Friday, November 23, 2007

Patterns in Nature: Making Connections in the Classroom

For the past six weeks, I have been teaching a unit to my gifted students called "Finding the Order in Chaos". In teaching chaos theory, I rely heavily on nature because fractals represent applied theory, and they are ubiquitous in natural settings. I get excited when one of my junior high students experiences an "aha!" moment and begins to understand the concept behind chaos theory.

In a nutshell, chaos theory states that natural systems are dynamic, and that small, seemingly insignificant random events reiterated over time produce significant outcomes. The most popular example is illustrated in the Butterfly Effect, which is described by saying that the flapping of a butterfly's wings can begin a pattern of atmospheric disturbances that could result in a hurricane in China six months later.
It is indeed something to celebrate when one of my junior high students begins to understand that even minuscule actions can have significant outcomes within dynamic systems like those on our planet. It's even better when he/she can discover and document examples on his/her own. But my joy isn't dependent upon the success of my students, because today I discovered and documented several examples of chaos theory in nature, and I share them joyfully with you.

This first example is probably my favorite because it is a dual example. Check out the venation in the leaf. Although the forces of nature (light, soil nutrients, temperature, cloud cover, winds, etc. ) that served as stimuli for the eventual direction and structure of the very first vascular tissue formed in this leaf were entirely random and unpredictable, an underlying order emerged as time progressed and we use that orderliness to identify this leaf as a representative of a particular species.

Now check out the frost that formed in the early hours this morning. What random stimuli combined to cause the initial water molecule to crystalize? Certainly, it was a synergy developed from a nearly infinite number of variables (temperature, humidity, winds, dew point, topography, soil type, etc., etc.). This original permutation iterated over time produces an orderly and intricate pattern of ice crystals on the surface of the leaf.
I was immediately drawn to the surface patterns in this photograph. Notice the random distortion of the water's surface caused by the presence of immeasurable stimuli; including underlying rocks, invertebrates, algae, and other countless biotic and abiotic factors. But enlarge the photo and pick out all the orderly patterns of spirals and waves that have emerged out of this dynamic biological system. There is an underlying order present in applied chaos theory.
My third example of applied chaos theory is illustrated to the left. Branching is a recurring pattern in Nature that is manifest in everything from trees, to watersheds, to circulation of blood through the body. Of course there are many other examples. But, again, see how the nearly infinite numbers of factors that determined the direction and structure of the leaf earlier in this post also have a say in determining the direction, structure, and shape of the tree itself..... or the direction, structure and shape of the river system........or the circulatory system in your body. You get the idea. By the way, there is a less than randomly marked Fox Sparrow in this picture!
It is here that chaos theory merges in my mind with the theory of natural selection to bring order to the natural world. What fascinates me about the two theories is that both begin with seemingly insignificant random events, occurring over time, and resulting in somewhat predictable outcomes. In nature, predictable outcomes emerge from initial random permutations which reiterate over time. Thus, the theories play out in the occurences of natural fractals, convergence, adaptive radiation, just to mention a few.
All natural processes are capable of being explained through scientific and mathematical processes. However, we must foster an atmosphere where science is alive, well, and respected in our society. The results of math and science inquiries so far seem to support a universe with rich mosaic of order emerging out of what we perceive as chaos.
"The important thing is not to stop questioning. Curiosity has its own reason for existing. One cannot help but be in awe when he contemplates the mysteries of eternity, of life, of the marvelous structure of reality. It is enough if one tries merely to comprehend a little of this mystery every day."
Albert Einstein


marvin d said...

A book on Chaos which, back in the late 80s, motivated me to do a lot of computer simulations of chaotic systems is James Gleick's CHAOS. It is accessible to almost anyone and was a joy to read.

Greg said...

Thanks, Marv,
I hope you'll be honest with me and let me know if I've misunderstood, butchered, or taken too much liberty with chaos theory! You know us bloggers! :)