Fractalina

Before using Fractalina, 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.

How to use Fractalina. To play a chaos game, you need to select a number of vertices in the plane together with their associated compression ratios and rotations. When you first open Fractalina, you will see three points at the vertices of a triangle in the plane, a red point, a green point, and a blue point. This is the configuration for the "standard" chaos game. To the right of the main window, you see the control panel, which is a list containing information about each vertex: namely,

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

We will explain how to change these in a moment, but for now, click the Start button. You should see the results of playing the chaos game with these three vertices, namely the Sierpinski triangle, gradually unfold. The computer selects a randomly chosen seed. Then it randomly selects one of the three vertices and moves the point half the distance (compresses the distance by a factor of 2) toward the chosen vertex. This is repeated over and over, using the endpoint of one move as the starting point for the next. The first 20 iterations are not displayed so you do not see any extraneous dots in the image. Click Stop to end the computation.

Magnification. You can magnify portions of this image by clicking on a point and then dragging with the mouse down to form a square in the window. When you then click Start, you see the portion of the image within the square magnified to fill the entire screen. To return to the original image, click Zoom out.

Changing vertices, compression ratios, and rotations. To move the vertices to other positions, you may click and drag on the center of the vertex to the new position. Be careful here: If you miss the center, the program will think that you are trying to magnify the image. Click Zoom out to return to the original configuration and try to drag the vertex again. For example, if you move the top vertex to form a right triangle, when you click Start, you will see the Sierpinski right triangle. Alternatively, you can input the coordinates of the top point directly by typing them into the control panel window. Both the x and y-coordinates run from -150 to +150, with (0, 0) at the center of the window.

You can similary change both the rotations and the compression ratios at each vertex.

To add a new vertex, click New point. A new vertex appears at the origin. You can then move this point to a new location and change the various quantities as above. The button labeled Kill point removes the last point on the list in the control panel.

Preset fractals. You can input the configurations for various preset fractals by pulling down the lower right menu. There you will find presets for the Koch curve, Sierpinski carpet, etc.

Changing colors. You can also change colors to black and white or other configurations in the Colors menu. Selecting Preset Colors always gives you our favorite next color when you click on New point. If you select Random Colors, your next point's color is generated randomly. Don't like that color? Just click Kill Point and choose a new point. Repeat this until you find a color you like.

Go to Fractalina.

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

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