. In
idealized conception, our understanding of simple harmonic oscillators,
pendulums, planetary motions, and such things fall into this category.
If one knows the equations it is not even necessary for the output
behavior to be simply periodic. It can be chaotic as well, and still be
totally predictable from the equations and the initial conditions. The
Lorenz attractor is a famous mathematical example of a set of equations
with a very beautiful chaotic output result that is trivial to compute
and follows from most any initial condition you choose.
3. The BEEF.
The third and last level of predictability is what is usually
referred to as UNPREDICTABLE but DETERMINED. This arises in the case of
equations with HIGHLY UNSTABLE input areas. Again, if you choose an
initial input value you will get a totally predictable output result,
either periodic or chaotic, but if you change the input value by an
INFINITESIMAL amount you will get a completely different set of output
results. It’s that word INFINITESIMAL that counts.
MEASURING the INPUT VALUES.
You see when an equation is applied to a REAL system, some living
breathing important operation of life and the cosmos, it’s all well and
good to have the equations ready at hand which totally describe the
behavior of the system under consideration, but you also have to specify
the initial input conditions. But this is a matter of DIRECTLY
MEASURING THEM AS THEY ARE IN THE REAL WORLD. The problem is that when
ever you measure a universe you usually have to use a part of that
universe to measure the other part. For example using a tape measure to
measure a sidewalk.
For this reason, in this universe, measurement is always
inaccurate. You might be able to get your measurement down to 1 part in
10 billion, which for most people would be good enough. A carpenter
would probably look at you weird if you gave him that kind of accuracy.
But for equations with fractal behavior and UNSTABLE INPUT AREAS, 1 part
in 10 billion does not cut it. In fact 1 part in 10 BILLION BILLION
BILLION BILLION a BILLION times does not cut it. Because no matter how
close you measure it, it is still a great big blundering error compared
to the INFINITESIMAL change necessary to change the output behavior of
your system COMPLETELY. You say, Completely? Surely NO equation is
THAT sensitive to ANYTHING. Well, you are wrong. Actually MOST
equations ARE that sensitive to EVERYTHING. So you see we are in a deep
pile of water here.
To the degree that the real world works in equations that are non
linear, and to the degree that the inputs to these equations just happen
to lie in HIGHLY UNSTABLE INPUT AREAS, you will never be able to measure
the initial conditions accurately enough to be able to tell what the
output will do.
The fact that the output does do something means that the input
must have had some value, but you won’t ever be able to know it
accurately enough to compute the output result. Only REALITY knows it
for sure, and if you talk to the quantum mechanic boys not even reality
may know. (See Footnote No. 2 QUANTUM MECHANICS. Read it after you
finish the rest of this.)
The BUTTERFLY EFFECT.
Of course it is not always true that reality is operating in the
unstable input area of a particular equation. In that case your
measurement of the inputs (initial conditions) will be close enough to
very accurately predict the result. In fact a whole mess of different
input values may go to exactly the same output result.
On the other hand if you ARE in an unstable input area, a single
butterfly may, by fluttering its wings in Timbuktu, be the cause of
Hurricane Gilbert 4000 miles away. Its all a matter of where you are in
the Mandelbrot Sets of life. On the inside, or on the outside, or on
the tendrils of chaos. No foolin’.
In summary therefore, any equation of the form Z = f(Z,A,B,C…)
is iterable and says so directly by having the Z both in the input and
the output. The variable that is in both the input and the output IS
the variable of iteration.
Because each equation has an INPUT and an OUTPUT, we can talk about
an INPUT AREA which is all the possible values any one of the input
variables can take on, and an OUTPUT AREA which is all the possible
values the output can take on. Each one of the input variables has its
own input area. STATE SPACES are input and output areas.
The behavior of the OUTPUT can be either PERIODIC or CHAOTIC.
Periodic means the output value settles down to an ever repeating set of
values finite in number although not necessarily finite in value. For
example the equation Z = 1/Z started at Z = 0 has a periodic cycle of 2
points consisting of 0 and infinity. 1/0 is infinity, and 1/infinity is
0, etc. Chaotic means the output value is forever new (thus finite in
value) never landing on the same point twice and never repeating itself.
Chaotic output is characterized by always new but reasonable activity in
a finite arena of operation.
The behavior of the OUTPUT is affected of course by the value of
the INPUT. An input area is called STABLE if large or ‘significan’t’
changes in input value cause little to no change in output behavior,
especially in KIND of output behavior such as periodic or chaotic.
