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MatheMUSEments
Chasing Arrows
By Ivars Peterson
Muse, January
1999, p. 27-28.
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.
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