Continuing on last week’s post, there were a couple of hands-on activities led by Joseph O’Rourke at the geometry minicourse that I attended. In one of them, you start by printing out the following sheet of paper (available at

The red and green lines indicate mountain and valley folds and at this point your job is to follow all of the folds, like in origami. With a little bit of playing around it will look like the following. The left shape is simply creased, and the dreidel-shaped object at right is the fully-folded version.

As an aside, the hands-on activities were a fantastic part of the mini-course. One trivial reason is that it was a long break from lectures. But a deeper thing that happened is that people were struggling, being confused, eventually gaining some insight and getting it to work, and then playing around! A good lecture is very seductive but there’s a limitation to the lecture format in that, without exercising your own mental muscles, it is difficult for the material to “stick” or hook in to your existing corpus of knowledge. In my Game Theory courses/math circles there’s the opportunity to actually play games which was fun. But for me, this was the first time doing origami in a mathematical setting.

Anyway, back to the task at hand. If you notice, there was a black line outlining the original A. If you have done your job correctly, all of that line will be neatly folded so that with one scissor-cut, you will chop the outline of the A, as well as the hole in the centre!

Not too bad, eh?

Unrelated joke. When they were naming the northernmost part of North America, the founders were very much inspired by the USA. It would be so great to have a three-letter acronym like USA! So they put all of the letters of the alphabet in to a hat, and the mayor pulled them out, one by one. “C, eh? N, eh?, D, eh?” And so, Canada was born.

Inspired by the alphabetic shenanigans, I tried to do a free-hand maple leaf afterwards. Here is how it turned out (left):

Is it a recognizable attempt? I would like to think so. But there is a more important question here. What if we also wanted to cut out the red bars on either side of the maple leaf? What if we wanted to do the US flag with all of its stars and stripes? How about an arbitrary line drawing? This elegant question was solved, then discovered to be buggy, and then fixed by two pairs of researchers (history here):

Theorem (Bern-Hayes ’09, Demaine-O’Rourke ’07). For any planar graph (made of straight line segments) embedded in a rectangle, there is a way to fold the rectangle, so that a single straight-line cut will exactly cut out that graph.

Pretty amazing! The proof is quite heavy, as indicated by the bug. At the same time, the methods are quite elegant, and there are two different proof methods.

Let me just briefly mention the other activity we participated in. Take an ordinary rectangular piece of paper. You can fold the left-centre point on to the right-centre point, and get a rectangle. Similarly, folding the top-centre point on to the bottom-centre point gives a different rectangle. But you can fold other shapes, in fact 3D shapes, out of this humble piece of paper! Start by folding any boundary point X on to the boundary point Y that is directly opposite. Then, you should think of the remaining edges as a “zipper” and tape them together both ways from the X-Y point. With a little patience, the paper will turn into a 3D polyhedron with triangular faces.


One Response to “Fold-Fashioned”

  1. 1 Imre

    You many know it already, but there is a great documentary about folding:

    Between the folds

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