Summary (Next Section)
Chaos in the Classroom (Cover Page)
Changing the rules in the (Previous Section)
BU Math Home Page
Another, more complicated type of chaos game results when we allow rotations as well as contractions toward specific vertices. For example, suppose we start with the vertices of an equilateral triangle. Again, when either of the lower two vertices are selected, we simply move half the distance toward them. But when the upper vertex is chosen, we first move 1/2 way toward the vertex, then we rotate the point about this vertex by 90 degrees in the counterclockwise direction. Note that we could equally well rotate first, then move 1/2 the distance toward that vertex. If we now play the chaos game with these rules, the image in figure 12 results. Note that this figure consists of 3 self-similar pieces, and each is 1/2 the size of the original, but the topmost piece is rotated by 90 degrees in the counterclockwise direction. Again, it is not so easy to predict what figure will result from playing the chaos game, but once we see it, there is no question what game we played to generate it.
Figure 12: The chaos game with 3 vertices, magnification factor 2, and
one 90 degree rotation about the top vertex
Figure 13: The chaos game with 3 vertices,
magnification factor 2, and
one 180 degree rotation about the top vertex
Rotations complicate the task of determining the chaos game that intuition to figure out some of these games. For example, how did we generate the fractals in figure 14?
Figure 14A: A test: how was this image generated?
For the answer, click HERE
Figure 14B: And how "was this image generated?
For the answer, click HERE
One rather fun exercise for students who have access to good computing equipment to perform is to make a fractal "movie. This can be done by computing the images of a number of fractals generated by chaos games where each "frame" of the film is generated by changing the parameters (rotations or magnification factors) in the previous frame only slightly. In15A-C we show a "clip" of such a film. How did we generate this movie?
Click on this icon to download the animation).
Figure 15A: How was this movie generated? You need to
have a QuickTime Movie player in order to view the animation. Choose
Loop in order to see the film continually.
Click on this icon to download the animation).
Figure 15B: How was this movie generated?
For the answer, click HERE
Click on this icon to download the animation).
Figure 15C (Dancing Triangles):
And how was this movie generated? Warning:
there is approximately 700K of data to download to view this film.
For the answer, click HERE
Summary (Next Section) Chaos in the Classroom (Cover Page) Changing the rules in the (Previous Section) BU Math Home Page