Sometimes a Great Notion: Turbulence and Ken Kesey

I recently began re-reading Sometimes a Great Notion, Ken Kesey's monumental second novel published in 1964, and one of my favorite books. Even though I read it over 20 years ago, I still remember many of the most famous scenes. More remarkable, perhaps, is the fact that I vividly recall the haunting descriptions of the Pacific northwest so magically captured by Kesey.

When I read about the chaotic aspects of turbulence, especially as described by James Gleick in Chaos: Making a New Science, I am always reminded of Great Notion's opening passage:

Along the western slopes of the Oregon Coastal Range ... come look: the hysterical crashing of tributaries as they merge into the Wakonda Auga River. ... The first little washes flashing like thick rushing winds through sheep sorrel and clover, ghost fern and nettle, sheering, cutting ... forming branches. Then, through bearberry and salmonberry, blueberry and blackberry, the branches crashing into creek, into streams. Finally, in the foothills, through tamarack and sugar pine, shittim bark and silver spruce -- and the green and blue mosaic of Douglas fir -- the actual river falls 500 feet ... and look: opens out upon the fields."

Many others feel the same way as I do about Great Notion. In a 1997 survey, the Seattle Post-Intelligencer published a list of the 12 Essential Northwest Books. Sometimes a Great Notion was #1, after being named the top book by over 1/3rd of the participants.

According to Kesey (who died in 2001), "I think 'Sometimes a Great Notion' is the best thing I'll ever write...Writing it was much different from 'Cuckoo's Nest,' which often seemed like filling in the blanks. 'Notion,' to my mind, is a great piece of work. People sometimes ask me why I don't write something like that again and I reply that I simply can't. I can't keep all that in my head at once anymore. Why, on 'Notion,' I used to work 30 hours at a stretch -- you've got to have youth to do that."

I don't know how the book fares on re-reading, 20 years after my initial reading. Maybe one must also "have youth" to really embrace the novel's sprawling, non-linear, 627 page narrative. I do know, however, that I will always be mesmerized by the "the hysterical crashing of tributaries as they merge into the Wakonda Auga River..."

Extending the Chaos Game: Determinism and DNA

HIV Color Game

I recently came across some interesting variations and applications of the Chaos Game.

One version is located at the Boston U web site and is credited to Johanna Voolich and Robert L. Devaney. Their version is very unique twist on the Game - instead of the randomness that forms the basis of the Game, they have replaced the randomness with a strictly deterministic game. In their version, the player tries to get a random starting point into a specific target area of the Sierpinski triangle by choosing a series of half-way jumps to specific vertices. The "winner" of the gme is the one who reaches the target area in the fewest number of jumps. (Click here to play the game on-line.)

An even more fascinating Chaos Game comes from Dan Ashlock at Iowa State University, where he uses a 4-cornered Chaos Game to display the sequence of bases in DNA molecules. (The image at the top of this post is from a segment of HIV DNA.) In this Chaotic Game of Life, points are drawn half-way to vertices depending on the next base in the DNA strand. The pictures look remarkably similar to playing the Chaos game randomly! See Ashlock's website for a set of images from different organisms, as well as some Markov chain models that help explain and interpret the images.

The Chaos Game: 3rd Grade vs. University Faculty

I just finished two separate presentations of the Chaos Game exercise that I typically use to start the Honors class to students at Arcdia University. Most of the students were education majors - and most of them were elementary ed majors. I am always pleased at the reactions to the game - especially from students who believe, or claim to believe, that they are not good at mathematics. The Chaos Game is such a rich exercise, especially given the approach of Understanding vs. Prediction, that most get caught up in the inadequacies of prediction, and the frustration of not being able to get a real comprehension of how the patterns (and colors) arise.

But the best part of the exercise, for me, is the ability to run it in almost any setting - from a 3rd grade classroom to a university mathematics department. Each group will get something different from the exercise. Personally, I prefer the response of the 3rd-graders, who , unlike mathematics faculty, don't try to understand what is happening at a deep mathematical level. Instead, they react purely to the shapes and colors, and their inquisitiveness makes for a wonderful teaching moment.

Education majors, and especially elementary education majors, are also a terrific audience for the Game as we play it. I believe that most of them could see how the exercise, or something similar, can be transported to their own future classrooms!

Chaotic Mixing

Stretching Field: lines of large past (red) and future (blue) stretching. Click to enlarge

Jerry Gollub of Haverford College is understandably quite famous for his pioneering work in measuring the onset of turbulence. With his conceptually clean (yet technically difficult because of their precision) experiments, he and his colleagues and students have produced a wide range of experimental and theoretical results that demonstrate the role of chaotic dynamics in fluid dynamics.

Gollub is at it again, this time with colleague Paulo Arratia of U. Penn. Gollub and Arratia designed a clever experiment in which they were able to observe the mixing of two fluids in a regime known as "chaotic advection," which is distinctly different from turbulence. (See the review article Mixing, Chaotic Advection, and Turbulence by J.M. Ottino for a good description of these different fluid regimes.

As described in the Feb., 2006 issue of Physics Today (and soon to be published in Phys. Rev. Lett.), Gollub and Arratia were able to measure the stretching field of their fluid. This field is the "local distortion of an infinitesimally small fluid element." This field, in turn, can be used to calculate the Lyapunov exponent for the fluid under different mixing conditions. (The Lyapunov exponent is a well-established measure of the tendency for the phase trajectories of chaotic systems to move apart.) Remarkably, Gollub and Arratia found that they could model the amount of chemical product formed from their mixing reactants as a function of Lyapunov exponent only for a large range of mixing conditions. This result is important because it demonstrates yet again one of the hallmarks of chaotic systems - universality, which is Feigenbaum's contribution to chaos theory (and which the Gollub/Swinney rotating cylinder experiments helped establish as experimental fact.)

