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The Möbius Strip: Math with a Twist

By Julie A. Swanson

Art by Patrick Wayne Britten

Want to experience something mind-boggling in the next few minutes? Give a strip of paper a half twist and tape its ends together.

You have just made a Möbius (MUH-bee-us) strip, a fascinating object created by German mathematician Augustus Möbius.

What's so mind-boggling about it? Unlike an ordinary band, which has two sides and two edges, the Möbius strip has only one side and only one edge! Don't believe it?

Imagine an ant crawling around the inside of an ordinary band. If it wanted to get to the other side, the outside, it would have to cross over an edge to get there.

But on a Möbius strip, the ant should be able to travel over the entire surface of the band without having to cross an edge.

To see for yourself, draw a line lengthwise around the Möbius strip. Notice that you encircle the whole thing. You return to where you started without lifting the pencil from the paper. By this time, you've drawn one continuous line over the entire strip, over what appeared to be both "sides" which you now know must be really only one side, since you never had to lift your pencil.

To show that the strip has only one edge, use a marker to color along the very edge of the band. You will return to your starting point without having to lift the marker from the paper. To color all the edges of an ordinary band, you would have to lift the marker.

And to see an especially strange property of the Möbius strip, cut it lengthwise with scissors, along the center line.

Instead of getting two thinner bands as you might expect, you get one band that is twice as long as the original!

Now you can see why scientists, mathematicians, artists, writers, and many other people are amazed and delighted by this curious case of math with a twist.

More Möbius Strip Tricks

~ Make a second lengthwise cut around the Möbius strip you have already cut. What happens this time?

~ Make a strip with two half twists. Is it a Möbius strip? Cut along its center and see what you get.

~ Make a strip with three half twists. Draw a line down its center to test if it is a Möbius strip. Cut along its center to see what happens.

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