CC MATHEMATICS CHAOS AND CALCULUSASSIGNMENTS


THE CHAOS GAME

Go to the web site http://math.bu.edu/DYSYS/applets/chaos-game.html.

When you open the chaos game applet, you see a game board that consists of the Sierpinski triangle computed down to level 2, i.e., with nine smaller triangles. One of these triangles is colored green; this is the Target. You also see a point colored red at the lower right hand corner of the triangle. This is the Seed. Your goal is to move the Seed into the INTERIOR of the Target in as few moves as possible. Each time you make a "move" in the chaos game, this point will move to a new location in the game board. This becomes your current location.

The Moves: There are three possible moves in this chaos game. If you click on the top (red) vertex, your current location moves half the distance to the topmost vertex. Similarly, clicking on the lower left (blue) or lower right (green) vertex, moves the current location half the distance to that vertex. By a judicious choice of moves, you should be able to move your point into the INTERIOR of the Target in just 4 moves. This is indicated in the Best Score window. Your score (the number of moves you have made) is recorded in the window called Your Score.  In the upper left corner of the screen your successive moves are recorded as a series of red, green, and blue dots. When you succeed in moving your current location into the interior of the Target, you receive a message telling you so.

Resetting the Target: To replay the current game again (without changing Target), click on Try Again. To change to a new Target, click on Restart. The computer randomly selects a new Target.

The Algorithm: There is an algorithm for moving the starting point into the interior of the Target in exactly 4 moves, no matter what Target you start with. Your job is to discover this algorithm. You should be able to explain in advance how to move the Starting Point into the Target in a couple of sentences, no matter where the Target is located.

More difficult Targets: If you click on the Medium, Hard, or Master buttons at the top of the game board, you find chaos games with a Target at deeper levels of the Sierpinski triangle. The Best Score panel is appropriately modified. The algorithm that you developed at the Novice level should help you maneuver through these more difficult games.

 

Your Mission: Play the chaos game at the Novice level until you understand the algorithm that allows you to hit the target in exactly 4 moves. Write up a clear explanation of the algorithm that would allow a person reading it to carry it out with ease. As much as possible, describe you algorithm in terms of general principles, so you do not have to explain each starting point separately. You and your partner should turn in this description of the algorithm and your best explanation of why you think it should work by 3PM on Tuesday, Sept 3. Hand in only one paper for the two of you.  Also explain how your understanding of the algorithm helps you see why the Chaos Game (Three Corners, ½ Version) described in your text on pages 22-23 leads to Sierpinski’s triangle. (For example, what would happen if you set as a target one of the triangles missing in Sierpinski’s triangle, and why?) If you have time and interest, try the Medium, Hard, or Master levels, and describe how your algorithm should be modified to fit these games. As part of this additional challenge, try to explain how to tell what the minimum number of moves required is for any target at any level.