However in an UNSTABLE input area even an infinitesimally small change
in input value can cause the output behavior to change wildly and
drastically from periodic to chaotic or visa versa. Or it can change
the periodic cycle of the output from one value like 4 to another like
50000 with out warning. Or it can cause the periodic points to change
from one set of values to a totally different set of values.
Finally the border line in the input area that divides periodic
from chaotic output behavior is usually infinitly complex (and often
quite beautiful). This kind of fractal behavior is manifested by the
fact that no matter how much you ‘blow up’ or magnify the border you
will never find the border straightening out or becoming more simple.
Instead you find more and more convolution and detail.
MANDELBROT SETS and JULIA SETS.
When studying the OUTPUT of an iterated equation, you are always
studying the behavior of Z or whatever the iterated variable is called.
However when studying INPUT values, one can study either Z or everything
that is NOT Z. Thus when studying the OUTPUT of an equation you are
always studying JULIA (Z space) images, but when studying the INPUT of
an equation you can study either JULIA or MANDELBROT (C space) images.
Of course you always study the input of an equation by studying its
effect on the output. Thus although the Mandelbrot image is a picture
of input values, it is colored by looking at the resulting output
behavior in Z for each input value of C. C gets colored by what Z does
starting at Z = 0 for that particular value of C.
We have come to the end of our discussion of the question ‘Do
Fractals Explain Everything?’. The answer is no, but it could be a good
bet. Of course some would say that God explains everything. But God
seems to have been a Mathematician.
Thank you for your attention.
Footnote 2. QUANTUM MECHANICS
Actually the quantum guys may have a real hard time with this. For
a long time scientists believed that if a given input gave rise to a
specific output, then all inputs in the same small region of the
original input would give a similar if not identical output. This seems
reasonable. But no one had the faintest dream that these equations have
INFINITELY UNSTABLE INPUT AREAS. Not until Lorenz came along and
surprised the hell out of himself one night. (Read Gleick, CHAOS)
The HEISENBERG UNCERTAINTY PRINCIPLE.
Quantum mechanics has two very important things to say about the
universe. One true and the other, well Einstein didn’t buy it. The
first principle is the Heisenberg Uncertainty Principle which says that
the more accurately you measure the exact position of a particle the
less accurately you can measure the velocity, and the more accurately
you measure the velocity of a particle the less accurately you can
measure the position. This is because the very act of measuring the
particle disturbs the particle. Thus the final result you get is not
only a function of what the particle was doing, but also of your
disturbance of it. It is impossible to determine what part is due to
disturbance and what part to its actual state, and so whenever you use
the universe to measure the universe you run into this inherent
inability to get an EXACT result.
Quantum mechanics handles this by dealing with particles as a
probability function that does not describe exactly where the particle
is, but describes a probability of finding the particle in a given area.
The particle’s probability ‘wave function’ has a general size for a
particle with a given velocity, so the particle does exist mostly inside
a well defined area, but there is only a probability of finding that
particle at any particular place in that area, and the probability falls
off as the distance increases from the center. More to the point, the
probability is NOT 100 percent AT the center.
Such ‘fuzzy’ particles are not considered to exist anywhere
exactly; not until an interaction takes place, at which point the
interaction ‘locates’ the particle in only one of its many possible
positions with probability determined by its wave function.
Now the first thing that can be said about quantum mechanics is
that it works. Up to a point. Much better than say Newtonian Mechanics
which also works, up to a point. (I have yet to find a pendulum clock
that kept good time.) However the quantum mechanic boys take this one
step farther to say what Einstein could not accept. They say that
‘because you can never MEASURE the exact position and velocity of a
particle, and because our mathematical model CLAIMS these particles
don’t HAVE an exact position and velocity yet works so well, IT MUST BE
TRUE THAT PARTICLES REALLY AND TRULY DON’T HAVE AN EXACT POSITION AND
As long as one assumed that ‘a given input giving rise to a
specific output, meant that all inputs in the same general area would
also give rise to approximately the same output’ this was fine. The
fact that the inputs were all ‘fuzzy’ particles without clearly defined
positions would not affect the output too terribly much because ‘all
inputs in the same general area would give rise to approximately the
However the discovery of INFINITLY UNSTABLE INPUT AREAS in iterated
non-linear equations may change all this. Over someone’s dead body I am
sure. If the output is doing something consistent, be it periodic or
chaotic, and the input is operating in an INFINITELY UNSTABLE area, then
all the inputs must have ABSOLUTELY EXACT VALUES (POSITIONS AND
VELOCITIES) with INFINITE PRECISION or else the output would rapidly,
wildly and randomly change from one behavior to another.
