2月 28, 2007 @ 12:00 am
When a Butterfly Flaps its Wings …
When a Butterfly Flaps its Wings …
—- An adventure into the world of Chaos,
Fractals, and Art
Prologue: Jurassic Park
—- A monologue with dinosaur-fan
boys.
In the far distance, a Tyrannosaurus
Rex roars and many Apatosaurs trumpet … Chaos reigns Isla Nublar
and Nature finally finds its way. Ian Malcolm has been right all
along: the Jurassic Park is a nonlinear dynamic system, which is
intrinsically unpredictable and unstable, and its safety is totally
at the mercy of a handful of computers. A little greediness by the
computer programmer Nedry (Nerdy?) has sent the whole park to its
doom. Now the mighty and efficient flesh-crunching machines are set
free, and in their eyes, their creators are but bunches of proteins
and nutrients …
Did everybody see Steven Spielberg’s mega-hit movie Jurassic
Park? You didn’t? Well, you should. Oh, you did? Good boy. Did you
like the dinosaurs? Good, Steven is awfully good at making those
huge monsters and giving people a few delightful scares, and he
always makes big bucks out of it.
Do you remember Ian Malcolm in the movie? No? Oh, it is that
rock-star-like guy with black shirt, black trousers, black socks,
black sneakers, black sun-glasses, black hairs … black
everything. Oh, not an African American, he’s white.
Remember him now? Good. What did he keep on murmuring about in
the helicopter when they were flying to Isla Nublar, the small
island off the coast of Costa Rica, where the Jurassic Park was
located? You remember? Good. “Chaos,” yes! “Nonlinear dynamics,”
yes!! “Strange attractors,” yes!!! “Butterfly Effect,” yes!!!! Good
memory, I wish I’d been that good. Don’t worry if you could not
understand any of those words. They aren’t supposed to be
understood anyway.
Fig. 1. Jurassic Park.
Can anybody tell Malcolm’s intention when he held Ellie’s hand
and put a few drops of water on her wrist, a little bit after their
doomed Jurassic tour had started? Anybody? What did you say? He was
flirting with her?? You nasty little boy! Stop giggling, you
good-for-nothing bunch. He was trying to explain to her a complex
system’s sensitive dependence on initial conditions! What’s a
complex system? Hmmmm … a girl is a complex system. Got it?!
Good, you’ll know as you get older.
Want to know something about the Butterfly Effect? Oh no, not
Madam Butterfly. I know your parents brought you to the opera. This
butterfly is not Japanese. This time it happens to be made in
China. It goes like this: a butterfly flaps its wings in Beijing,
and the stock market on Wall Street flounders. Don’t think so? OK,
let me modify it: a butterfly flaps its missiles, oops, wings, in
Beijing, and the stock markets in Taiwan and Hong Kong flounder.
OK? That’s more like it. You know, Nedry to Jurassic Park is like
the Butterfly to Taiwan and Hong Kong.
Iteration One: Chaos
Fig. 2. A nonlinear dynamic system. (Cartoon
(c) 1988 by Jacques Boivin).
Who is Ian Malcolm then? Oh, he’s one of the
leading new-wave Chaos mathematicians who emerged from obscurity
and had quickly risen into stardom in the ’80s, riding the tide of
the explosive growth of computer techniques. They, unlike
traditional mathematicians who despise any connection of their work
to the secular world, are openly interested in “how the real world
works.” Despite what the almighty and omnipotent Albert Einstein
said about mathematics, “As far as the laws of mathematics refer to
reality, they are not certain; and as far as they are certain, they
do not refer to reality.” These Ian Malcolms have managed to do
well. See, they are even portrayed in one of Hollywood’s big
movies, a treat few physicists, even the very Albert Einstein,
have. Maybe they are quite certain that their mathematics is
uncertain enough to be relevant to the real world.
These new-wave mathematicians work almost exclusively with
nonlinear equations, either just for the fun of it or using them as
toy models for real-world complex phenomena like the stock market,
the weather, and … the Jurassic Park. They also heavily use
computers — a practice REAL mathematicians frown on — and
computer graphics to generate dizzying and often intriguing
colorful pictures which some REAL artists don’t regard as art.
Yes, their mathematics may be uncertain; their usage and
sometimes reliance on computers may be frowned upon; the computer
images generated from their work may not regarded as art. But, they
introduced a new way to look at how the real world works and how
Mother Nature is organized and structured. They induced a paradigm
shift. From their work, the word “Chaos” has gained new meanings
and lost some of its mysteries.
Fig. 3. Create chaos from order using simple rules. (Cartoon
(c) 1988 by Jacques Boivin).
