The Butterfly Effect – When Science meets the Pop Culture

No, I haven’t seen the 2004 Hollywood movie The Butterfly Effect that “Change one thing, Change everything”.  I am talking about the serious mathematics and science of Chaos Theory and Fractals that have caught so much public attention and imagination for the last quarter century.  Have you ever seen or owned posters or T-shirts with graphics like these “fractal arts” below?

Did you know that these beautiful pictures represent what is known in science and math world the Mandelbrot Set which was introduced in 1975 by Benoir Mandelbrot, a highly regarded mathematician using an innocent looking quadratic polynomial z_n+1 = z_n² + c in complex plane?

Fractal, as Mandelbrot called it in his 1975 book Les objets fractals, forme, hasard et dimension (an English translation Fractals: Form, Chance and Dimension was published in 1977) [Wikipedia] is often referred to as the geometry of chaos theory.   It aptly provides, among other contributions, incredible visualization of some intricate structures found in nature that contain tiers of miniature sub-structures that look exactly the same as the whole.  But what is chaos theory anyway?
 

It all began when Edward Norton Lorenz (1917-2008), a MIT professor in mathematics and meteorology, published in 1963 a very important paper Deterministic Nonperiodic Flow in the Journal of the Atmospheric Sciences.  It is not clear however if the paper had attracted much attention outside his research area at the time.   Almost a decade later in 1972, Lorenz gave a presentation before the American Academy for the Advancement of Science.  The title of the talk was "Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”  Being a grounded scientist and mathematician, according to the legend, the choice of the butterfly metaphor was made by the session convenor Philip Merilees and Lorenz originally was using sea gull.   

The vivid contrasting image of a beautiful but fragile butterfly penetrated and stuck to people’s mind and the reference of “butterfly effect” began to be widely used and abused ever since.  Can you imagine what would happen if it was called “sea gull effect”?  Although one had never ruled out the possibility of what Lorenz had discovered accidentally, it was Lorenz’s who demonstrated clearly and called people’s attention to the fact that realizations or trajectories of some nonlinear systems could be dramatically different given ever so slight a difference in the system’s earlier state - such as if an unknown butterfly in thousands of miles away flapped its wing, or not. 

It is not clear how it happened but the literary reference of the butterfly effect often suggests a “scientific theory” that a slight different action or choice can cause a huge impact down the road all by itself. This cannot be further from the truth and is in fact the opposite of what Lorenz’s discovery says.  Ignoring the technical reasons of such a erroneous statement, the significance and the implication of Lorenz’s work is that there is an inherent limitation (like the famous Heisenberg’s Uncertainty Princicplein Quantum Mechanics) in predicting accurately the future behavior of a complex (nonlinear) system, in this case, the weather system – arguably the most complex yet observable natural system to each of us, given the possible high sensitivity to some of the variables.  This is a perfect demonstration of the traps and hypes that sciencentific and mathematical discovery can run into when people got carried away with wild imaginations and misinterpretations.>/p>

The story did not stop there. Few years later, two Mathematics professors, James Yorke  and Tien-Yien Li (李天岩) published a paper in 1975 on the American Mathematical Monthly a paper entitled Period Three Implies Chaos that marked the first time the term chaos was used to refer to the behavior of this type of systems.  The name Chaos Theory became a household vocabulary when James Gleick published his bestselling book Chaos: Making a New Science in 1987.  The book made it possible that a reader can feel and see the discovery without knowing anything about science and mathematics.
 Merriam-Webster dictionary tells us the familiar word chaos came from Greek and means:

1 obsolete : chasm, abyss
2 a) often capitalized : a state of things in which chance is supreme; especially : the confused unorganized state of primordial matter before the creation of distinct forms — compare cosmos b) : the inherent unpredictability in the behavior of a complex natural system (as the atmosphere, boiling water, or the beating heart)
3 a) : a state of utter confusion  b) : a confused mass or mixture.