Of course there are a lot of stunningly interesting experiments in
physics that will keep this controversy going on for a long time. It is
hard not to be charmed by the particle nature of light in one experiment
and the wave nature of light in another. I am sure we will be
scratching our heads for years. However, fractal instability may be
another moment in the history of science when the nature of pure
mathematics determines the possible end nature of reality, and throws
into discomfiture one of the grandest and most entrenched theories of
Of course it may be that reality never operates in the unstable
areas of equations. In which case the little fuzzy particles will get
along just fine. Just remember however, that Quantum Mechanics was
created before anyone knew about fractal instability, so one would
expect this data to have some influence. And to be met with some
THE CELL AND THE WOMB.
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.
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
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 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 environment. Identical cells become very different
when allowed to grow in different environments. The human body is
living proof of this. The internal genetics do not change, but 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, it grows into a different
cell. From this inflow 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
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.
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 JULIA PLANE.
(First thing, let’s get something straight. 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 are lost.)
The progress of a cell and what it becomes (its STATE) can be
plotted on the inside 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.
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 no change at all, or 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
The INSIDE PLANE is the JULIA PLANE.
The JULIA PLANE is the STATE SPACE of the INSIDE of the CELL.
The OUTSIDE PLANE.
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 Los Angeles 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 limited.
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 overpopulation.
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.
JULIA PLANE – INSIDE – CELL
MANDELBROT PLANE – OUTSIDE – ENVIRONMENT
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
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.
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 offspring.
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 cell.
The red dot is allowed to go anywhere but infinity. Infinity means
too much change in the cell and this means death.
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 cells
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 black area quickly.
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 got 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 process.
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
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. Z changes.
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. We know this 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 call 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
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 7containing the particular
object it was positioned next to.
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 others
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.
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
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 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. -HWS
THE LAST LAUGH
A Mandelbrot fractal is a statement of how well something survives
in a given environment. If it does NOT survive well you give the
environment some color of the rainbow specifying how quickly the thing
died. If the item in question DOES survive well or lives forever you
color that environment black.
Mathematically speaking infinity does not mean infinite survival.
Infinity means infinite change, which means death. Thus in the
Mandelbrot plane (C) if the iterated variable Z goes off to infinity
that means it died and you give the environment (C) some color
representing that fact, and how long it took to go out of bounds.
What the variable does in most of the black area of the Mandelbrot
set is not much better as far as survival is concerned. In much of the
black area the iterated variable attains a steady state of no change at
all, or a periodic change from one state to another and back again.
This too is not life.
Life is not a rock or a statue. It is a constantly changing
system. You grow, you change, you shed your entire skin every so often,
you replace broken parts, you age, you split in half, or you procreate
and then die.
This is very unlike the behavior of the variable going off to
infinity and it is very unlike the variable hanging out at a fixed point
or fixed period cycle. Life much more resembles the iterated variable
involved in a chaotic attractor, those areas of constant change within
These areas of constant change, of chaotic attraction, happen for
environments (C) that are on the boundary of the Mandelbrot Set. But
they also happen if the environment (C) is constantly changing from
iteration to iteration. A fixed point for one value of (C) will not be
a fixed point for another value of (C). So if Z is zooming in on a
fixed point or dull periodic cycle for a particular value of (C) and you
keep changing (C), you will change the value of the fixed point out from
under the variable that is homing in, and it will attain a state of
constant change within reasonable bounds never settling down to a dull
or rigid cycle. This is life.
One last point should be made here. Once the iterated variable (Z)
reaches a fixed point or cycle, it stays there BECAUSE the environment
(C) is NOT changing. But in real life the environment is always
changing and in fact it is often changed by the very production output
of the iterated item in question. The cell that splits in two while
iterating CREATES a new environment for itself consisting of its newly
formed sister cell. Thus what may be a fixed point during one iteration
may no longer be a fixed point during the next iteration because the
environment variable (C) has changed. Thus the iterated variable, the
inside trying to survive in an outside, may constantly skit around
looking for stability to find it always eluding its grasp. This keeps
you from becoming a rock or a statue. This is the constant ebb and flow
of biological life, and keeps the wheels of progress, production and
consumption, always turning.
This can have a negative side too. People trying to find health
and happiness by DRIVING to work everyday may be foiled by the
productive output of their automobile engines.
A lot of people laugh at ‘The Cell and the Womb’ or the idea that
fractals have anything to do with insides surviving in outsides.
However the equation Z = Z*Z + C directly says that what happens to Z is
a function of what Z was just a moment before and EVERYTHING ELSE IN THE
WORLD THAT IS NOT Z. If that is not something ‘surviving by changing’
in an environment not itself, then I don’t know what is.
