Vol. 7, No. 7: March 15, 2001
The Chaos Game: Stimulating Math Curiosity with Interactive Software
by Richard O'Malley,
Professor, Department of Mathematical Sciences,
University of Wisconsin-Milwaukee
In the early 1990s, I was asked to create a math seminar for incoming freshmen at the University of Wisconsin-Milwaukee. The seminar needed to be accessible to students with minimal math background and high levels of math anxiety. Additionally, the course should act as an introduction to mathematics as mathematicians view it: an area of research full of open questions. The course should also develop the students' critical thinking and writing skills. This is how I summarized my problem:
Purpose of the Freshmen Scholar seminar in Contemporary Applications of Mathematics:
Method of achieving these goals:
Fortunately, I had encountered this challenge earlier in a course for our honors program based on chaotic dynamics. I had used a freeware package called "Fractint." However, it was not designed as a teaching tool, and I felt it wasnít meeting the first three purposes outlined above.
I decided to design my own software and course material simultaneously. Thus, the freshmen seminar came along at the right time. With the objectives above, I worked with a graduate student from our department, B. Douglas Ward, to design a sequence of Pascal modules based on the Chaos Game, which I had seen demonstrated in the mid-1980s. In reviewing what I wanted as an accessible topic for freshmen with math anxiety, I realized the fractal (IFS) images that the game generated had the visual impact and complexity that I wanted. There is also a "surprise factor" that draws the students into the activity.
Here is how I presented the activity to my students, asking them first to try doing the task on paper (I suggest you try this yourself):
Rules for the Chaos Game
Proceed in this fashion, repeating shuffling and placing new dots until you have done 100 dots.
I really would like you to try this. However, if youíre like my students, then you will: a) be working hard at finding the midpoint, b) have lost interest by the third dot, or c) not see any pattern. Surprisingly, this is what I expect; the students will see no purpose to the exercise.
However, below is an illustration of what you will obtain if you persist in the Chaos Game for approximately 1000 or more iterations:
The figure involved is called the Sierpinski Triangle, a IFS fractal, and it will always be the object obtained if you follow the rules. If youíre like my students, you'll be asking, "Why or how can chaos always lead to such a figure?" To challenge my students and to nudge them in the right direction, I ask them for their opinions on what would happen if we switch to a right triangle or a square or a pentagon. They will formulate guesses, but they donít want to plot the 1,000+ points to find out. Here is where technology comes into play: the mechanics of steps 4, 5, and 6 are perfect for a simple computer program. Letís try and identify the real parameters of the Chaos Game.
First, there is the shape of the figure on which it is played. For our purposes, that can be reduced to deciding on the number of vertices and their location. Second, there is the randomizing device that represents step 4. In this case, we picked what is called a uniform random variable that will pick the Jack, Queen, and King with equal (one-third) probability. You can ask yourself how the game would have been affected if we had had four cards with two of them being Jacks, but still only three vertices. Third, there is the location of the new dot in each iteration. We could position it four-fifths of the way to the vertex or one-quarter or three-quarters, etc. We call that "selecting the ratio" for the game. Later, we add even more parameters.
The software I designed allows the students to experiment with varying these parameters to gain mastery of the underlying concepts. First, we experiment to learn why the holes appear in the figure. Next, after they understand the interaction of the parameters, they jointly set out to determine an algorithm for "reverse engineering" these figures. They must determine a set of concrete rules whereby, when given a fractal that is the result of some choice of parameters of the Chaos Game, they can identify those parameters just from the image. See figures 2 through 4 for example of fractals that they can solve. At first, students canít believe they can do it. After a few attempts, they quickly discover that trial and error will not help unless they include reflection on the process and a thoughtful understanding of what is driving the process.
They experiment, extrapolate, share conjectures, retest, and communicate. In essence, they are employing the scientific method without knowing they are doing it--aided by the fact that the software is kept transparent. Because the students donít spend much time learning how to use the software, it provides them a prime opportunity to interact with each other and get the results they are seeking. In the end, complex mathematic concepts are made understandable, and students learn to approach mathematics not as just simple computation but as a field of open inquiry.