xfl7.memo
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'The Paths of Lovers Cross in the Line of Duty.'
DO FRACTALS EXPLAIN EVERYTHING?
Copyright (C) 1988 by Homer Wilson Smith
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
PAGE 2
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
PAGE 3
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
PAGE 4
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.
PAGE 5
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
PAGE 6
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.
PAGE 7
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.
PAGE 8
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
PAGE 9
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
PAGE 10
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.
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