You got to remember something about fractals. A long time ago Ben
Franklin was playing around with electricity and he came up with some
interesting theories to explain some very interesting phenomenon. They
really knew how to zap people in those days, what with Leyden jars and
kites on strings and all. Everyone had to ‘feel the spark’, that was
part of their initiation into the inner conclave of Electricians. If
you failed the initiation, either by accident or otherwise, they buried
you and found someone else.
No joking, Leyden jars ganged in parallel could store millions of
volts and throw sparks 2 feet long. But they still thought electricity
was a liquid that you could dissolve in water. That’s HOW they
discovered the Leyden jar. ZAP!
This was all very impressive to everyone involved, but does this
mean that Ben was right? No of course not, Ben’s ideas of electricity
were near ridiculous. It took a guy by the name of Faraday to make any
real sense out of the matter, but to hear HIM talk of it Ben was a
That’s because before Ben’s time people were pretty much passing
banana’s back and forth in the trees as far as electricity was
concerned. Ben made the first bold steps towards making electricty a
respected and controllable subject of knowledge.
Just so today with fractals. Chaotic dynamical systems is a VERY
VERY NEW field of math and it is VERY DIFFERENT from anything that has
gone before. People have no idea what is to be found there, and people
have no idea what it could be used for. They might as well be passing
bananas back and forth in the trees when it comes to fractals.
So you could say that these ideas about the ‘Cell and the Womb’ and
so forth are in the Ben Franklin stage of discovery. No one has even
proven them wrong yet. They are still LAUGHING at them, don’t you see?
The Faraday stage of development, where they get ripped apart and put
back together again correctly, is still way down the road. But in 200
years people will know what this was all about and not only will they be
able to prove that their understanding is correct they will be able to
use it to incredible ends.
Of course we may not be able to recognize these original ideas in
the final useable version, just as Ben’s original ideas have been lost
in the upgrades. But the people who LAUGHED at Ben were not the ones to
make the upgrades, and Ben himself would have been the first to embrace
the improvements even if it had meant leaving his own name in the dust.
And with that attitude you can be sure his name never will be left
in the dust.
THE BEAUTY PRINCIPLE
‘This love affair with fractals is disturbing to mathematicians
like myself who see too many people believing that this stuff is serious
mathematics. Fractal geometry has not solved any problems. It is not
even clear that it has created any new ones.’ Steven Krantz, Research
News, 27 July 1990.
The Beauty Principle states that if a theory or idea is beautiful
chances are it is useful. It also says in reverse that if something is
useful, chances are beauty will be found in it.
Beauty is surely in the eye of the beholder, and as such, beauty is
surely connected to the fundamental nature of that beholder.
Life consists basically of survival, of winning, of besting the
elements, even of dying in such a way as to live better as a species.
Thus the deer is beautiful because it can run, and the tiger is
beautiful because it can chase. Those that could not run and those that
could not chase long ago passed away, and what is left are those that
won the game of survival.
As for dying, grass is edible because manure helps it grow. Being
eaten (dying) helps grass grow better. Thus the deliciousness of grass
is not an evolutionary failure of grass to out survive its enemies. If
grass had wanted to be poisonous it could have done so, easily. The
tasty stuff survived better BECAUSE it was tasty. Get it?
Everything in nature, with few exceptions, whose form has been
molded by success in its own element, has beauty in it for the eye of
the human beholder. From the wing of the bird hovering in the halcyon
winds of summer, to the flight of the Challenger as it lands on the run
way, there is beauty in the form BECAUSE THE FORM WORKS, form that has
been developed over millions of years of evolution when all the non
working ones were left behind as dust in the wind.
That the form of a bird’s wing could be turned into the wing of an
aeroplane, does not diminish the wisdom contained therein, and the
beauty that results either in the wing or the plane, is directly
proportional to its rightness in the game of survival in a given
The human body itself (although maybe not the conscious unit
within) is also built on the same principles, and its internal
structure, from its bones to its brain, it based on the simple
mathematics of survival that everything in this physical universe is.
It is not unreasonable to assume that when the body is placed near
or in communication with another entity built on similar principles also
having withstood the tests of time, that a certain resonance would take
place resulting in an awareness of the beauty, functionality and
inherent rightness in the design of the other thing.
It is also reasonable to assume that such a human body, put into
direct communication with the basic mathematics of its own internal
structure, would elicit a similar reaction of aesthetic appreciation.
Being able to appreciate the wing of a bird, or your own body is no
different, each is the workmanship of ages of survival, and one would
assume that, given enough intelligence, each would be able to appreciate
the ideas and mathematics and design principles that went into each
other and their present state of success with existence.
