Spirolaterals: A Math + Art Mashup
love a good cross-curricular connection. When I stumbled across the idea of mixing math with art, I couldn’t resist. At first glance, I found some beautiful patterns. I was drawn to their geometric nature and symmetry. Not to mention I’ve always been a sucker for geometry. On further inspection, though, I found a complex and interesting math pattern that I simply had to share with my students.
Spirolaterals are geometric designs made by repeating numbers following a simple rule. At their most basic, they are formed by drawing a repeating series of line lengths, turning the page by a set angle between each segment. There are infinite combinations that can create spirolaterals, and each is interesting in its own way. If you have time, take a few minutes to try some combinations! Some of my favorite generators can be found here or here.
So what’s the math/art link?
Easy answer: line segments (measurement) and angles (geometry). You could totally stop with those two and still be fascinated! I decided to dig a little deeper and found something unexpected: MULTIPLICATION! I know, I know, the link here isn’t as obvious. But wait–there’s more! 🙂
Take a minute and try this: write down the products for a set of basic multiplication facts (try your 1 or 2 facts, starting with x1). When you start getting two-digit products, add the two digits together to get a single-digit sum. Write down your sequence of products, continuing until you start to repeat. For example, your 2 fact product series would be: 2, 4, 6, 8, 1, 3 , 5, 7, 9. After the 9, it repeats back to 2, so you could stop there.
Now, on a sheet of grid paper, choose a starting spot. Draw a line segment the length of the first number in your series. Turn your paper 90˚, and make a segment for your second length. Turn your paper 90˚ again, and make a segment for your third length. Turn your paper 90˚ again… you get the gist! Keep on keeping on until you return to your starting point and you should have yourself a super, symmetrical, satisfying design!
Super cool, but why do it with the kids?
Extensions, of course! Ask students to think through further here to increase the depth, complexity, and rigor:
Why does the sequence of products produce a spirolateral in which the end meets the beginning?
Does this always happen? (And my personal favorite… prove it!)
If you use multiples of numbers 1-9 and make each into a spirolateral, what patterns emerge? Why?
What happens if you choose random numbers? Why?
What happens if you change your turn from 90˚ to 45˚? 60˚?
The possibilities here are endless, but all of them are super interesting. We would love to see what you can make happen with your spirolaterals! Post your pictures in the comments to show off your work!
By Anna and Emily