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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

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PRESS RELEASE

‘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

Julia image.

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

Sets.

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.

INTRODUCTION

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

understood something.

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

mathematical overtones.

‘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

years.

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

distribution.

Where the corn is still growing at the end of the experiment color

it black.

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

think?

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

OUTSIDES.

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

in existance.

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

the road.

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

themselves in.

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

until now.

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

infinitum.

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

survival.

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

output behavior.

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.

UNPREDICTABLE DETERMINISM.

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

FREE WILL?

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.

1. OBSERVATION

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

system.

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

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.

FINAL FAREWELL.

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.

PROBABILITY WAVES.

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

VELOCITY.’

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

same output’.

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

our time.

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

resistance.

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

traits.

When a cell divides in two, each half is smaller but identical.

There is a time for growing before the cell is allowed to divide again.

A cell grows by two way interchange of chemical substances across the

cell wall. The cell grows by taking in MASS from the outside world. In

different environments it ‘eats’ different masses so it becomes

something different. The 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

cell.

This might seem to be hopelessly confusing, but it becomes simpler

if we study the grow and divide cycle of just one cell.

One ITERATION is one cycle of grow and divide in the on going life

of a cell. Cells can live or die. If they live they continue to

iterate: grow and divide. If they die they stop iterating. Usually

they die while trying to grow. Cells often will not divide unless they

have grown enough, that is attained a large enough mass through eating.

If they do not have enough proper food they will not grow to the

dividing stage and so die of starvation or poisoning. If they make it

to the divide stage, they usually have enough food energy to make it all

the way through the division. Cells do not eat while they divide.

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

maximum change).

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

chance.

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

iterations.

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

OUTSIDES.

Division however is the sign of SURVIVAL, a sign that the cell made

it. After division there are two of them after all and surely that

means it survived. But it made it because of the correctness of its

INSIDES in their ability to function properly in the environment given.

Division is sort of a reward for having successfully made it through the

growth phase to maturity. Thus division is a sign of survival. Thus

SURVIVAL happens during the DIVIDE phase BECAUSE OF viable INTERNAL

CONSTRUCTION (INSIDES). However the result is two IDENTICAL cells. No

CHANGE takes place (except in size) during the division process in the

internal nature of the cell. Thus GROWTH is associated with CHANGE

which is a form of NON SURVIVAL, and DIVISION is associated with

SURVIVAL which is a form of NO CHANGE.

When CHANGE takes place during DIVISION a mutation occurs. What

was is no longer, although it might fare better. When SURVIVAL takes

place during GROWTH, the cell has failed to differentiate properly due

to its surroundings.

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

becomes C1.

Recognizing that any equation may be used to model the growth and

division of a cell, we may write this in general mathematical terms as

follows. Let Z1 = Z0*Z0 + C0 be generalized to Z1 = f(Z0,C0) and let C1

= C0/2 + Z1 be generalized to C1 = g(C0,Z1). The iteration of Z1 =

f(Z0,C0) relates to the GROWTH of the cell in environment C0 from the

smaller Z0 to the bigger but different Z1. The iteration of C1 =

g(C0,Z1) relates to the change in environment from C0 to C1 caused by

the DIVISION of cell Z1 from the larger Z1 to two smaller but identical

Z1’s. The sister copy of Z1 becomes part of the new environment C1 of

the first copy of Z1.

The MOTH and the FOREST.

The question naturally arises, is it reasonable to represent the

entire inside state of a cell by one number? Or even more ridiculous

the entire rest of the universe by one number? Consider a population of

moths in a forest. Here the inside is the system of moths trying to

survive in the forest environment which is the outside. The number of

moths in the forest at any time is a function of the number of moths

just prior plus the environment.

Clearly the population of moths in the forest at any one time can

be represented by one number. But can the entire rest of the forest be

represented in this same way? The forest is a large system of

interacting subsystems, like the number of trees, the number of birds,

the number of oxygen molecules in the air. It would seem that if you

broke the forest into its parts you might be able to represent the

forest as a system of things each of which can be represented by one

number. Hence representing the entire forest at any one time as a

function of many single numbers resulting in one overall number is not

so wild. Just so with the inside and outside of a cell. Of course the

arrangement of things can be important too. But in large systems, the

DENSITIES of things can be more important that exact positioning. And

where positioning becomes important, it would be taken into account by

the number representing the environment 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

and F.

Fractal math is a way of looking at equations and physical

phenomena. Just like calculus is a way of looking at equations and

physical phenomena. Calculus deals with related rates. Fractals deal

with 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

reasonable boundaries.

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

pioneering genius.

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

environment.

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

image.

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.

It happens.

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

mathematical relationships.

Apparently God was a mathematician first, and a purveyor of

damnation second.

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

them.

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

integrals.

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

all along.

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

equation.

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

large.

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’

was God.

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’

Part 1

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

explain this.’

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

spaces.

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

of ITERATION.

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

CHAOTIC behavior.

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

acceleration.

No problem. Further a graph of D vs T will show a non linear plot

basically like Y = X*X.

Being non linear one might immediately wonder if there is potential

for fractal behavior in this system.

The answer is no.

To start with, the system as modeled here is not iterated, it is

merely evaluated. We can however turn it into an iterated system by

choosing a unit of time to match the unit of iteration. Let’s choose

our unit of time to be 1.

Thus the preceding equation can be remodeled and iterated as

follows.

D = 1/2A + V + D

V = A + V

D is the iterating variable in the first equation, and V is the

iterating variable in the second equation.

If D and V both start off with value of 0, and A is a constant

acceleration, then each iteration will give us new values for D and V

for each second down the road.

This is an iterated system and not an evaluated system because you

can’t just plug in the number 10 and get the final distance 10 seconds

down the road. You have to operate the pair of equations 10 separate

times to get the final answer.

However you will notice that both equations are now linear. Thus

there is no fractal behavior evident.

The equations have to be non linear in the ITERATING VARIABLE in

order for fractal behavior to be manifest. Our original equation was

non linear in T but iterated in D. That is why it is non fractal.

However consider the situation where the acceleration is no longer

constant but is a function of D itself such as in a spring system or a

gravity field. If A is a non linear function of D, then indeed the

equation in D is non linear and will show fractal behavior. If we

consider relativistic effects, it is possible that the acceleration will

also be a non linear function of V too. Then BOTH equations have non

linear terms in the iterating variables D and V and will show dualistic

fractal effects.

THE THEORY BEHIND ‘THE CELL AND THE WOMB’

Julia Sets

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

iteration.

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

them alternately.

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

primary equation.

Since,

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

pattern.

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

Set.

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’

Recapture

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

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