Fractal Tool

The bottom of the work area shows Initial, Now, and Next. Initial is the original shape from which a fractal will be created. Now and Next describe the process that will be used to create the fractal — any element that looks like Now at the current stage will be transformed to Next in the following stage.

There are four options for the shape. They are the Koch Snowflake, a fractal tree, a reduced square, and the Sierpinski triangle.

The New button resets the current fractal to Stage 0, where it looks like the Initial state shown in the bottom left. The +/– buttons can be used to step through the fractal, stage by stage. The Play button runs the entire animation through from the current stage to the last.

Note: Each fractal will be animated for only the first several stages, due to computer limitations. However, statistics about stages beyond those shown graphically will appear in the table of values. The orange rows represent those stages for which the the image can be displayed; the blue rows represent those stages for which there are no images.

Choose one of the fractals at the top of the dashboard. This will place that fractal at Stage 0 in the display. It will look like the Initial image in the bottom left.

Notice the Now and Next pictures at the bottom of the work area. Now and Next describe how the fractal will be changed after each stage. For each additional stage, any part of the image that looks like the Now image will be transformed to look like the Next image. Use the + and – buttons to watch the figure step through stages.

Think about the following questions, based on the table of values.

• As you go from Stage 0 to Stage 1, how many times was the Now/Next rule applied? From Stage 1 to Stage 2? 2 to 3?
• When the table shows blue rows, the image is no longer changing. What would the image look like in the next stage?
• In Stage 4, can you find a smaller portion of the image that looks like the whole thing? Can you find more than one?

Think about the following questions, based on the table of values.

• How does the number of segments/squares/triangles change from stage to stage?
• How does the total length or area change from stage to stage?
• What values will appear in the row for Stage 16?
• What values will appear in the row for Stage n?