In other words for those to whom survival is beautiful, one would
expect them to find their own design principles beautiful too.
Thus one imagines a direct harmonic resonance between the human
central nervous system and fractal images. The communication line
between the two could not get more direct, as the optic nerve and the
brain are MADE of the very mathematics that are displayed in the fractal
If you like fractals, it is because you are made of them.
If you can’t stand fractals, its because you can’t stand yourself.
This is not a matter of some artistic, philosophical or religious
mumbo jumbo. We are talking physics at the level of tuning forks on a
sounding board. Absolutely ground level.
Such a statement will surely be held against me all the rest of my
life, but it will be well worth it as our future history books will
record who was the fool and who wasn’t.
If it IS true that fractal mathematics has produced no new problems
or solutions it is only because the people who work with fractals are as
yet too dense to figure them out, which is very too bad, because much of
world is still out of control and there is a crying need for increased
understanding, especially in the field of how things do and do not
survive in their environments.
It is unlikely that the underlying mathematics of this universe is
utterly complex in nature. In fact one finds the simplest non linear
equations abounding in everything you study in classical physics, such
as planetary motion.
Fractals are not something special, anymore than the parabola is
something special. The parabola is a visual representation of the
EVALUATION of the simplest non linear equation Y = X*X + C, and the
Mandelbrot Set is a visual representation of the ITERATION (repeated
evaluation) of the same SIMPLEST POSSIBLE non linear equation.
It is also unlikely that iteration is any less important than
simple evaluation, especially for systems that are a function of
themselves a moment before plus their environment, and so it is unlikely
that the Mandelbrot Set is any less important than the parabola.
And to claim that the parabola is unimportant would be unwise at
this time. – HWS
IN DEFENSE OF THE MASTER
‘In the preface to The Science of Fractal Images, Mandelbrot
suggests that fractal geometers also use computer graphics to develop
hypotheses and conjectures. But the difference is that the hypotheses
and conjectures are (like the objects which they study) self-
referential. One generates the pictures to learn more about the
pictures, not to attain deeper understanding. That the pictures have
occasionally inspired fine mathematicians to prove good theorems seems
serendipitous at best.’
Steven G Krantz, The Mathematical Intelligencer, Vol 11, No 4.
I have probably burned more CPU cycles than most in the search for
pretty pictures, thus I can sympathize with Krantz’s deploring such use
of computer time, however I have also worked long and closely with Dr.
Hubbard during many of the years that he was first interested in fractal
mathematics and there is something that needs to be said here.
In the first place Dr. Hubbard is the first to tell his graduate
students that pretty pictures are a waste of time unless they can PROVE
something about them. Proof, it would seem, is the coin of the realm.
I know this for a fact because I have had to listen to the endless woes
of poor graduates students who can produce the most amazing pictures,
but who have a very hard time proving anything. ‘Hubbard wants me to
PROVE something’, they complain.
Further my own experience working with Dr. Hubbard has given me a
direct and personal insight into the relationship between pretty
pictures and deep mathematical cognition or proof.
In the first place, no picture can ever prove a conjecture, just as
one example can never prove a hypothesis. However one measly picture
CAN DISPROVE a conjecture in no time flat. Dr. Hubbard is a fine one
for coming up with endless conjectures about iteration theory, and he
uses the images that I make for him to scan anxiously for the one that
will prove the conjecture wrong in an absence of immediate analytical
ability to prove or disprove it formally.
It is an enormous waste of time trying to prove something right
that is indeed wrong, and if one can bypass that effort by making a few
pictures to see if the disproof is easily forthcoming, then making such
pretty pictures is well worth the time.
Secondly, we were recently working on a problem of the
intersection of two quadratics and the behavior of Newton’s Method in
this space to find the points of intersection. Dr. Hubbard directed me
to make a whole slew of movies, each movie had a 1000 frames, that
scanned the parameters from low to high.
During the viewing of these movies it became apparent that there
was a line in the space that Dr. Hubbard could not immediately prove
should be there. Thinking about it some he suddenly came up with the
conjecture that this line was invariant under Newton’s method which
meant that any iteration starting on the line would forward iterate to
another point still on that line.
He got very animated suddenly, dragged me over to a table and said
‘This is amazing, I can’t believe how stupid I am to not have thought of
this before, I wonder if I can PROVE it is true!’
He then proceeded to drag me through the entire procedure of
working through the proof, possibly for the first time in human history.
Now THIS is the stuff of real mathematics, all of Steven Krantz to
the contrary. And I was there.
