Saturday, May 07, 2005

The Möbius strip

Today, I decided to do a little Möbius strip project that I have been thinking about for a while. While I was doing this project, I realized that I didn't understand exactly how the Möbius strip works, so I decided to do a little experiment.

("What are you doing?" asked my S.O. "I'm doing a math experiment," I said. "There are no math experiments," said my S.O., "math is not an experimental science." "Then I'm performing a miracle," I said.)

I took a strip of paper and taped it together to make a Möbius strip, with one half-twist. Then along the left side of the strip, I made a purple line, and next to it on the right side of the strip, I made a yellow line. Note that this means that along the whole strip, there were adjacent purple and yellow lines, and left and right mean nothing; they're just to orient you.

I was wondering what would happen to the lines when I cut the strip the long way down the middle, because I figured that it couldn't possibly just end up with the yellow line switching to purple or something; it had to be that the yellow line would be on one side and the purple line would be on the other. And that is what happened! Wow.

So I figured the next thing should be to see what happens when you add more stripes. So to the right of the purple line I drew a blue line, and to the left of the yellow line I drew a red line, both all the way along the whole strip. So now on one side of my strip there were parallel blue and purple lines, and on the other strip there were parallel yellow and red lines.

("Logical fallacy," you say: "A Möbius strip has only one side." Correct, but once you cut it in half the long way, it ends up having two sides, and it's not a Möbius strip anymore. Pity, I know, but it's the truth. Instead of one half-turn, a cut-in-half Möbius strip has two full turns (four half-turns), I think.)

The idea of doing this was that you could not end up with red, yellow, purple, and blue each on their own side, because there are only two sides. So something drastic had to occur. This is probably common knowledge, but here is what happens when you cut an already-cut Möbius strip in half again: You get two interlocking twisted strips! They each have four half-turns. And they're not just singly linked, like the Olympic Rings, but are kind of doubly linked.

Naturally I tried cutting them in half again, and I got four rings which are all interconnected in a kind of knotted-up way. That wasn't very interesting, but I knew you'd be wondering. I should try it with a bigger piece of paper next time. For a more interesting experiment, try this, suggested by Mathworld (link below): "In addition, two strips on top of each other, each with a half-twist, give a single strip with four twists when disentangled." It's true! It happened to me! Whoa.

For anyone who is very interested in the Möbius strip (and you had better be interested, Ms. 3.95) check out the animated Möbius strip on Mathworld. You have to have some sort of Java to see it, I think, but if you do, you can click and drag to spin it around, and if you kind of drag and let it go as though you are throwing it, it will spin in the direction in which you threw it, so long as you keep your cursor within the boundary of the graphic. Awesome. I think that I was talking about this awesome graphic when someone gave me the idea for the project that inspired me to investigate the particulars of the Möbius strip -- a project that I will soon be sending to you, Ms. 3.95, which is why you had better be interested, because today, I decided to do a little Möbius strip project that I have been... yes, you see, it all comes together at the ends, just like a Möbius strip.


Becca said...

I hope you are not referring to me when you say Ms. 3.95, because I go by no such false name... Only the math part of me is Ms. 3.95, and the other Miss 3.71 part of me doesn't like being left out.

Diana said...

In fact, I was appealing to the math part in this post, which is why I used the title Ms. 3.95.

Anonymous said...

it's easy to see all this stuff if you use the model of the MS as a square with opposite sides identified. "cut up" your model and see what's identified to what in the quotient space.