I was flipping through James Tanton’s Solve This last night and found a project I thought would be fun. I had no idea!
We’ve done a few paper cutting projects before. For example:
Tadashi Tokieda’s “World from a Seeht of Paper” lecture
and this list also need this incredible project from Joel David Hamkins:
Math for nine year olds: fold, punch and cut for symmetry
So, having see a few folding / cutting project for kids previously the projects about Möbius strip in Chapter 8 of Tanton’s book caught my eye. As I said above, though, I had no idea how cool this project would turnout to to be!
We started with the standard project of cutting a Möbius strip “in half”. What happens here??
Oh, and before getting to the move – most of the movies below have a lot of footage of us cutting out the shapes. I was originally going to fast forward through that, but changed my mind. The cutting part isn’t that interesting at all, but I left it in to make sure that anyone who wants to repeat this project knows that the cutting part (especially with kids) is a tiny bit tricky. You have to be careful!
Now, once we’d done the cut the boys were still a little confused about whether or not the result shape had one side or two. I thought it would be both important and fun to make sure we’d resolved that question before moving on:
With the Möbius strip cutting out of the way we moved on to what Tanton describes as “a diabolical Möbius construction”. All three shapes start as a thin cylinder with a long ellipse cut out. You then cut and twist the strips outside of the ellipse making a Möbius strip-like component of the new shape. Hopefully the starting shapes will be clear from the video.
Try to guess what the resulting cut out shape(s) will look like prior to them getting cut out 🙂
The first shape involves putting one half twist into one of the strips left over after cutting the ellipse out of the cylinder.
The second shape also starts as a cylinder with a long ellipse cut out. This time, though, we make put half twists (in the same direction) in both of the long strips that are left over after cutting out the ellipse.
Sorry about the camera being blocked by my son’s head a few times – oops!
The final shape for today’s project is similar to the second shape, but instead of two half twists going in the same direction, they go in opposite directions.
All I can say is wow – what an incredible project for kids. Thanks James Tanton!!
Mike's Math Page 
I’ve had a hard time getting a copy of this book, so really appreciate these projects.
I paid a fortune for it on Amazon, but it looks as though there are several copies for sale now for under $50 (though shipping to you may cost more).
There’s also a “print on demand” option at the MAA:
http://www.maa.org/press/ebooks/solve-this
I’m pretty sure the last two resulting surfaces define isotopic embeddings into R^3.
