Fold Your Way to Glory

Yesterday, I led a meeting of a Teachers’ Math Circle about the fold and cut theorem. This theorem says any region with a polygonal boundary can be folded and cut from a sheet of paper using only one cut. I learned about the theorem last year when Numberphile posted this excellent video featuring a virtuoso performance from Katie Steckles, who folds and cuts every letter of the alphabet from memory.

I was a little nervous about leading the program because I had prepared almost nothing to say. Everything I thought about saying was boring, so I decided the best way to approach the activity was to just get people started on it. Luckily for me, the group was ready to jump right in. I dumped a bunch of paper into the middle of the table, and people started folding.

I encouraged people to try the most symmetric shapes first, but other than that, I didn’t have to give them many suggestions. I was prepared for some frustration when they started trying the scalene triangle because it’s a big step up in difficulty, but several of them got the scalene pretty quickly, and no one seemed to give up. In general, the strange shapes people got when you mess up were amusing rather than frustrating.

My favorite fold-and-cut mistake. I was trying to make a rectangle with a smaller rectangle inside it. Image: Evelyn Lamb.

My favorite fold-and-cut mistake. I was trying to make a rectangle with a smaller rectangle inside it. Image: Evelyn Lamb.

Participants almost immediately started asking mathematical questions and trying to extend the activity: do we have an existence theorem? Must we always fold along every angle bisector? Is there a general theory of folding? I liked Anna Weltman’s suggestion of trying to make things without drawing on the paper, and I spent some time trying to fold stars without drawing them, but the teachers didn’t really bite on that. Instead, some of them started thinking about minimal folding numbers for different shapes, and some of them worked on developing a folding algorithm.

Erik Demaine is one of the pioneers of fold-and-cut theory and the mathematics of paper folding in general. His page about folding and cutting has links to all the gory mathematical details as well as some templates. I ended up bringing copies of his swan to the teachers’ circle. They are beautiful, but I had mixed feelings about bringing them because they have the fold lines marked on them already. I didn’t hand them out until one group had started talking about how to use angle bisectors and perpendiculars in their folding algorithm, and I thought the swan template might give them some ideas. Because I gave them only the template, not any explanation of how it was made, I think it didn’t take away too much of their fun.

My fold-and-cut swan now enjoys pride of place on my new hexagonal shelf. Image: Evelyn Lamb.

My fold-and-cut swan now enjoys pride of place on my new hexagonal shelf. Image: Evelyn Lamb. Swan template: Erik Demaine.

In addition to Demaine’s swan, I brought templates for lots of different shapes from Patrick Honner and Joel David Hamkins, who uses hole punching symmetry activities as a warm-up for cutting. I also got ideas from Mike Lawler, who has done fold and cut activities with kids, and Kate Owens, who ran a fold-and-cut workshop for teachers.

I’ve done a little bit of origami, but I’ve never gotten good enough to feel like I had geometric intuition for doing it. I’m still at the level where I follow directions and get what the book says I should. Making these fold-and-cut shapes, though, is an easy way to start thinking about paper folding mathematically and creatively. Thanks to the resources I mentioned above, you too can easily introduce people to the joys of mathematical paper folding.

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