Before the new theories of Chaos were developed,
“chaos” had not been an auspicious word, and its ominousness was
deeply rooted in many religions and cultures. Even today, when
people speak this word, most of the time they mean disorder,
confusion, something incomprehensible, something indescribable,
something out of control, something bad, something that should be
avoided at any cost. Although Chaos is also referred as the state
of the world before Genesis, and that state may not be necessarily
bad, it is still for sure that nobody would ever want to live
there. “Chaos” had been used like the trash bin on a MacIntosh
computer — one can dispose into it anything that doesn’t fit in
with one’s frame of mind and thus cannot be explained; then a purge
takes care of it all.
Yet Chaos remains. You know it when you are caught by an
unforecast rain storm, on a country highway, in a pitch-black
night; you know it when the stock market dips suddenly and some
broke stock brokers jump over the edge of a skyscraper; you know it
when three years ago your skills were at the top of your profession
and now you find them to be out of date and that you are out of
job. How much human beings want to know the future and control
their own fate! What should one do?
Some people resort to the almighty God; some people go to see
crystal balls; some people hide their heads in the sand. But
scientists, as always, call upon their intelligence and scientific
methods. Scientists may oversimplify; they may just get a small
glimpse of a tremendous natural force; they may only approximate;
and along the path of search for truth and understanding, they may
fail, but at least they have tried. Had God really existed, he must
have favored scientists the most, since they bother him the least.
God put intelligence in every human being he created and he wants
people to use it by themselves when facing fate, so that he can
have more time to drink. What do you think can make God angrier
than interrupting his feast when he gave you the very intelligence
not to bother him?
Now let’s first think like a scientist and ask a few questions.
Is Chaos really orderless? Could there be a hidden order in Chaos
that regular methods fail to unveil? If we can not tell where a
system goes next, can we tell where it might go? Does Chaos have to
be in a complex system? Could it be that when we see only disorder
and unpredictability, we might be too localized and narrow-sighted,
so when we go up one or few levels higher, we might be able to see
order and the whole picture?
Chaos theory’s answers: No, Yes, Yes, No, Yes.
Fig. 4. Chaos — Orderly disorder.
Think about a hurricane. When you are aboard a ship amid a
hurricane, all you feel is Chaos, even disaster. That’s because you
are too much inside it. If you happen to be in a weather satellite
above the hurricane, then what you see from there is an orderly and
majestic swirl moving along a certain path. Different view points
sometimes do give qualitatively different understandings.
Chaos theories treat real world complex systems as nonlinear
dynamic systems. A nonlinear dynamic system (see Fig. 2) is a
system which evolves through time according to certain rules and
which, when disturbed by external forces, doesn’t respond
proportionally with the degree of perturbation. In contrast, a
linear dynamic system responds to perturbation linearly or
proportionally. Take some real-life examples. A new car is a linear
dynamic system; when you depress the accelerator pedal harder, the
car runs faster; when you don’t depress the pedal, well, it stops.
A well-worn grandpa car, on the contrary, is a good old example of
a nonlinear dynamic system; when you depress the pedal gently, it
may jump like an excited horse; when you depress the pedal harder,
it may hardly move; sometimes when you do nothing or very little
things, it may tremble or simply disintegrate.
Chaos theorists use nonlinear differential equations to model
the rules of nonlinear dynamic systems. Sometimes there can be just
one simple rule; sometimes there are many complex rules. Simple
rules do not automatically give rise to simple system behaviors
(see Fig. 4). What a nonlinear dynamic system does is to follow the
same set of rule(s) over and over again through time and space. The
present state of the system is determined by the last state and in
turn decides the next state, and so on. One of these spiral cycles
is called an iteration.
If one simply follows one iteration after another too closely,
he may get lost easily, just like being trapped in a hurricane. It
requires a leap into a global view to see the whole picture, which
is the result of all the iterations. Chaos theorists don’t care
about one particular iteration and where it will lead to; they care
about the global picture — the phase portrait. Fig. 4 is an
example of phase portraits. In Fig. 4, if you plunge too much into
the details, you may be like a mountain traveler trapped in a
morning fog and don’t know where you are and where you are going.
Jump out and jump high; you will see an unspeakable order before
your eyes!
Iteration Two: Fractals
When wandering at the vegetable department of a supermarket, did
you ever pay attention to a fresh and clean cauliflower and get
intrigued by it? If not, take a look at Fig. 5 now, or simply buy a
fresh and clean cauliflower, then zoom in and out using your eyes
at its surface structure. Despite the elegant spiral arrangement of
the small buds, what more can you see? The whole cauliflower
consists of smaller cauliflowers, and the smaller cauliflowers in
turn consist of even smaller cauliflowers, so on and on …! If, as
you look closer, your size shrinks according to the size of the
cauliflower buds you focus on, can you tell whether you are looking
at the whole cauliflower? A small bud of it? A smaller bud on a
small bud? … Most certainly you cannot, because you are looking
at a self-similar structure, a scaling-invariant object, … a
fractal.
