)
)
fall 2001 course description
fall 2001 syllabus
Instructions for Mathplay
study tips
course evaluation form
To understand mathematical chaos, you first need the idea of a dynamical system. That is just a mathematical situation that changes with time. For example, count the number of bats in a particular bat cave. As bats reproduce and die, their population size changes. Or imagine an object falling through the air. The distance to the ground changes as time passes. These are two examples of dynamical systems.
Mathematicians have studied falling objects at least since Galileo (1564 - 1642), and biologists have more recently used mathematics to model populations. Up until 40 years ago, the equations used in these studies were thought to behave in prescribed, easily understood ways. In the 1960's, to everyone's surprise and dismay, some simple equations were found to exhibit behaviors that did not fit the old paradigms. When viewed in the traditional ways, the behaviors seemed to lack pattern and order, and in everyday language, this is what chaos means. But if the mathematical systems are viewed in new ways, a "hidden order" emerges. These behaviors, seemingly random on the surface but possessing hidden order, comprise mathematical chaos.
Fractals are geometric objects with interesting and complex structure. Like mathematical chaos, they were disturbing when they were first discovered. People referred to them as "monsters." With time, their properties became familiar and better understood, and now they are considered beautiful and exciting.
Here are a few pictures of fractals. The most evident characteristic is their "fractured" appearance. Another fascinating property is their lack of scale. This means that if you look at a small piece of the image, it is similar to the whole. The image is repeated over and over at smaller and smaller levels. This property is called self-similarity. A third characteristic is their fractional dimension. While lines and arcs are one-dimensional, and triangles and circles are two-dimensional, the Sierpinski triangle is 1.585...-dimensional!
| The Sierpinski Triangle: | The Koch Curve: | |
|
|
|
| The Mandelbrot Set - click to enlarge | An Associated Julia Set - click to enlarge | |
|
|
Euclidean geometry, named for the Greek Euclid (325 - 265 BC) who wrote the first geometry textbook, studies simple figures such as circles, triangles, cones, and spheres. These shapes are ideal abstractions of what we see in nature: the sun and full moon seem to be circles, mountains are conical, and some fruits are spherical. But nature comes in other forms, and it is interesting to contemplate why we were so long in abstracting those. It was Benoit Mandelbrot (1924 - ) who called our attention to the fact that
"Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does lightning travel in a straight line."*
 |
The lungs and blood vessels show a fractal structure, with larger passages branching into smaller passages, which branch again into yet smaller ones. | |
 |
Heart and brain rhythms are dynamical systems. Interestingly, a healthy heart beats in a periodic (nonchaotic) pattern, and healthy brain waves are chaotic. Conversely, the dangerous fibrillation of a heart in trauma shows chaotic patterns, and the brainwaves seen during epileptic seizures are periodic. |
fall 2001 course description
fall 2001 syllabus
Instructions for Mathplay
study tips
course evaluation form
back to Margie's home page