Topology With Paper and Scissors

Yesterday, I introduced an old friend—the Möbius strip—in the context of topology, the study of objects independently of their size and specific shape. Since I’m currently away from teaching for the longest period in my life since 1999, I even assigned homework, which (typically) none of you did. I am very disappointed with you all. You’ll pay for your laziness when the final exam rolls around.

However, I’m a kind soul, so here are the “homework” solutions, with bonus mathy goodness thrown in. I used white typing paper, so unfortunately the pictures may not be as clear as would be ideal.

1. “Make both types of Möbius strip, and show that you can’t turn one into the other without cutting or breaking the tape—just like your right and left hands aren’t equivalent.” The images below show construction of one type of Möbius strip, and then the completed left- and right-handed versions. I’ve colored one edge of the paper to guide, but I think you’ll be happier if you just do it yourself.

Making a Mobius strip in the version where the right edge is turned toward you. I stapled the finished product because I don't remember where I put the tape.

The left- and right-handed versions of the Mobius strip, showing chirality.

2. “Naively, you might have thought that you could make a sphere by just taping the top and bottom of the paper together. Here’s the square representing that situation, which also is the universe of the classic video game Asteroids(and its innumerable imitators). Figure out what shape this actually makes and leave your answer in the comments.” This shape is a little hard to make, so I decided to try a trick: cutting little squares out of the top and bottom. I think it will work better with a bit more experimentation, but the answer is…a torus, AKA donut shape. As you can see from the figure, that’s the same as making a cylinder, then connecting the top of the cylinder to the bottom.

My first attempt to make a torus wasn't 100% successful, but you're probably able to figure out an improvement on my design. I started by cutting squares out of a strip of paper, then making a cylinder. The final step connect the ends of the cylinder together, which in my case wasn't very donut-like (alas). In the game Asteroids, leaving the screen on any side brings you to the same point on the opposite side, which means the game takes place on the surface of a torus! (Pac-Man, on the other hand, takes place on a cylinder.)

3. “Now for a really fun one, which you won’t be able to do with paper, but perhaps can figure out anyway. Imagine taping the left and right sides together like a cylinder, but taping the top and bottom together like a Möbius strip. Draw the result (or locate a picture online that illustrates it), and leave your answer in the comments.” The shape you get is called a Klein bottle, and the ideal geometrical version would involve passing paper through itself. However, my bloggy colleague Greg Gbur pointed out a way to construct this one with paper, if you don’t mind that it isn’t strictly a perfect version. My Klein bottle is shown below.

To make a fake Klein bottle, cut a piece of paper into a cross shape, with one leg signficantly longer. Use that long leg to make a Mobius strip, then use the remaining two legs to make a cylinder around the Mobius strip. A "real" Klein bottle doesn't have edges like this, so the Mobius strip part would pass through the surface of the bottle at some point - it's mathematically just fine to do that, but not really possible to make using paper!

Again, I think you’ll be happier (and possibly more successful) making these shapes yourself. The fun in this stuff is never in looking at what others have made—go and mess around, and if you create something awesome-looking, send me a message and I’ll post it in a follow-up article!

Bonus Features

4 Responses to “Topology With Paper and Scissors”

1. December 29, 2011 at 16:04

Ahhhhh, that’s how you do it (use a stapler!!). I was confused! Thanks Dr. Francis

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