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?

Activity

Instructions

Modes

Koch Snowflake

Fractal Tree

Reduced Square

Sierpinski Triangle

Create your Own

Features

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.

How to Use

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 RESET 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.

Exploration

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?

Objectives and Standards

NCTM Standards and Expectations

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.