THE WHY IS GOD
One of the great discoveries of the 19th century is that everything
in the physical universe (except maybe consciousness) follows
mathematical laws. All of the relationships between objects in space
and time can be measured in terms of numbers, and all of the causal
influences between them can also be described in terms of equations and
Apparently God was a mathematician first, and a purveyor of
Thus the study of math becomes of paramount importance in the
process of understanding and controlling the universe around us.
It goes without saying that if the universe is controlled by
mathematical laws then the qualities and conditions found in those
mathematical laws will be manifest in the universe that is controlled by
Thus the study of pure mathematics is beneficial to us all, for as
we gain deeper insight into the nature of the math we use, we are more
able to find these same things manifested in the exterior physical
universe. If it were not for the mathematician making the discovery on
the purely theoretical level, these manifestations would go forever
unnoticed or unexplained in the physical universe.
It is quite apparent that the mathematics that governs the macro
scale world is based mostly on the simple force laws of Gauss, Maxwell
and Co. namely the 1/R*R laws that relate to the propagation of force
lines through out the surface of a surrounding sphere. This one idea
alone brought science into the 20th century and with good reason. It is
basically correct enough.
We find this simple mathematical equation expressed not only in the
laws of gravity, but also in electromagnetism and electricity without
which our world would still be cold and dark.
Thus it would seem that if the mathematicians are to stay ahead of
the game they should spend some time studying the ins and outs of the
1/R*R laws and their related equations, such as derivatives and
For the longest time, the missing part to this whole story was the
concept of iteration. For example in the force law F = GMM/R*R we see
Force expressed as a function of mass and position, but we do not see
POSITION expressed as a function of mass and POSITION. Thus the use of
the F=GMM/R*R law is strictly on an evaluation basis. You find the
inputs and you get one output. There is no place to stick the output
back into the input because the Force term is not in the input, and the
input terms are not in the output.
Most of physics went this way for eons although it need not have.
People have long known of differential equations wherein the present
CHANGE in state is a function of the present state. But really it took
some very bright people looking into living biological systems to start
coming up with equations that expressed the PRESENT STATE directly in
terms of the PRESENT STATE (just a moment before!)
This of course gives the needed conditions for ITERATION, or
repeated evaluation of the equations involved.
It is quite possible to then go back to all the old familiar stuff,
like planets moving around the sun, and REWRITE the equations in terms
of iteration, wherein the present position is a function of past
position. In fact the rocket guys do it all the time, as simple
algebraic solutions to multi bodied planetary systems are either
impossible or difficult. Actually the use of iterated differential
systems has been with us for a long time.
So if we go back and look at the world in this new light, all of a
sudden we find we have a plethora of iterated expressions with little to
no understanding of what their MATHEMATICAL nature is.
So once you start looking into the matter you find a wonderful
world of structure and ‘nature’ in the subject. You find there are
fixed points, and period cycles, and chaotic cycles, there are repulsive
and attractive things with basins of attraction, and there are open and
closed Julia Sets which give rise to all sorts of fantastic Mandelbrot
Sets. One finds that all these things depend on the values of the
parameters involved in the equations, and that all equations manifest
these things in one form or another.
One also finds that for many of them the various boundaries
surrounding basins of attraction or what have you are quite FRACTAL, an
incidental but very important new thing to know about the matter.
IT COULD NOT BE SO PROMINENT AND NOT BE IMPORTANT, DON’T YOU SEE?
Not as long as the equations which manifest such things are key
causal descriptors in the physical universe in question.
First you find out what is true MATHEMATICALLY, and THEN you find
out where those things are applied in the real universe. I assure you
these things do not exist in a vacuum of vanity. If they exist
mathematically, they ARE applied.
Now the important equations of the physical universe have not
changed, Y = X*X + C is still Y = X*X + C, but the CONCEPT of what you
are going to DO with it has changed, from simple evaluation which
produces parabolas, to multiple iteration which produces Mandelbrot Sets
and Julia Sets. And all of a sudden this entire new landscape of
structure and nature appears to unfold as if out of nowhere.
It may seem like it is new, but you and I know it has been there
AND YOU CAN BET THE PHYSICAL UNIVERSE KNOWS IT TOO.
The physical universe has been using these same equations to march
its wheels of time forever, and you can be sure that if these equations
show some inner structure or ‘nature’ then that same nature will be
manifested outwardly in the physical universe expression of that
The physical universe is basically an iterated system, so actually
it is surprising we have made the progress we have, using only simple
evaluation. The equations have been around forever. The physical
universe has been USING them almost forever. The equations have as part
of their nature things like fixed points, period cycles, chaotic cycles,
basins of attraction, etc., so you can be sure all these things are
manifested in the physical universe INCLUDING FRACTALNESS.
