Fractanimate

Before using Fractanimate, you should be familiar with the notion of an iterated function system or, as it is more commonly known, the chaos game. For more details, see the book Fractals, especially Lessons 5-7. You should also be well-versed in the use of the applet Fractalina, since this applet provides the basic building blocks for working with Fractanimate. To go to Fractalina, click here.

This JAVA applet is designed to allow you to make "fractal movies." To see several examples of these movies go to the Dynamical Systems and Technology Project movie theater. Each frame of these movies is a different "Chaos Game."

How to use Fractanimate. When you start up Fractanimate you should see three different windows. One is called Starting Frame, one called Ending Frame, and the Control Panel, included in your browser's main window. You may have to move one or more of these windows around to see all three. Often the Ending Frame covers the Starting Frame completely.

To make a movie, you must specify the starting frame, the ending frame, and how many frames to interpolate in between. Each frame of the film is a different chaos game, so you need to specify for each frame the number of vertices, which must be the same for each frame. This is accomplished exactly as in Fractalina. For each vertex you determine:

• The color of the vertex
• The rotation about the vertex
• The compression ratio
• The x- and y-coordinates of the vertex.

You should do this first for the Starting Frame. Then change certain of the parameters for the Ending Frame. Finally specify in the Control Panel the number of frames to compute. Fractanimate will then interpolate linearly between the starting and ending frames to produce an animation of the changes.

An Example. When you first open Fractanimate, both the Starting Frame and the Ending Frame contain the parameters corresponding to the "standard" chaos game, i.e., the game that produces the Sierpinski triangle. Let's make a film in which we change the rotation number about the top vertex by 30 degrees at each stage, ending with a 360 degree rotation. That is, the final frame of the film will contain 3 vertices, compression ratio 2, and rotation of 360 degrees about the top vertex. Input this information into the Ending Frame now. (All you need do is change the rotation about the top (red) vertex to 360.) Click Start in this window and you will see the Sierpinski triangle. Of course! Rotation of 360 degrees about the top vertex is the same as no rotaition at all, so you are really using the original chaos game rules. Don't forget to click Stop to finish the computation. Otherwise, you keep on playing this chaos game and waste a lot of your computer's precious resources.

Now click to see the Control Panel window. Enter 13 into the Frames window and then click Make Animation. You need 13 frames since the first and last frame are the same. Thus frame number N rotates 30 (N - 1) degrees around the top vertex where N = 1 is the starting frame.

The computer then should make each of the frames in succession. When finished, click Play to see your film.

Some hints. Some people have reported bugs on certain platforms. While we have not been able to duplicate them here, we have heard these reports several times

• On some platforms, you always seem to get animations with only 6 frames, no matter what number you enter into the Frames window. Apparently, if you do NOT hit the RETURN key before clicking Make Animation, this problem does not occur. If this happens to you, simply refrain from hitting return after entering the number of frames. I know that this is tough to remember---but hey, think of it as saving your RETURN key from overuse!!!

• On some platforms, if you change from making a movie with colored frames to one with black and white frames, you end up creating a movie that consists of white dots on a white background. That's not very satisfying! If this happens, simply return to the Colors option in both the Starting Frame and Ending Frame.

• Remember that each of these frames must be stored in your computer's memory before playing. Want to crash you computer? Make a movie with lots of frames. To be safe, start with short animations and gradually progress until see how much chaos your computer can handle!

• Do NOT forget to click Stop in both the Starting Frame and Ending Frame before trying to make an animation. If you do not, you will continue to compute each of these chaos games, and this will result is a much slower computation of your movie.

• Find a problem? Let us know at the e-dress below.

Go to Fractanimate.

Created by Noah D. Goodman, Adrian Vajiac, and Robert L. Devaney.

For comments and suggestions write to Robert L. Devaney at bob@bu.edu