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**

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.

This JAVA applet is designed to allow you to play the chaos game with a variety of different configurations. You may enter as many vertices as you would like (up to 15), move them around the plane, choose various compression ratios and rotations, and then display the resulting image.

** 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