To say therefore that fractals have nothing to with anything and
have not explained or proven useful in our understanding of the universe
is more a statement about the people who are working with fractals
rather than a statement about the pertinence of fractals to the world at
Fractals are so pertinent to the universe no one can see it yet.
Long time ago, they thought math did not pertain either. The ‘why’
The ‘why’ may still be God, but if it is, then clearly God is a
mathematician of significant merit, and no doubt a fractal enthusiast.
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‘The Paths of Lovers Cross in the Line of Duty.’
MANDELBROT SETS AND JULIA SETS.
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
FIXED POINT for INSIDE the JULIA SET, 1 is the FIXED POINT for ON 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
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
ATTRACTIVE CYCLE POINTS in the middle.
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.
In BACKWARD ITERATION the JULIA SET ATTRACTS all points not ON it. The
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
BACKWARD IMAGES. Both green BACKWARD IMAGES would FORWARD ITERATE to the
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.
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.
THE THEORY BEHIND ‘THE CELL AND THE WOMB’
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
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
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
heading of STABILITY.
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
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
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’. I just wonder why people CARE so much.
THE THEORY BEHIND ‘THE CELL AND THE WOMB’
An example of a NON Fractal system.
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
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
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
THE THEORY BEHIND ‘THE CELL AND THE WOMB’
A Julia Set is a closed (connected) boundary in the Z plane that
separates the Z’s that go off to infinity under forward iteration and
the Z’s that go inward towards a single or multiple cycle fixed points.
In rare cases Z can forward orbit into a chaotic cycle called a Siegel
disk. Points directly on the Julia Set (the boundary) forward iterate
to other points directly on the Julia Set.
Each C taken from the Mandelbrot Set has a characteristic Julia
Set. For C’s taken from outside the M Set, the Julia sets are open
(dusty) and are called Cantor Sets. In this case the inside and outside
of the Cantor sets are contiguous and all starting Z’s go to infinity
except those directly on the Cantor Set which still go to other points
on the Cantor Set.
In case anyone wonders how people get the names for these things,
Mandelbrot, Julia and Cantor were all people. Just as were Hertz, Volt,
Ampere, Ohm, Couloumb, Faraday and Gauss.
Thus for each Julia Set there are 3 regions of interest each having
its own fixed point. A fixed point is a starting Z that forward
iterates directly to itself. Since every Z has TWO points that forward
iterate to it, (every Z has two backward images), a fixed point also has
two backward images one of which is itself, and the other is another
point somewhere on the Z plane.
The first region of the Julia Set is OUTSIDE where Z’s go off to
infinity. Infinity is an attractive fixed point because it attracts
under forward iteration all Z’s in its general area. It is a fixed
point because INFINITY**2 + C is infinity. It’s backward images are
plus and minus infinity both of which go to infinity under forward
The second region of the Julia Set is ON the Julia Set where Z’s
forward iterate to other points ON the Julia Set. For this region you
find the fixed points by solving the quadratic equation:
Z = Z*Z + C or 0 = Z*Z – Z + C.
Using the quadratic formula this becomes
Z = (1 + SQRT(1 – 4*C))/2 or Z = (1 – SQRT(1 – 4*C))/2
For the 1 cycle case (C chosen from the main cardioid of the M set)
one of these fixed points is ON the Julia Set, and the other is INSIDE
the Julia Set.
The fixed point ON the Julia is a REPULSIVE fixed point as it
repels all Z’s away from it except those ON the Julia set. Take for
example the number 1 on the real axis. 1*1 is 1. So 1 is a fixed
point. But 1.1*1.1 is 1.21 which is further away. And .9*.9 is .81
which is also further away in the opposite direction.
When considering the complex plane the number 1 becomes the entire
Julia Set. Points ON the Julia Set iterate to points ON the Julia Set
just as 1 iterates to itself. Points INSIDE the Julia Set iterate to
points further inside, and points OUTSIDE the Julia Set iterate to
points further outside. This is repulsive behavior. The Julia Set is
basically a repulsive item, attractive only to itself.
This is very much like a mountain range where a marble balanced on
the top stays put, but off to any side starts to roll down the mountain
way from the top. Repulsiveness is the mark of unstability.
Attractiveness is the mark of stability. The second fixed point
inside the Julia Set is attractive and sits inside something that is
very much like a valley or a basin, in fact the area immediately
surrounding this attractive fixed point is called a basin of attraction.