For definition, a fractal object is self-similar in that
subsections of the object are similar in some sense to the whole
object. No matter how small a subdivision is taken, the subsequent
subsection contains no less detail than the whole.
Fig. 5. Cauliflower — a living
fractal.
Fractals are ubiquitous in nature. Look up, you can see the
forever changing shapes of fractal clouds. If you have rich
imagination beyond your eye sight, your mind will see the fractal
distribution of matter in the universe. Look around, you see
fractal trees, fractal mountains, fractal coast lines. Look inside
your body, you see the fractal architecture of arteries and veins,
fractal bronchi, fractal nerve system … all perfect examples of
Nature’s efficient and optimal use of space and materials. Look at
the hierarchical structures of a bureaucratic system and a society;
the word “fractal” never fails to pop up.
“Fractal” is a powerful word, a powerful concept, an unifying
theme of a vast and diversified set of shapes, structures, and
organizations adopted by Nature.
Fig. 4. Mandelbrot set — The Icon of
Fractals.
It is Benoit B. Mandelbrot, a mathematician at IBM, who first
put all the dangling and scattering pieces of fractals together in
the late ’70s and ’80s, coined the very word “fractal,” and founded
a new geometry — Fractal Geometry, the Geometry of Nature.
Rightfully, he has earned his position in the Hall of Scientific
Giants. The first fractal image from one of his toy mathematical
models, the Mandelbrot Set, has become the icon of fractals (see
Fig. 4).
As a typical Western scientist, Mandelbrot didn’t stop at the
metaphysical level of simply formulating concepts, as many ancient
Oriental sages had often done. He had developed and revitalized
many rigorous mathematical methods and had heavily relied on
computers and computer graphical techniques. Because of the latter,
he broke ranks with conventional mathematicians, and was quite
often at odds with REAL mathematicians.
One mathematical method he employed is iteration, which is one
of the connections between Chaos theories and Fractal theories.
Mandelbrot views a fractal object as a set of points in a space,
and these points are connected and bonded by certain relations or
rules. If the rules are known and one wants to unveil the fractal
object, one can simply pick up a random point in the space, apply
those rules to it, and observe its trajectory. This is analogous to
discovering the phase portrait, or global picture, of a chaotic
system by repeating the same set of equations again and again. If
the rules indeed correspond to a fractal object, after many
iterations, the trajectory of the test point converges to a finite
object and the search is ended. If one needs a real life analog,
think about the digging of a dinosaur skeleton fossil.
Fig. 5. Barnsley’s IFS fern — a successful
example of modeling Nature with mathematics.
Mathematician Michael F. Barnsley is just such a paleontologist
of fractal “fossils.” He developed the so-called Iterated Function
Systems (IFS) to model and generate both real-life and abstract
fractal images. Shown in Fig. 5 is his benchmark fractal fern. Only
28 numbers are needed to contain the rules of generating this fern;
this means succinct storage of information. Barnsley realized this
potential and further developed a system to compress computer
images — a good example of the transition from pure ideas to
practical applications.
Fig. 6. Fractaiji — a hybrid of Fractals and Taiji.
I have just said a few bad words about ancient Oriental sages.
Now let me make some compensation by hybriding Lao-tzu’s Taiji with
Mandelbrot’s fractals. The result is Fig. 6, a fractaiji.
Iteration Three: Arts
Enough has been said, and it is time for some pictures. In the
following, I will just make three quotes as regards whether fractal
images, and computer graphic arts in general, are really art. The
first quote is a con and the remaining two are pros. See the
pictures and form your own opinions. Enjoy.
“Fractal images are incomplete art, of course, since
they are abstract and not culturally rooted.” — P.W. Atkins,
1990
“The distinction between art and science is contrived.
Both are processes of discovery and both use a variety of tools and
techniques.” — Computer Graphics World, 1989
“Nature is relationships in space. Geometry defines
relationships in space. Art creates relationships in space.” — M.
Boles and R. Newman, Universal Patterns, 1990
Reed
Window
Complex
Dragon
Spike
Epilogue: Fleas
So Fractalists observe,
Great fleas have little fleas
upon their backs to bite ‘em,
And little fleas have lesser fleas,
and so on ad infinitum.
And the great fleas themselves, in turn,
have greater fleas to go on;
While these again have greater still,
and greater still, and so on.
–
Jonathan Swift
A maverick who sniffs at all genes and memes.
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