# #MathArtChallenge 78: Ferozkoh Jaali & perspectives

The Challenge: Create this design, and then once you have the basic underlying grid, play around with the possible interpretations of how to color and “finalize” it.

Materials Needed: compass, straight edge, colors; could use graphing software.
Math concepts you could explore with this challenge: angles, circles, geometric construction, geometry, Islamic geometry, lines, symmetry, tessellations

This one is a bit philosophical – but does get to the heart of my passion for math art.

I use math art as a way to open doors normally closed in math classrooms. In my experience, often those students who carry “high status” in a math classroom (think about who bestows that, how or if it is challenged, what the implications are for each student surrounding that “status”) are those students most intimidated by math art. Traditional math classrooms are often centered around a correct or best answer, and not the path leading to it. Art is a way for us to slow down, think deeply, and create avenues for our student’s brilliance to shine that are not as available otherwise.

I’m currently reading Francis Su‘s extraordinary Mathematics for Human Flourishing and geez is that title accurate. One of his mathematical problem vignettes comes from this image from Catriona Shearer.

I immediately recognized the pattern as similar to the Ferzokah Jaali (it’s in Samira Mian’s first online course you should definitely sign up if you have any interest in playing with Islamic geometry).

It’s a well known design among Islamic geometrists, likely because of the relative simplicity of the underlying grid, although I really struggled to find an origin picture. This image (bottom) the best that I’ve yet found, which came out of this online PDF. I cropped it to include the page number so you can find it for yourself.

Based on internet sleuthing, “Ferzokah” refers to the capital of the 12th-century Ghorid Dynasty in Afghanistan, but I couldn’t find any images of this construction dating to then. Unclear to me if this pattern originated then or if a style/location is simply being referenced. The image in the book above references the pattern’s appearance in Agra Fort (India), Iran, and Cairo. I wonder if this pattern originated in the 12th century and spread to these other locations, or if it was discovered independently in any of them. Makes me want to dig in a lot further to the origins of these designs. I refer to them as “Islamic design” or “Islamic geometry”, but the history of them seems far more complex.

After discussing some of those origins with students, I would invite them to create the underlying grid for this pattern and then let them loose. With a little exploration, you can work it out from the images below.

One of the reasons I like this pattern is that it’s easy to mess up with some interesting results:

And, as the video at the top shows, there are so many ways to interpret the underlying grid. There’s a lot of room for students to play here, and mathematical play is one of the best ways to invite students to dig deeper into deep, rigorous mathematics.

• How many different ways can you design this pattern and still have it be recognizable?
• Does your brain naturally intepret this a a 3, 6 or 12 fold rotational pattern? Or something else?
• What is an ideal ratio for negative to positive space in this? Why do you think so?
• What other ways can this grid be interpreted?

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!

2.) Reason abstractly and quantitatively. In the images that turn this into linked rectangles, is there a best way to order the intersections where 3 links overlap? Is it possible to make the links alternating?

7.) Look for and make use of structure. What are the core components of this construction? At what point is a design no longer “true” to the construction and becomes something else?

## Author: Ms. P

Math Teacher in Minneapolis, MN.