![[fractal image]](resource/frctlogo.gif)
Table of Contents
1. Introduction
The laws of nature are often chaotic due to their non-linear makeup. That is, they
exhibit
seemingly random behavior with an underlying order. Applied scientists do not usually
have a strong mathematical background, so they try to model complex systems with
linear
approximations, thereby introducing error.
2. What is chaos?
Figure 1: The Lorenz Attractor
Deterministic chaos is a set of seemingly random results produced, usually, by a
system of
non-linear equations. Real-life systems often behave according to a set of
non-linear
rules,
but scientists tend to try to model and predict their behavior with linear approximations.
The resulting model appears to contain "noise," which is usually ignored, due to the error
in the linearized equations.
3. What is a fractal?
In order to define what a fractal is, it may be beneficial to start with what a fractal is
not.
In evaluating a shape it is often desirable to find the capacity dimension of the shape.
The
concept of topological dimension is a familiar one. We all know, for instance, that a line
is
one-dimensional, a filled square is two-dimensional, and a cube is
three-dimensional.
The
capacity dimension happens to equal the topological dimension for most shapes in
Euclidean geometry.
3.1 What are some examples of fractals?
3.1.1 The Cantor Set
Figure 2: The Cantor Set
The first example of a fractal is a simple figure called the Cantor set, shown in Figure 2.
The Cantor set is constructed by drawing a line of length 1, then removing a piece of
length, say, 1/3 and leaving two pieces of length 1/3. Next these pieces are cut the
same
way, leaving 4 pieces of length 1/9. Repeating this process n times results in 2n line
segments of total length (2/3)n.
3.1.2 The Koch Triadic Island
Another example of a fractal is the Koch Triadic Island, shown in Figure 3. It is
constructed by starting with a triangle with sides of unit length. A piece 1/3 long is
removed from the center of each side and replaced with two pieces each 1/3 long, for
a
total new length of 4/3 per side. This object, resembling a star of David, is then divided
the same way.
Figure 3: Fifth Generation Koch Island (from Froyland)
The topological dimension of the Koch Triad is one, because it is composed entirely
of
line segments. Yet the length of the outline approaches infinity as iteration n approaches infinity. Its
fractal dimension can be calculated to be D=1.2618. It is interesting that the Koch curve
encompasses less area than a circle scribed around the original triangle despite its
infinite
boundary.
3.1.3 The Mandelbrot Set
The Mandelbrot Set is the graph of a recursive algorithm in the complex plane. To
calculate the Mandelbrot set, start with z=0+0i and with c=a+bi, the point you are
graphing Iterate znext=z^2+c. If the point converges, mark it. If it blows up to infinity, do
not mark it. The result, shown in figure 4, is the Mandelbrot set. The dendritic patterns
surrounding the familiar lemniscate of the Mandelbrot set carry self-similar versions of
itself as the scale is reduced.
Figure 4: The Mandelbrot Set
PC programs such as FractInt and WinFract are available to enable anyone to
experiment
with the Mandelbrot Set. It is often graphed in color to dramatically show the rate of
convergence.
Figure 5: Close-up of the Mandelbrot Set Showing Self-Similarity
3.2 Where are fractals used?
Fractals and chaos theory have been used on the fringes of science in many
different
disciplines. Over the years there has been a tendency to ignore "bad data" or "noise,"
but
these unpredictable effects are often due to the order-in-chaos of non-linear error terms
in
the experiment. Because of the difficulty in solving non-linear equations, most physical
scientists preferred to put them aside. The very few who decided to tackle the problem
did so in isolation. There was no money for pure research into something
mathematicians
considered "interesting but useless." Even if they wanted to publish their results there was
no forum for fractals and chaos: the first symposium on chaos was not held until 1977.
3.2.1 Edward Lorenz, Weather Prediction
In 1960 mathematicians still distrusted computers, and they certainly were unlikely to
read
scientific journals pertaining to weather prediction. Edward Lorenz, however, was a
meteorologist with a strong mathematical background who had an early computer at
his
disposal. In an effort to understand and predict the weather, Lorenz created a "toy"
weather system based on twelve numerical rules that represented real factors such as
the
east-to-west warming of the earth as the sun rose.
Figure 6: Divergence With Time (from Mullin)
When he came back he was stunned to find that the new data initially matched the
old
data. But after a period of time it diverged. Figure 6 shows an example of this
divergence
with time. When he thought about it, Lorenz realized that the numbers from the printout
were rounded off from the numbers in the actual computer calculations. He thought
that
the small amount of rounding error would be insignificant, but in fact,
non-linear systems
can show a great deal of sensitivity to initial conditions.
