# Exercises for Chaos Under Control

## Chapter 1

1.[E] Here's a nice way to demonstrate that the starting point in the Chaos Game is irrelevant. It requires access to a bit of equipment and is best done as a group project. You will need a reproducing machine, some indelible markers, some overhead projection foils, and enough dice and rulers to go around to everyone in the group. Mark three points A, B, and C on a master foil. Run off a number of copies on foil blanks. Distribute a copy to each member of the group. Have each member play the Chaos Game (Three Corners, One-half Version) on his or her foil, but in each case do not record the first few new points -- e.g., have each player mark the first few (about ten) points with a light pencil that can be erased. Each person should play the game about thirty times making about twenty indelible points in each case. Of course, each should be encouraged to pick a different starting point. Now collect all of the foils and superpose them so that all the A's, B's, and C's, respectively, exactly overlap. The superposition of indelible marks from the entire set of games should form a piece of the Gasket. The more players, the more complete will be the superposition. Try it. You'll be amazed.

2.[C] Another way of proving to yourself that the starting point of the Chaos Game is irrelevant is to use the ChaosGame Option of the TreenessEmerging Program with the option "three corners" and 0.500 for "scaling factor." Place "corners" near the upper left corner, near the lower left, and near the lower right. Place a starting point anywhere you like. Watch what picture emerges. Now, repeat with another starting point. And so on.

3.[C] Run the ChaosGame Option of the TreenessEmerging Program with four corners. Place the corners at the vertices of a square. What picture emerges? Now, run with four corners again, but this time make an irregular 4 sided figure. What picture emerges this time? Try a few more with 5 sides and higher. Can you see any pattern in what results?

4.[C] Run ChaosGame Option of the TreenessEmerging Program with three corners, but now use "scaling factor" equal to 0.667. What do you observe? With "scaling factor" equal to 0.333? How about with four corners?