The Challenge: Create a self-repeating pattern – a fractal. You may choose your own design, or perhaps you recreate some of the ones from Ron Eglash’s survey studying the fractal formation of African villages. I did both of these looking at the applet at his website.
Materials Needed: paper and pencil, likely, but you can probably get more creative than that, too! Maybe using sculptural materials?
Math concepts you could explore with this challenge: angles, fractals
The History: Ron Eglash noticed aerial footage, in 1988, showing that the layout of African villages built in fractal formations. He got a grant to travel the continent for a year studying these fractal formations, and wrote a book (a PDF available here). This article does a nice job of outlining what he discovered, and his TED talk is below:
I really love the depth with which he’s studying how the creators of these villages consider them and their structure. I must admit, I have not yet read his entire book – just passages here and there. It is certainly on my “currently reading shelf”.
His conclusions are fascinating, and raise all sorts of new questions for me. For example, he hypothesized that all (or most) indigenous cultures would replicate these fractal patterns, but found that to not be the case. These fractals are largely unique to Africa.
“But if fractal architecture is simply the automatic result of a non-state social organization, then we should see fractal settlement patterns in the indigenous societies of many parts of the world…we will examine the settlement patterns found in the indigenous societies of the Americas and the South Pacific, but our search will turn up very few fractals. Rather than dividing the world between a Euclidean West and fractal non-West, we will find that each society makes use of its particular design themes in organizing its built environment.” (Eglash, p. 39)
Chicago’s WBEZ has a great piece (~50 minutes) about this from last year that led me to Eglash’s fantastic computing website https://csdt.org/ There, you can play with fractals, read some about the history of various ethnomathematical concepts and explore how and where mathematics appears in culture. I made this using an applet on the website:
The website and WBEZ piece go into bit about African hairstyles and pulls in Nnedi Okorafor‘s brilliant book Binti, which is an Afrofuturistic novel about an African girl who is a mathematical genius. The mathematics behind cornrows also appears in Dr. Shelly Jones’s book, Women Who Count.
In sum, I’m grateful for the depth of Eglash’s work highlighting these structures, although I do take issue with at least one of his premises:
“Unconscious structures do not count as mathematical knowledge, even though we can use mathematics to describe them.” (p. 12)
I think of these “unconscious structures” as the foundation of human mathematical knowledge. I want to celebrate my student’s inborn mathematical tendencies as the first layer of mathematical understanding. To honor the idea that teachers should never be the sole “knowers of things” in a classroom we need to celebrate the inborn mathematics our students bring to our classrooms. We want to invite and encourage our students to build upon their inherent mathematical tendencies. That said, I do appreciate the distinction Eglash makes between those structures built mathematically without conscious effort and those that are built conscious of mathematical structures. They are worth studying differently.
So, go, explore the website, read the book, and play with fractals. And ABSOLUTELY, when you bring fractals up in class, make sure that if you’re sharing that Mandelbrot “discovered” them in the 1970’s, share that he definitely wasn’t the first. Let our Black students revel in and explore their mathematical history.
Depending on how you use this activity, you may engage with different mathematical standards. I’ve listed possible connected math content above. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!
3.) Construct viable arguments and critique the reasoning of others. What is a fractal? How do we recognize them in society?
6.) Attend to precision. How do we describe fractals? How do we decide who is the “discoverer” of a piece of mathematics? Who should get credit for mathematic structures?