3.2.2 Robert May, Biology and Ecology
In studying populations, biologists have always noticed that populations vary from
year to
year. Robert May studied one of the simplest equations of population growth,
Xnext=rx(1-
x). This logistical equation dates back to Malthus and models the feedback loop of a
population, in theory allowing for growth to settle down and reach a stable level.
Figure 7: Bifurcation Diagram (May, from Gleick)
Figure 7 shows May's graph of parameter value versus final population level. At low
values of the parameter the population tended towards extinction, while at higher
values
the population tended to stabilize. May did not anticipate the bifurcations seen at
values
above 3, and the bifurcations on bifurcations of a fractal structure.
3.2.3 Arthur T. Winfree, Physiology
Arthur T. Winfree had a background in both biology and engineering physics, and this
led
him to study circadian rhythms in terms of chaos theory. Circadian rhythms are the
biological clocks that tell an organism when to eat and when to sleep. Jet lag is a
common
example of disrupted circadian rhythms.
3.2.4 Benoit Mandelbrot, Jack-of-all-Trades
Benoit Mandelbrot was a mathematician who worked in the pure research wing of
the
International Business Machines Corporation. His primary contribution to IBM was in the
field of information theory. The engineers were looking for ways to eliminate data
transmission errors and set Mandlebrot to work on the problem.
3.2.5 Michael Barnsley, Image Processing
Michael Barnsley is a mathematician who began exploring period-doubling and
bifurcations in 1979. He had the idea that the bifurcations might be the real part of a
shape in the complex plane. He was correct, but unfortunately he had only managed
to
rediscover the Julia Sets.
Figure 8: Fractal Fern (Barnsley, from Gleick)
Barnsley's editor put him in touch with Mandelbrot and from there Barnsley began
exploring recursion in general. He came up with an idea that he called "The Chaos
Game," in which a set of rules and a randomizer, e.g. a coin toss, generate familiar
shapes.
The first shape Barnsley created was a fern, shown in figure 8.
Nature exhibits chaotic, that is, seemingly random behavior with an underlying but
unpredictable order. Applied scientists do not usually have the mathematical
background
to solve systems of non-linear equations so they model complex systems with linear
approximations.
Appendix A: Bibliography
Frøyland, J. (1992). Introduction to Chaos and Coherence. Philadelphia, PA:
Institute
of Physics Publishing, Inc.
Appendix B: Computer Program in GWBASIC
Created by
Leslie Ellis,
April 6,1997.
Modified July 30, 2002.
Fractals are a special case of non-linear equations. The exponents are fractional,
and the
shapes formed often show self-symmetry as the scale is changed. Fractals are an
important tool in chaos theory.
Computers have made new investigations into non-linear systems and chaos theory
possible. The PC in particular, with its graphical capabilities and ease of use, has made it
possible for even high school students to investigate and understand fractal geometry.
This new understanding of fractals and chaos theory has made advances possible in
many
fields of Applied Science. They include Meteorology, Biology, Ecology, Physiology,
Medicine, Economics, and Image Processing, to name just a few.
Figure 1 shows the Lorenz attractor. This image represents a system of three non-linear
equations. The map shows a certain order, always rotating around the attractor,
occasionally flipping over to circle the other attractor, yet it never repeats itself.
For example, start with a filled unit square whose sides are length 1. A square is
drawn in
two-dimensional space, so it has a topological dimension of 2. Now form a square
whose
sides are of length 2 and measure how many unit squares fit in it. The answer is 4. Since
4=2*2, the capacity dimension of a square is 2. The capacity dimension of a cube may
be
calculated in a similar manner.
Fractal geometry concerns objects whose capacity dimension is fractional rather
than
integer. Much of nature contains such shapes. Clouds, for instance, and mountains are
fractal shapes rather than Euclidean.
Notice each successive generation can be overlaid on the previous generation by
a scale change and translation. This makes it possible to use one definition for fractal dimension:
L = constant + l1-D
We can eliminate the constant from this equation by substituting in the ratios of
terms for
successive generations and solving for the fractal dimension D as follows:
The total length of the line segments approaches zero as n approaches infinity, implying a topological
dimension of zero, while the total number of "motes" approaches infinity. The Cantor set
becomes a dust, not a single point, and therefore still shares some characteristics of a
line.
The Mandelbrot Set has captured the imaginations of people who have seen it.
The
central figure resembles the mandalas found in Oriental rugs and ancient Indian
tapestries.
The paisley boundaries resemble the imagery of the 60's drug culture. It has been
suggested that the fractal nature of the Mandelbrot Set somehow mimics the complex
structure of the brain. See figure 5 for a close-up of the Mandelbrot Set showing
dendritic connections and an embedded copy of the larger set.