How much attention would it have drawn if Gleick chose a book title like Order and Properties of Nonlinear Dynamic Systems – a new development in science?  Although it is a much more accurate description of what Chaos Theory really is, how many copies do you think it would sell?  To give you an idea of how visible chaos theory is to the society, some people apparently felt it is necessary to look into its implication on religion, see e.g. the 1997 article Theological Reflections on Chaos Theory by John Jefferson Davis.  Indeed the irony is that the real driving force of all these explorations by the mortal scientists and mathematicians is to discover orders and to make sense of the seemingly chaotic world. 

I have not studied Chaos Theory previously other than few brief encounters with others’ research work on self-similar traffic and “long tail” studies and analysis in data networks in early 90’s.  I do know enough about nonlinear dynamic systems to be dangerous though.  Recently, I got a chance to view the 24 thirty-minutes lectures series on Chaos recorded in 2008 by Steven Strogatz, a Professor of Applied Mathematics and of Theoretical and Applied Mechanics at Cornell University.  The November 6, 2008 Chronicle Online, the daily newspaper of Cornell, had the following to say: “… Strogatz designed a course targeted to a generally well-educated audience -- light on the math, but including a wide variety of angles he hadn't explored before. That meant brushing up on the chaotic paintings of Jackson Pollock, learning about ancient Babylonian culture, and pondering the text of the Declaration of Independence.”  Wow! Isn’t that exciting? A theory that cut across physical and life sciences, engineering, arts and culture!  Let me tell you what I have learned and what I thought after watching these 12 hours of videos on my laptop with great interest.

The good: I did learn a great deal about Chaos Theory from the lectures and began to get a sense of what it really is.  I also learned a few rather interesting scientific discoveries and speculations through the applications and examples Strogatz uses in the lectures.  One such an example is the Kleiber’s Law in lecture 17 “Fractals inside us”.  Biologist Max Kleiber claimed in 1932 that for the vast majority, an animal's metabolic rate is proportional to the ¾ power of the animal's mass which suggests some efficiency of scale (i.e. more “energy” per pound is needed to sustain the life of a lighter animal).  Or, algebraically, the law can be written as (presumably in some averages) metabolic rate ~ (body mass)^3/4 , an example of the power law.  Waht is amazing is that this ¾ power law appears to hold true over 27 orders of magnitude of body mass from enzymes in respiratory complex to mammals!   The open question is why do most animals “obey” such a law despite their distinctly different mass and metabolic rate? 

Chaos theory and fractals may offer an answer to it, according to Srogatz.  Noting the networks of the pulmonary arteries and airways of the lung are branching networks architecturally, Strogatz argues why evolution or Natural selection might favor “fractal networks” which 1)is an extremely efficient way to distribute and communicate in a 3-D space using a branching network in 1-D tubes, 2)has enormous surface areas, 3)is easy to “program” as it involves a recursion of branching or splitting.  He cited the 1997 work by the theoretical physicist and biologists Geoffrey West, Brian Enquist, and James Brown in which they proposed a mathematical model and showed under certain simplifying assumptions that fractal network is the optimum branching network solution.  Further they showed since the terminating capillaries are the same in size for all mammals, the ¾ power law rises out of their mathematical models! Chaos theorists are excited about such a result as it suggests that chaos theory may hold the key to explain some mysterious properties in nature.  Is such a belief correct? Are we getting close to be able to understand something fundamental and common about life forms?  Many scientists remain (legitimately) unconvinced about the validity and applicability of those claims and models.   For one, wouldn’t it contradict the observations that Darwin’s natural selection that does not suggest or require optimality as mentioned in my last blog?

Another interesting example is the Feigenbaum Numbers mentioned in Lecture 23.  Anytime, if you find a rare universal constant that pops up in many seemingly unrelated systems, you know you are onto something.  Mitchell Feigenbaum did just that.  He was awarded the prestigious Wolf Prize in Physics in 1986 for his discovery of the Feigenbaum Constants, that have been observed in many natural systems and phenomena possessing a period-doubling bifurcating process - a fancy way of saying “splitting into two”.   The first constant is 4.6692…, the limiting ratio of each bifurcation interval to the next.  The second one is 2.5029…, the ratio between the width of a branch and the width of one of its next sub-branches.  Next time, if you run into these two numbers but not sure why, you might be staring at one such a mysteries form!