To view more ongoing Gollub projects (as well as interesting applets showing chaotic mixing), visit the Nonlinear Physics and Fluid Dynamics Lab of Haverford College.

Sudden Climate Change

Butterfly/Lorenz Attractor from the Chaos and Culture class website of Prof. Tom LoFaro, Gustavus Adolphus College

In the final article of the 3-part Inquirer Series on the Gulf Stream that appeared in December 2005, the connection with Chaos was at least implied. (See my earlier post on the surprising lack of chaos in the first two parts of the series.)

The article chronicles the discovery by Richard Alley in 1992 of extremely rapid temperature shifts of up to 15 degrees within a 30 year time span. Such a rapid temperature change today would lead to world catastrophe (regardless of whether the earth was cooling or warming).

Alley was investigating core ice samples in Greenland. Remarkably, based on the depth of the samples, he measured this rapid temperature shift as occurring approx. 12,000 years ago . The culprit? You guessed it - Ben Franklin's Gulf Stream.

The connection with chaos is obvious to anyone who understands the separate wings of Lorenz's butterfly. As the trajectory that describes the state of the weather moves around one wing, it will suddenly flip over to the other wing - a sudden reversal that "just happens."

As the Inquirer article points out, "until the 1950's, climate was viewed as essentially a stable system."

Lorenz, Alley, and many others have certainly shown otherwise.

Classroom Chaos Redux Final

Originally Posted by Sean Houlihan

My final project was a computer program that dealt with recreating the classroom chaos concerning the Logistic Growth map and producing new a various forms of analysis for it. I felt I was successful in producing my goal. However, some aspects are not as accurate as i would have like them to be. Hopefully, I will be able to work on it in the future in order to create a truly refined program. I would have like to put up a download of the program for everyone, however, I do not know how to do that. So if anyone could tell me how, it would be greatly appreciated.

Maybe, one-day, RAD will use my program in his class.

Please let me know what you (the class) thought of my project!

Reflections of Chaos and Fractals

Originally Posted by Jeremiah Noll

At the beginning of this course I was much more certain of things! I have learned quite a bit, but with this learning came new mysteries. Studying chaos has expanded my mind and taught me things I never would have dreamed. When you sit and think about the concept and the principles which founded chaos it just makes sense. Butterflies can start El Niño's and stock market is predictable to a degree. But sensitive dependence on initial conditions is not the only principle of chaos, there is also the fact that extremely simple rules can produce infinitely complex results. Who would have thought a couple lines of computer code could produce the Serpinsky Triangle or model the population growth of a small nation.

But the coolest things I have ever seen are fractals. Through this course I have developed a great understanding of the image, mapping, and product of fractal code. Simple functions and an elementary Cartesian plane can model infinity in the most accurate and beautiful way. I enjoyed fractals so much I decided to make them my semester project.

The value of this course is a tremendous amount of insight into how the world works and how things thought to be random can really be very simple. I feel as if I have been shown a piece of how God put this universe together and how he made it so complex in only six days. Like deciphering the human genome code chaos makes us more able to predict, heal, and understand the way things work and why events happen.

Chaos and Nature

Originally Posted by Joe Cosella

In my project, I described how chaos is involved with different natural disasters. The examples I looked at were turbulence within the Earth's crust that causes earthquakes, flooding patterns in Southern China, and hurricane statistics.

I also introduced a sandpile game which can be found here.

Chaos, Consciousness, & Dreaming

Originally Posted by Rachel Hensey and John Sehi

In a dream, Paul McCartney (1965) created the melody to the popular song, "Yesterday."  Likewise, Tse Wen (2003) was inspired in a dream in which he developed a breakthrough drug that greatly reduces the risks for people who suffer from peanut allergies. These two men are examples of scientists, musicians, athletes, mathematicians, writers, and artists who have reported accounts of moments of inspiration or breakthroughs during dreaming time.

This construction of nonlinear meaning that occurs during dreaming is an example of chaos theory at work in the brain. The brain is a complex, chaotic system, and small shifts in its input during any state can dramatically alter how it operates. Inputs of order and chaos cause a tension in the brain that is mandatory for proper growth. Without the input from order, the brain would dream too much and thus fall into irrationality. Without the input from chaos, the brain would no longer dream and thus would function like a robot/automaton, without any real creativity. The output (our thoughts, feelings, and behaviors) depends upon a proper blend of chaos and order. For additional information on chaos in the brain, click here.

Modeling the Arms Race

Nuclear Proliferation game box-top. Click to enlarge.Following up on Tom and Meredyth's presentation and post: the Saperstein article referenced dealt with modeling interactions among nuclear states. A recent article by William C. Potter, director of the Center for Nonproliferation Studies at the Monterey Institute of International Studies, entitled The Second Last Chance: American Power and Nuclear Nonproliferation points out the need for non-proliferation modeles and theorists to consider the effects of regimes and terrorist agents on our notions of non-proliferation.

Potter's article is an extensive review of the book Nuclear Terrorism: The Ultimate Preventable Catastrophe by Graham Allison (Harvard Kennedy School of Government). Because of its call for the U.S. to change its current theories and practices of approaching non-proliferation, the connection with Saperstein's model is an important one. I don't know whether Saperstein, or any think-tanker worrying about such modifications has the ear of the government. At the very least, the chaotic behavior of nuclear-arms possessing states as predicted by the Saperstein model will certainly be more prevalent as the highly non-linear interactions of terrorist agents and rogue states are factored into the model.

(Note: the image at the top of the post is from the card game Nuclear Proliferation by the FlyingBuffalo company. From the blurb on the box: "It's a sarcastic, humorous look at the futility of Atomic Warfare in the post-cold war 1990s.")