Perhaps the best way to cover the many uses of fractal geometry and chaos
theory is to
give a brief history and some biographies of some who have made important
contributions
to the field. This list is by no means comprehensive, and fails to include some important
theorists for the sake of covering practical uses of chaos theory and fractal geometry.
Lorenz' computer program provided a chaotic output: it was seemingly random
but with
an underlying order to it. Lorenz learned that certain patterns were usually followed by
other patterns and believed he was on the path to completely accurate weather
prediction.
Not so.
A computer program is by nature deterministic, that is, the same data in yields the
same
data out each time. One day Lorenz wanted to start his toy weather program later in
the
run. He flipped into the back pages of one of his hefty printouts and typed in a line of
numbers, then went out for coffee.
Lorenz realized that to accurately predict weather in the long term would require
almost
infinitesimal sensing and readjustment of initial conditions. The flapping of a butterfly's
wings could completely change the course of the weather, a phenomenon known as
"The
Butterfly Effect."
As an aside, the weather satellites used by NOAA do not have the resolution
required.
Meteorology as practiced by the network weathermen is still somewhat of a Black Art.
Lorenz later simplified his model to find the least complicated chaotic system
possible, one
consisting of three simultaneous equations. The result was the Lorenz Attractor,
discussed earlier.. The Lorenz attractor may apply to the earth's magnetic field, which
varies chaotically. Geological evidence found in the Kaap Valley, South Africa indicates
that it flipped in the earth's past: this would correspond to the Lorenz attractor's flip from
one side to the other. The Lorenz attractor may also explain the sudden fall of the
average
temperature that results in an ice age every 10,000 years or so.
Appendix C contains a short GWBASIC program that prompts for the value of r and
performs 1000 iterations of the logistical equation. This program demonstrates bifurcation
and chaos in the logistic equation.
Nor did he anticipate the chaotic region found at even higher parameter values.
Up until
then it was always assumed that if a population seemed to vary randomly, it only meant
that the researchers had missed some anomalous environmental factor.
May applied the chaotic model to epidemiology and found that it applied to that
field as
well. Since epidemics of rubella, measles and polio tend to be somewhat cyclic, he
predicted correctly that mass inoculation programs may cause short-term rises in the
diseases they were intended to eradicate. Such short-term rises in the disease rate might
otherwise have convinced health department officials that an inoculation program was
unsuccessful.
Winfree was experimenting with mosquitoes and found that in the 24-hour-a-day
light of
the lab environment, mosquitoes switched from their normal dawn-to-dusk, 24-hour
activity cycle to a 23-hour cycle. Apparently this cycle was reset every day, somehow,
by
the sun.
Winfree's experiment consisted of applying precisely timed bursts of light to his
mosquitoes and observing the effect that the bursts had on the mosquitoes' circadian
rhythms. Some bursts delayed the rhythms and others reset them, but as Winfree
graphed
his results he noticed a singularity in the graph and predicted that a certain number of
photons at a certain time could obliterate the mosquitoes' circadian rhythms altogether.
He applied the burst and found that the mosquitoes no longer followed the 23-hour lab
day, but buzzed and slept at random intervals.
In inducing permanent jet-lag in mosquitoes, Winfree had demonstrated that
biological
systems can be examined using an empirical approach and the tools of chaos.
Next Winfree applied non-linear dynamics to the rhythm of the heart. Fibrillation is a
stable but fatal state where the heart muscles beat out of synchronization, literally like a
bag of worms. It has long been known that defibrillation could sometimes be effected
by
applying a large DC voltage across the chest of the victim. Compare this approach to
Winfree's application of light to mosquitoes, and to May's theory that vaccination
programs may cause a short-term rise in the disease rate.
Within four years May's ideas had all been proven, with the potential of saving tens
of
thousands of lives each year.
Mandelbrot examined a data stream by dividing it into a given length in time and
counting
the errors. He was trying to come up with an average bit error rate, but was having
trouble with it. The error rate did not follow any known distribution, i.e. gaussian. In
fact, the error rate was infinitesimally small. It seemed that no matter how large or how
small the time interval, he would still find intervals with no errors and other intervals with
bursts of errors. He had seen this similarity in scale in other systems and realized that
clean data transfer is virtually impossible because noise is inherent. In fact, Mandelbrot
was dealing with a Cantor Set, which was explained previously. He suggested that the
engineers concentrate on redundancy checking and error correction rather than on
trying
to eliminate noise altogether.
Mandelbrot's duties at IBM allowed him to peruse the literature for interesting
side-tracks
in mathematics and the sciences, and provided him with the computer tools to explore
them fully. But his knack for discerning pattern in chaos came from his own
mathematical
ability, which he applied to many other fields.
