A little over a year ago, the Museum of Mathematics opened in the heart of New York City. One of my favorite exhibits at the museum is the Human Tree. When you stand in front of the Human Tree screen and wave, your arms are replaced by images of your body. This structure repeats, with more and more copies of your body linked together in a tree-like form.
If you’d like to experiment with a similar effect, check out the interactive Web Sketchpad model of a Pythagorean tree below. In its starting configuration, the tree appears to be simply a geometric illustration of the Pythagorean Theorem with a flower in each square. Drag the red point labeled n to change its value from 1 to 2. Notice that the model has sprouted two more right triangles with accompanying squares. You can drag any of the red points to experiment with the model or press the Animate button to watch the model move.
Now, make the value of n larger. As you do, you’ll see this Pythagorean tree grow more branches. It’s pretty cool!
Here are some questions for you to consider: How many new squares are added to the tree when n increases by 1? How many squares are there in total? How much new area is added when n increases by 1? How much area is there in total?
If you’d like to use Sketchpad to build a Pythagorean tree of your own, check out this online tutorial.