MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

March 18, 2007

Chasing Arrows

Recognize the symbol on the garbage bin? It looks like three bent arrows chasing each other around a triangular loop. You often find it printed on cardboard cartons, envelopes, greeting cards, packages, and trash containers. It stands for recycling.

It's also an intriguing mathematical object. Imagine joining the tip of one of the arrows to the tail of the next all the way around to create a continuous band. What's unusual about the resulting shape?

You can make the same shape from a long strip of paper. Just twist one end halfway around before gluing or taping the strip's two ends together.

Although the resulting band looks like it has two sides, there's really only one connected surface. Try drawing a line down its center until you return to your starting point. You won't be able to find a "side" that has no line. Cut the strip along the line you have just drawn. You end up with a new band twice as long as the original. It has two twists—and two sides!

The one-sided object is called a Möbius strip, named for the German astronomer and mathematician August Ferdinand Möbius, who discovered it in 1858. When you lay it down and flatten it into a triangle, you end up with a shape that has the same geometry as the recycling symbol.

Möbius strips even have practical value. Take a look at the belt that drives a car's radiator fan. Ordinarily, friction wears the belt out more quickly on the inside than the outside. But if a belt were made with a half twist, like the Möbius strip, it would have only one side and wears more evenly and slowly.

If you examine printed recycling symbols, you'll sometimes see a version that doesn't look like a standard Möbius strip. How could that happen?

One possibility is that someone drew just one bent, twisted arrow, made two copies of it, and put the three arrows in a triangle pattern.

In this case, the chasing arrows form a band that includes three half twists instead of just one. If you were to lay a string along its edge until the ends met and pulled the string tight, you would end up with a knot in the string. If you did this with a standard Möbius strip, you wouldn't get a knot.

You never know what sort of cool math you'll find in a trash heap or anywhere else around you.

Muse, January 1999, p. 27-28.