The bad:
The part of lecture that I found most distrubing and does not help the credibility of the story is the second half of the Lecture 18, the Fractal Art, albeit the discussion is very enjoyable and entertaining.  Professor Strogatz, apparently prodded by the Teaching Company who produces and sells the DVD, used

Jackson Pollock’s abstract painting as an example of an application of chaos theory.  In case you did not know who Jackson Pollockis, he is one of the most influential and revered American painters of the 20^th century, having left deep marks on Abstract Expressionism.  Just because the appearance of Jackson’s drip and splatter paintings appears “chaotic” to the naïve eyes of viewers, you might wonder, like I did, how is that having anything to do with the Chaos Theory?  How could Strogatz possibly pose the silly and ridiculous question on an art that Are these drip paintings random or fractals?!  Strogatz’s story is based on the controversial fractal analysis by the physicist and fractal enthusiast Richard Taylor who claimed to have found “pattern” or consistent fractal dimension (think of it as an embedded signature of a figure if you don’t know what that is) of Pollock’s paintings in progression by year.  The purported applications of fractal analysis thus include authentication and identification of “chaotic” paintings of Pollock’s.  Even if the claim turns out to be valid, I hope Chaos Theorists have higher goals and would not waste their time like this!  Of course, it could be about money.  Jackson Pollock’s drip painting entitled “No. 5, 1948” (see the photos to the right) was rumored to have been sold in 2006 for $140 million dollars that would make it the most expensive painting on record in history so far.  By the way, Jackson Pollock’s work preceded the Chaos Theory.  He created his original drip painting techniques in 1940s.

In addition to the above What!? There are some “so what’s” here and there such as the story of the unexpected horizontal oscillations and repair of the newly built London Millenium Bridge in 2000.  All in all, to be fair, Strogatz is an excellent speaker and hard to imagine one can deliver the stories much better, no wonder he had won so many teaching awards in top universities.  Further, chaos theory and fractals has made exciting new advance in the study of nonlinear dynamic systems and I should not be overly critical just because it is popular or misrepresented by some.  The research did and will continue to offer a new way to examine the behaviors of some complex systems and help identifying common characteristics of things found across many disciplines from physics, chemistry to biology and ecology.  The fact that validated governing models are based on same type of equations do tell us a lot about the systems; it clearly suggests that the similarity more likely are derived from some common features rather than the chance or coincidence.  After all, mathematical model is an invaluable and indispensible tool in science and engineering and good models do reflects our understanding of the subject it models after.  But based on what I have learned from the lectures, I do not think Strogatz’s claim is valid that Chaos Theory is the third revolution in science in 20^th century, after Relativity Theory and Quantum Mechanics.  The claim is at best premature and at worst, a hyperbole. 

In sum, it is a fact that we often found ourselves limited by our own experiences and language and that we have to borrow existing words, notions and analogies to describe something new.  We often forgot models are limited and necessarily simplistic.  Validated theory such as Newtonian mechanics may fail when operated outside the regions they were designed for.  The misconception and misuse of the words and notion of chaos theory is a perfect example to remind us what can happen when a difficult and intricate subject could be overwhelmed in sensations and excitement. Strogatz did insert carefully words like seemingly or appearance of in front of the word Chaos at places.  Unfortunately, these critical qualifying words tend to get dropped and forgotten and messages got severely distorted as they propagate in time and space.

By making it so easy to understand, haven’t we introduced and suffered from the very butterfly effect: that ever so little simplification of the complex truth could be amplified exponentially in time till we are lost in the intricate world of fractals?  It is critical to be intellectually curious but it is equally important that we stay truthful somehow in every step along the way. How can we guard against the danger of amplifying the little butterfly’s flutter when science meets the pop culture just because something sounds better and cool?