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# What is a Mobius Strip?

If you took a strip of paper and taped the ends together, you would most likely end up with a belt. It would be a loop with an inside surface and an outside surface. But what if you took that same strip of paper and gave it a half twist before taping the ends together? The result would be a fascinating geometric oddity called a mobius strip.

A mobius strip is an example of non-Euclidean geometry made real. Most of the time, non-Euclidean designs can only be imagined, or drawn like optical illusions. They could never exist outside of an M.C. Escher dream world. Yet the mobius strip is indeed a three dimensional object with only one side. The strangeness doesn't end there, however.

To construct a mobius strip, you will need a length of paper at least two inches wide for best results. A strip of newspaper cut lengthwise will suffice. Take the two ends of the strip in both hands and give one end a half-twist. Bring the two ends together and bind them with tape.

What you should have is a belt of paper with a half-twist. This is now an official mobius strip. Find a pair of scissors and a marking pen to perform the rest of the experiment.

The first principle to demonstrate with a mobius strip is the concept of a single surface. Using a marking pen, start drawing a line down the middle of the mobius strip without stopping. Your continuous line should eventually meet up with your original starting point. This proves that the mobius strip does indeed only have one side. Performing the same action on a normal paper loop would only mark the inside or outside surface.

Using the scissors, cut along the line created by the pen. Instead of becoming two separate loops, a mobius strip will form a single loop twice as large as the original. Cutting down the new mobius strip will result in two interlocking loops. If you use a wider strip of paper, the mobius strip will continue to form continuous or interlocking loops. You can also vary the experiment by cutting the loop into three equal sections or sections of varying lengths.

A mobius strip is an excellent way to introduce students to the worlds of science and geometry. The experiments are simple enough for young children to perform, but the science behind the illusion should fascinate older students as well.

## Discussion Comments

This is mind blowing.

Because mobius strips only have one side that repeats over and over again if you were to follow its path, I wonder if maybe there would be a way to utilize that somehow.

I don't know what it would be exactly. I have seen videos, though, of people who make mobius strip music. There are small "music boxes" you can buy that use a strip of paper with holes punched into it to make sound. The concept is the same as a player piano or record player. If you twist the paper into a mobius strip, the song will play in a loop until you stop it. Like I said, though, I don't know how you would turn this into a practical use.

@matthewc23 - I'd be interested in someone explaining more about non-Euclidean geometry, too. As far as uses of a mobius strip go, I would say its shape really limited the way it can be used in functional things like machinery or tools.

Mobius strips have a very striking shape, though, and I have seen them used all the time in artwork and things like mobius strip bracelets. As a matter of fact, I was just at a museum not too long ago, and they had a sculpture garden outside. Someone had used a huge metal beam and twisted it into a mobius strip. It was an interesting piece.

@titans62 - I had the same thought you did until I actually got a piece of paper and tried it for myself. Because of the twist in the middle, the ends of the line do not meet up after the first pass around the mobius strip. You have to go around twice, and then the lines do match up, and you have a mark on both of the original sides of the paper. I tried out the rest of the experiment, and both of those parts work, too.

What is the significance of non-Euclidean geometry? I am not a math person by any means, so I'm sure I don't even fully grasp why a mobius strip is important. It is just a neat experiment as far as I am concerned.

Do mobius strips have any sort of practical use in daily life? Is there anything that uses the shape?

I have always seen mobius strips, but never understood the significance of them. The seem to be a popular display in math departments.

I don't have any tape around me to try out the experiments that were described in the article, but I don't understand how you can prove a mobius strip has only one side by drawing a line on it. Even if you drew the line, you would still have an empty side. Wouldn't that prove that it is two sided?

I don't know about cutting down the middle and what effect that would have, so I'll have to assume the article is right. That kind of mental thinking is a little too complex without having a mobius strip in my hand.

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