ART MATRIX PO 880 Ithaca, NY 14851-0880 USA
(607) 277-0959, Fax (607) 277-8913
'The Paths of Lovers Cross in the Line of Duty.'
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 minus 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 particularly after a few
interations.
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. This would be an example of the
chaotic output hitting a spot of highly unstable input, which then
sudden shifted the output into a whole new direction.
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 periodic 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, each however also 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 of Z 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, but within a limited arena of change, otherwise the
living human would become dead, and the apple would become an orange.
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 (everything not
itself.)
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.
Thus the greatest survival happens when the output is chaotic
within a fininte arena of operation.
CAUSE and EFFECT.
Whether or not the output of a system is periodic or chaotic
depends on the initial input conditions, either the environment or the
thing trying to survive or both.
Some input values will cause periodic behavior in the output
result, while other input values will produce 'chaotic' behavior in
the output result.
We really have three things in operation here, the value
of the thing trying to survive, the value of the environment, and
the equation that relates the two from input to output.
Whether anything survives well depends on its own value, the
value of its environment and the equation itself relating the two.
Change anyone of the three and the output can go from a cozy chaotic
homebody (finite arena of operation) to a wildly changing roller
coaster ride to ending up (dead) at infinity itself.
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 significant 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 significant 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.
In this case C is the environment (Conditions) that Z (Zygote) is
trying to survive in. So the Mandelbrot set is a picture of every
possible external Condition an egg could find it self in trying to
survive.
If you pick a C inside the main cardiod of the Mandelbrot Set and
follow the forward iterates Z = Z*Z + C with Z starting at 0, 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 but to two wildly
different outcomes, one near 0, the other at infinity.
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. Actually MOST
equations ARE that sensitive to EVERYTHING. So you see we are in deep
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 the input
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
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 'significant'
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.
Mathematicians like to say such infinitely complex borders have
fractional dimension. For example a straight line has dimension 1,
and a square has dimension 2. But a line that is infinitely
convoluted might have a dimension around 1.5 say, and it is this
'fractional dimensiona' that gave rise to the term 'fractal'.
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, and
that may be true, 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.', merely and only because we can never
know them.
As long as one assumed that 'a given input results in 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'.
Thus the effects of quantum fuzziness in the input might not
be all that noticeable in the output.
However the discovery of INFINITLY UNSTABLE INPUT AREAS in iterated
non-linear equations may change all this.
If the output is doing something consistent, experiment after
experiment, be it periodic or chaotic, and the input is operating in
an INFINITELY UNSTABLE area, then all the inputs, experiment after
experiment, 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 in each
succeeding experiment.
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.
On the other hand maybe reality does operate in the unstable
areas of equations, in which case the probabilistic nature of quantum
events may cause wildly incongruous results from experiment to
experiment.
It's something worth looking for.
Homer