## Fractal Tool

Play with shapes that grow, shrink, and change over several stages. Explore self-similarity and patterns in fractal measurements. Can you see how these shapes are fractals?

Begin by selecting a fractal from the top menu. Your choices are:

Koch Snowflake | Fractal Tree | Reduced Square | Sierpinski Triangle | Create your Own |

**Initial**, **Now**, and **Next** images are shown below the work area. **Initial** is the original shape (Stage 0) 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.

Stepping through the stages:

- To step through the stages of your fractal, use the
**+**and**–**buttons. The fractal will move 1 stage forwards or backwards, and the table to the right of the work area will be updated. - The
**PLAY**button automatically steps through all stages, starting with the current stage. - The
**NEW**button resets the current fractal to Stage 0 (the image shown in the**Initial**image).

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

Choose a fractal from the top menu or create your own. Observe how the fractal image and table values change as you step through the various stages.

How does the image change with each step? If you step one stage forward, what about the fractal image changes and what stays the same?

Fractals have a property called *self-similarity*. This
means every part of the fractal looks like the whole fractal. Looking at
the highest stage of your fractal, try to find parts that look like the
whole. How many can you find?

As you go from Stage 0 to Stage 1, how many times is the Now/Next rule applied? from Stage 1 to Stage 2? 2 to 3? Can you see how this is reflected in the table values?

When the table rows turn blue, the fractal image will not change anymore. What would the image look like in the next stage?

Looking at the table, see if you can answer the following questions:

- 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?
- Is there a relationship between the second and third columns?
- What values will appear in the row for Stage 16?
- What values will appear in the row for Stage
*n*?