It is winter in Michigan, and as I’m writing the snow is accumulating on the ground and covering the tree branches outside my window. So naturally, my thoughts turn to–fractals. Or, more specifically, to this guest post from Melanie describing creative ways to teach fractals, including fractal snowflakes. Here’s Melanie.
Fractals are mathematical sets that demonstrate self-similarity. Each iteration of the pattern, or number of times the pattern is repeated, makes the fractal more and more detailed. Some classic examples in nature include trees, sunflowers, broccoli and blood vessels! The Museum of Mathematics in New York City even has a camera that allows patrons to see themselves projected on a wall as a fractal tree with iterations of tiny versions of themselves on each arm branching out into a human fractal tree.
[Note from Alane: Imagine an art activity involving graphic fractals. Students could create branching human fractal trees from photographs (a great chance to practice computer skills with graphic images), or choose animals or objects to “fractalize.” Makes me want to try!]
Some more traditional fractals you can make in class would be Koch’s Snowflake, Sierpinski’s Triangle and Menser’s sponge. Julian sets and Mandelbrot sets are incredibly beautiful but impossible to recreate by hand.
A fun fractal to make in the winter time is Koch’s snowflake. This snowflake is created by using three of Koch’s curves, one of the first fractals described by Swedish mathematican Helge von Koch in 1904. It starts with an equilateral triangle. Each side of the triangle is cut into thirds and the middle segment is replaced with an equilateral triangle. After drawing in each triangle, erase its base, which is also the middle section of the line. The first iteration of a Koch’s Snowflake is a six-pointed star, or technically a hexagram. Each iteration recursively alters each line segment revealing a more and more complicated snowflake with an infinitely increasing perimeter. You can see the steps here.
The math of fractals is the stuff of science fiction: inter-dimensionality and finite area and volume contained by infinite perimeter and borders. With older students you may be able to calculate the area of Koch’s snowflake even though it has an infinite perimeter. Younger students will still enjoy trying their hand at recreating some of the classic fractals: Koch’s Snowflake, Sierpinski’s Triangle and Menser’s sponge. Once they’ve mastered the basic fractals, encourage your students to create their own self-similar patterns. Challenge your students to create an image that demonstrates their understanding of iterations. This snowflake actually has six little snowflakes of 1-iteration, one medium sized snowflake of 2-iterations, and one large snowflake of 3-iterations.
Last winter, I took a few days to discuss fractals as we began our geometry unit. I showed the students several examples of fractals in nature and fractals discovered by notable mathematicians. We spent a couple of days constructing the three classic fractals. A few students unimpressed asked when they would ever use fractals, but they quickly changed her mind when I suggested the doctors, engineers and artists work with fractals all the time. For the final project, I gave them a small assessment defining fractals, listing examples of fractals in nature, matching mathematicians with their biography, and I asked them to make their own fractal with at least two iterations. Many students recreated one of the fractals we had constructed in class, but several took the opportunity to make their own cactus like claws or checkered patterns that were self-similar. Creative students can take fractals in many different directions calculating limits of infinity, exploring iterations, or just creating new fractals.
Note from Alane: There are at least two things I like about the creative potential in this topic. First, of course, students have the opportunity to create original fractals that allow them to both demonstrate their understanding of content and utilize creative thinking. But in addition, creative teaching with fractals allows students to envision math as something beyond calculations. A glimpse into the beauty and wonder that are at the core of mathematics can give future creative mathematicians a peek into their future.
If you’d like to see the Museum of Math’s fractal trees in action, take a look.