Lab #5

Goal: The goal of this experiment is to see how iterated function systems produce fractal images. You are asked to find the appropriate iterated function systems which produce the given fractals.

Part One: Consider the following four fractals:

Fractal 1

Fractal 2

Fractal 3

Fractal 4

Each of these fractals was generated by using Fractalina, available at the Dynamical Systems and Technology home page. For each of these fractals

• Find the coordinates of the points and the compression ratios that produced the resulting images. You should give the coordinates using "screen coordinates" running from -150 to 150 as in Fractalina. You should use the program to experiment with different coordinates and magnifications until you find the one that most nearly resembles the pictures provided. There was no rotation in each of these systems, so the "Rotation" should be kept at 0.
• Compute the fractal dimension of each of these fractals.

Part Two: Now consider the fractal in this figure:

Fractal 5

How do you think that this image was produced? This time there was a non-zero rotation involved at each point. Again give both the coordinates and the compression factors as well as the rotations involved that were used to produce this image.

Part Three: Now consider the images in this figure:

Fractal 6

Fractal 7

How do you think that these images were produced? Again give both the coordinates and the compression factors involved that were used to produce this image. In a brief essay, explain why certain portions of the images are darker than other portions.

Part Four: On the web, open the "chaos game" Java applet located at math.bu.edu/DYSYS. As discussed in class, find an algorithm to win this game at each level (novice, medium, hard, master). Your algorithm should apply to all targets; not just specific cases. Describe your algorithm in an essay in proper English. Remember that the goal of this game is to move the red starting point into the interior of (not the boundary of) green triangle using only the moves of the chaos game.

Part Five: Open the "movie" by clicking here. If you cannot view the movie, see below. How do you think this movie was created? Each frame of the film was created using an iterated function system. Using Fractalina, see if you can figure out how each frame was generated. In an essay, describe the number of transformations used, the associated contraction factors, and the rotation angles. Hint: The contraction factors are not all the same.

Note: You will need a QuickTime viewer to see this animation. Click here to download a QuickTime player for a Mac or PC.

PS: These last two parts are difficult!