Fold and Cut

Brendan — my geometry coconspirator — and I went to the Museum of Math recently to see Erik Demaine give a talk about math and magic. It was a special lecture for me because I saw Prof. Demaine speak at the very first Museum of Math lecture (before the museum was built), and this was the five year anniversary of that talk.

Prof. Demaine and his father Martin Demaine both are mathematical artists — playfully using mathematics and art in search of higher truths. The most mindblowing thing that he discovered was that by folding paper however you want, and making only one single cut, you can cut out any polygon. Evenmoreso, the theorem goes further: “Thus it is possible to make single polygons (possibly nonconvex), multiple disjoint polygons, nested polygons, adjoining polygons, and even floating line segments and points.” [1]

Whoa, right? So say you want to cut out each letter of the alphabet? Done.

Or you want to cut out a swan or jack-o-latern?

You can do it. It boggles my mind.

When we went to Prof. Demaine’s talk, on each chair was a packet of paper and a pair of scissors. We were challenged to “fold and cut” each of the shapes out. The shapes were scaffolded well, and so I got pretty far along and was figuring things out. At that time, Brendan and I realized that both angle bisectors and perpendicular lines were key for much of what we were doing. We also realized that the puzzle nature of the challenge got us obsessed. We both were stuck on a single page [I’ll write about that in the P.S.] and as I was waiting for the subway home, as I rode the subway home, and all throughout the next morning, I grappled with it. I still have no clue how to solve it.

In any case, we both wanted to expose our geometry students to this puzzle. We figure next year we could turn it into a lesson — having them play and then have them analyze what they figured out. But for this year, we wanted to just see what happened if we gave our kids the puzzles.

I faintly recalled my friend Bowman doing this in his class and blogging about it, so I found that post and used his recommendations about what to have the kids cut out in which order, with the scaffolding that Prof. Demaine used in his packet, with some ideas that Brendan had, to create our own packet of fold and cut puzzles.

Fold and Cut Figures [PDF download]

What happened? Well, we gave kids 25-30 minutes. We had extra copies of pages for if kids messed up and wanted to try again. And we said “go at it.” Of all the kids in my class, only one seemed not to get into it… at the beginning. That student was trying too hard to have a “method” and their intuition wasn’t as strong as the others… but they showed me proudly at the end when their star! All the other students were addicted. Paper flew about. Kids called me over to proudly show me their successes, and wailed in frustration when their cut didn’t work (and then hurriedly asked me for another copy of the page they messed up on ). It was exciting to see kids focused but also having fun playing with math. I would say that 25-30 minutes was the right amount of time, because at that point, I saw kids start to fade. (It could also be that we met at the very end of the day, and this was the last 30 minutes of a 90 minute block…) No kid in the time given was able to get the scalene triangle (many got close) or the last quadrilateral. But almost every kid was able to get all the figures before ’em.

Next steps from here? I want to turn this into something more formal. I like the play. I love the play. But then we need to come up with some general conclusions and talk about why they work. Why are we doing lots of folding along angle bisectors? [Hint: the answer has to do with reflections!] Why are we doing lots of folds perpendicular to the lines of the polygons we’re trying to cut out? [Hint: if we imagine a “vertex” at the place where we have a perpendicular fold, we can consider our fold an angle bisector — bisecting the 180 degree angle of the vertex!] If kids understand those two principles (and the scalene triangle is the most perfect shape to make them both come alive!), I will have a way for kids to tie their puzzling to our geometry curriculum.

What most impressed me was how much intuition kids already had with regards to these. It was amazing to see them take to it as adroitly as they did.

And who knows? Even though I say we should tie this to the curriculum formally next year, maybe I’ll get to it this year after we complete our mountains of salt investigation. Because heck if they aren’t perfectly related to each other!

P.S. So… Here’s where we got stuck. We were given the following paper… no polygon, just a line segment that we had to cut.


You might say: duh, fold it vertically in half and make a half cut. But here’s the thing: you have to make a COMPLETE cut. So once you start cutting, you have to keep cutting until you have completely hit the end of your paper. And BOOM! Suddenly I am perplexed.



  1. Hi Sam, great post and puzzle!

    Does the line segment have to be purely one dimensional, or can it be a rectangle with a teany-tiny width. The latter case is pretty straight-forward I think!

    1. Hihi- I think it should be one dimensional… Otherwise you’re right — too easy! It is so awesomely frustrating. Maybe it isn’t possible, and I misunderstood the problem? It FEELS impossible. But I bet it IS possible.

      1. I’m also assuming there’s a rule that you can’t cut any part of the paper that is not a side of the figure?

        was this true for all of them? (i.e. when cutting out the pentagon the rest of the paper must remain intact) would it make a difference to have that constraint or did all of the previous ones work without this issue coming up?

  2. Sam
    Have you seen the beautiful origami documentary called Between the Folds? The Demaine’s are featured in that film. I show it near the end of the year – after the AP test – to show some intersections in art and math.

  3. This! This belongs in the book (Playing with Math)! You have gotten me excited. I hope we can play with this at TMC. Now I wish I taught geometry. Hmm…

      1. What public speaking? You just hand out the paper, like you did with your students, and encourage us while we go wild.

  4. Hey Sam, my friend and I have been following your enthusiastic blog for a few weeks now. We are constantly amazed by your effort of finding new creative methods to catch the attention of your students!!

    We are both students from a German University and are very grateful to have come across your blog. Otherwise we would have missed this great opportunity to share our ideas on the topic.
    Unfortunately, German schools often lack the tendency to work with more vividly descriptive and understandable exercises. Of course they try, but it is obviously not enough when we compare your ideas. We would be very thankful if you could tell us where you get your inspiration from? Do you have any other tips for us?

    We are very much looking forward to hear your reply! :)

    1. Howdy Sarah and Alex!

      Are you students working towards teaching mathematics, and thus interested in where I get my inspiration for lessons? Or are you talking about inspiration when approaching and solving math problems?


  5. Hi Sam,

    We are very excited to hear so soon from you! Well, we don’t consider us experts of maths just yet, but yes we just began our math studies and already have the urge to improve our exercise courses :) we were inspired and even a little bit jealous of your great variety of differnt exercises ;) Our courses sometimes give the impression that they could use a few new strategies to really get their students excited about maths. Exactly you might have a few good ideas about how to create a profitable lesson from which everyone gains something. It is so frustrating sometimes leaving the lecture hall and not really feeling satisfied. For example, how would you start your lesson? Or how would you approach difficult exercises? Do you give guidance to your students while they try to solve the Tasks? Or do you leave them rather to themselves and then talk about the solution afterwards?

    We are still hopeful to spice our lessons a little bit more up! :)

    Can’t wait to hear your ideas!!

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