For instance, Mandlebrot was able to demonstrate that cotton prices in New York
followed the same sort of distribution as the data errors, and exhibited the same
self-similarity when scaled.
Mandelbrot also turned his attention to data on river flooding. He reviewed literally
millennia of data on the height of the Nile and invented terms for two effects he saw in
operation. The first effect he called the Noah effect, and it means a jump discontinuity.
The second effect he called the Joseph effect, and it means persistence or consistency.
He
then analyzed stock market data and found that it followed the same rules.
In "How Long Is the Coast of Britain?" Mandelbrot argued that as the length of a
coastline approaches infinity as the length of the unit of measure approaches zero. The
logic is similar to that of the Koch Triadic Island. He had read an obscure paper from the
1920's that touched on chaos-related topics such as numerical weather prediction and
flow dynamics, topics that simply were not solvable without computers.
In examining coastlines and the Koch curve and other paradoxical shapes from the
literature, Mandelbrot began looking for a way to describe them. Inevitably he found
that
many of these shapes shared a common property, that of fractional dimensionality. He
coined the term fractals to describe a shape with fractional dimensionality. Self-similarity
as scale is changed is another property of fractals.
Mandelbrot made forays into many fields. He suggested that the veins, arteries and
lungs
have a fractal structure, contrary to the then-prevailing medical belief that they were
exponential in nature. It has since been determined that many organs in the body have
a
fractal structure.
Mandelbrot recognized that the self-similarity of fractals lent itself well to the
recursive
methods of a computer program. He began playing with Julia Sets, recursive algorithms
that were developed during World War I and abandoned for lack of the tools to
compute
and visualize them. Recognizing the fractal nature of Julia sets, Mandelbrot attempted
to
catalog them by varying a different parameter and graphing the convergence in the
complex plane. The result was the Mandelbrot set, discussed above.
Mandelbrot's fractal geometry also proved useful to physicists, chemists,
metallurgists,
polymer chemists, seismologists, and probability theorists.
Barnsley theorized that the DNA of the fern does not carry the entire shape of the
fern,
but rather an equation for generating the fractal fern shape out of randomness. This
idea
explains how life forms as complex as human beings can form: many of the internal
structures are fractal in nature and thus can be coded in a few simple rules.
Barnsley also created an algorithm for exploring and coding the fractal nature of
images
stored in the computer. His algorithm saves space over some other computer formats,
and
interestingly enough, it eliminates much of the resolution error normally seen when
enlarging an image.
4. Conclusion
Fractals are objects with fractional dimension. They show self-symmetry as the scale
is
changed. Fractals are an important tool in chaos theory.
Computers have made new investigations into non-linear systems and chaos theory
so easy
that even high school students can investigate and understand fractal geometry.
Fractals and chaos theory are now being used in many fields of Applied Science.
They
promise to revolutionize man's ability to understand the world around him.
Glieck, James. (1988). Chaos. New York: Penguin.
Iseke-Barnes, Judith M. Enacting a chaos theory curriculum through computer
interactions. The Journal of Computers in Mathematics and Science Teaching v. 16 no1
('97) p. 61-89
Layer, Paul W., et al. An Archean geomagnetic reversal in the Kaap Valley pluton,
South
Africa. Science v. 273 (Aug. 16 '96) p. 943-6
Mandelbrot, Benoit B. "How Long Is the Coast of Britain?" The World Treasury of
Physics, Astronomy and Mathematics, ed. Timothy Ferris. (1991) Boston: Little,
Brown.
May, Mike. Fractal image compression. American Scientist v. 84 (Sept./Oct. '96) p.
440
Mullin, Tom, ed. (1995). The Nature of Chaos. New York: Oxford University Press.
Rainville, Earl D. and Phillip E. Bedient. (1981). Elementary Differential Equations Sixth
Edition. New York: MacMillan.
50 REM LOGISTIC EQUATION
51 REM Xnext=RX(1-X)
52 REM L. Ellis 5/7/97 53 REM This program demonstrates bifurcation and period doubling
54 REM in a population as a feedback parameter is varied.
55 REM VALUE RESULT
56 REM .9964 extinction
57 REM 2.7 stable population
58 REM 3.5 period 2 oscillation
59 REM 3.52 period 4
60 REM 3.55 period 8
61 REM 3.565 period 16
62 REM 3.569 period 32
63 REM 3.57 chaotic region
64 REM 3.83 period 3
65 REM 3.842 period 6
66 REM 4 chaotic region
100 INPUT"r";R
110 X=.1
115 FOR I=1 TO 1000
120 X=X*R*(1-X)
130 PRINT INT(10000*X)/10000;" ";
140 NEXT I