Fractal Tree -- Geometry Class Project


One of my goals this year is to bring fractals into my teaching, especially in geometry class. Today I am going to share the fractal tree. It can easily be done on a computer but is also great for practicing measuring with a ruler and protractor. Now a fractal tree is probably one of the most basic fractals and well-known or at least its cousin, the fern is certainly seen as a fractal. 

To understand the fractal tree, watch this video.

Now today I am going to share with you two different noncomputer methods to creating a fractal tree. One is by hand and then I am sharing an easier method using wax sticks (as well as a Koch snowflake out of wax sticks too). 

I used Sharpie pens, a pencil, eraser, ruler, and protractor to create these fractal trees. I had started one and got to the eighth iteration, but it got stepped on and ripped. Here it is. You can see that on the left side I was working on the eighth iteration and on the left side I only got to the seventh.

To draw a fractal tree you begin with a segment measured precisely. From the endpoint you measure a chosen angle from each side of the segment. Then draw the next segment which will be a picked fraction of the first segment. Then from the endpoints of the new segments you repeat this process. I have a project assignment sheet for you

I wrote in my measurements so you can see what I am doing in each iteration. Above is two iterations or stage 1. We start with stage 0, which is the trunk. Below is the next stage.

As you can see it requires much measuring to do each iteration. Each iteration adds more segments exponentially. It gets to a point where it becomes difficult to measure and draw the short segments. 

For me I was stopping at stage 7. I wanted to find a way to share fractal trees with younger kids as well and had some wax sticks hanging around. I decided to try to make a wax stick fractal tree.

I used the entire stick for the trunk. Then folded one in half for the next iteration. Then I cut the next one into two pieces and folded both. I kept repeating this procedure. I used the stickiness of the sticks to stick it to the paper. However, if you want to keep it you will need to glue them. I found some of the sticks fell off the paper over time. This went much quicker, and I did not worry about measuring the angles. It was fun to put it together, that I decided to try the Koch snowflake as well. 

As you can see the Stage 3 wax stick was hard and didn't really work. I love being able to bring some fractal lessons to different age groups as well as to my class. Fractals are the way of the future. The more that is learned about them the more useful they become. They are being used in medicine, geography, movies, computers, and so much more.