The marble when placed in the center stays put, and when put up the side
of the slope a bit, rolls right back to the center.
Slight perturbations to the marble on the top of the mountain will
cause it to loose its position entirely to which it will never return.
This is unstability. Slight perturbations to the marble in the basin of
the valley will cause the marble to settle back down to where it was.
In summary therefore let’s consider the case of C = 0 for only the
real number line.
0 is an attractive fixed point because 0*0 + 0 = 0 and all points
near by are attracted to it. Infinity is an attractive fixed point
because INFINITY*INFINITY + 0 = INFINITY and also attracts all points
near by. 1 is a repulsive fixed point because 1*1 + 0 = 1 and repels
all points near by to either 0 or infinity.
If C is chosen from one of the other balls on the Mandelbrot Set
that are NOT in the main cardioid, then the forward orbits of Z have a
period cycle greater than 1. For example if C = -1 and Z starts at 0
then 0*0 – 1 = -1, and -1*-1 – 1 = 0. Thus every TWO iterates Z comes
back to where it stated. Observing this on the Julia plane, we see that
there are two fixed points inside the Julia Set, namely 0 and -1, and
any Z starting off inside the Julia set will forward orbit to BOTH of
Therefore in general every C taken from the Mandelbrot Set has
associated with it a compete Julia Set, and each Julia Set (if closed)
will have an attractive set of one or more ‘fixed points’ inside it that
iterates will go to if they start off inside the Julia Set. If iterates
start of outside the Julia Sets they will go to infinity and of course
this is true for all Julia sets whether open or closed.
In order to find these multi period fixed points inside a closed
Julia Set you first must know what period you are looking for. This is
determined by knowing which ball of the Mandelbrot Set you have taken
your value of C from.
Once you know the expected period of the Julia Set you can easily
find the exact values of the period points themselves.
Take for example C from the 2 ball of the M set where the cycle is
2, then you find the equation that is equivalent to 2 iterations of the
F(Z) = Z**2 + C
it must follow that two iterates of this is
F( F(Z) ) = (Z**2 + C)**2 + C
= Z**4 + 2*C*Z*Z + C*C + C
Remembering that a fixed point is a value of Z that iterates right
back to itself after (in this case) two iterates we can write the
required equation as follows.
Z = Z**4 + 2*C*Z*Z + C*C + C
which is the same as
0 = Z**4 + 2*C*Z*Z – Z + C*C + C
Being a 4th degree equation this has 4 answers. Two of these
answers are the previously discussed ONE cycle fixed points found in the
earlier discussion. This is because a one cycle fixed point is also a
two cycle fixed point. It comes back to itself after one cycle so it
certainly comes back to itself after two cycles!
The other two answers are the 2 cycle fixed points that lie INSIDE
the Julia Set. They return to themselves after two cycles. Thus we
call them 2 cycle fixed points.
Each ball on the Mandelbrot Set has its own cycle count, and
distinctive Julia pattern. The Julia Set is the boundary of the basins
of attraction that contain the fixed point cycles in the middle of them.
Thus a 5 cycle Julia Set will have a 5 fold basin of attraction and this
will determine the basic shape of the Julia Set.
Thus each ball of the Mandelbrot Set (meaning the C’s taken from
the balls) has its own Julia pattern. Because each ball has other balls
connected to them and further balls connected to THEM the Julia patterns
can get very complex. However each ball retains its own distinctive
This is so much so that Julia Sets taken off of balls off of balls
will have both patterns in a clearly recognizable mix. For example the
2 ball off the 3 ball has the 2 ball pattern inside the 3 ball pattern
and total cycle of 6. Likewise the 3 ball off of the 2 ball has the 3
ball pattern inside the 2 ball pattern and also a total cycle of 6.
This of course has to be seen to be believed, and the reader is
directed to the sheet entitled ‘Mandelbrot Sets and Julia Sets’ for a
visual confirmation of these facts.
Being able to draw Julia Sets, their 1 cycle fixed points, and
their period cycle ‘fixed points’ is much more important than being able
to draw the Mandelbrot Set. It is the Julia Set where all the life is
and it is the Julia Set that determines the nature of the Mandelbrot
Julia Sets taken from properly chosen points on the Mandelbrot Set
can be stunningly beautiful. Julia Sets however tend to be very self-
similar and scale independant. Thus zooms tend to be boring after a
point. Once you have seen part of it you have seen all of it.
Mandelbrot Sets however are not strictly self-similar and so zooms
take you into ever new territory.
THE THEORY BEHIND ‘THE CELL AND THE WOMB’
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?
The answer is quite simple. Lets say we start with a particular C with
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