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Playing the chaos game in class

There is a wonderful idea due to a number of individuals [Peitgen, et. al] provides an easy way to play the chaos game without using technology. Give each student (or group of students) an overhead transparency onto which you have copied the vertices of the original triangle (so they all are playing the same chaos game). Have the students choose the colors of the vertices and the original seed. Instead of using a marking pen to indicate the seed, have the students use the small circles that form the output of a three-hole punch. This allows them to perform the first few iterations by simply moving the ``dot'' rather than erasing.

Now have one student roll a die and another take charge of moving the ``point'' half the distance to the indicated vertex. A wonderful idea for moving the point exactly half the distance is the ``half-way ruler'' invented by Henri Lion. See Figure 5.

Figure 5: The half-way ruler

After seven or eight iterations with the movable point, have the students begin to track the orbit by recording the points on the orbit using a marking pen. Depending upon the class-size, 10 iterations of the chaos game by each group should be sufficient. Gather all of the transparencies and align them correctly on the overhead. Provided nobody has goofed, you will see the beginning of a good representation of the Sierpinski triangle. (It is important to use transparencies that are ``see-through''; some transparencies become opaque when several are piled together.)

Another interesting and valuable classroom activity involves target-shooting. Draw one of the images in the deterministic construction of S on the board or the overhead projector. Actually, it is best to begin at a relatively low level, say the second or third stage (so either 9 or 27 triangles remain). Fix a seed, perhaps one of the 3 vertices. Then color one of the triangles that remain at the lowest stage of the construction (the smallest triangles that have not yet been thrown out or subdivided). This triangle is the target. Then the game is to move the seed into the interior of the target in the smallest number of moves. This is not a random game any longer: the students have to determine which rolls of the die put the seed in the target in the smallest number of rolls. See Figure 6.

 
Figure 6: A target for the chaos game

There is no unique most efficient method to hit the target, as there are several alternatives. However, there is a least possible number of moves that cannot be beaten. At first, students seem to have a great deal of fun challenging one another with this game. Later they turn their attention to finding the algorithm that yields the most efficient solution. It is a wonderful exercise for students to try to find this winning strategy, and an even more beneficial exercise for them to explain their strategy once they have found it.

As part of the Dynamical Systems and Technology project at Boston University, there is a Java Applet available that students can use to play this game interactively. The URL is {\tt http://math.bu.edu/DYSYS/applets/ }.

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