It's hard to miss the triangle of three bent arrows that signifies recycling. It appears in newspapers and magazines and on bottles, envelopes, cardboard cartons, and other containers.

But have you noticed that there are two versions of this ubiquitous symbol? The difference between them lies in the direction of the twist in one of the three chasing arrows that make up the figure.

The late Cliff Long, who was a mathematics professor at Bowling Green State University in Ohio, had an eye for such details.

In his mathematical studies of surfaces, Long often found it inspiring to carve certain forms out of wood. "The carvings . . . significantly improved [my] understanding of the surfaces . . . and the mathematics behind them," he once remarked. His first sculpture, carved more than 30 years ago from a piece of white pine, was of a mathematical figure known as a Mobius band.

A Mobius surface has only one side and one edge. You can make a Mobius band by gluing together the two ends of a long strip of paper after giving one end a half twist.

Alternative (incorrect?) rendering of the recycling symbol.

Some time ago, Long noticed that the arrows of the usual symbol for recycling are twisted in such a way that if they were joined together in a continuous ribbon, they would form a Mobius band. Then, one day, he happened upon a second version of the recycling symbol, boldly printed in color on the front page of the Toledo Blade, that differed from the one he had originally encountered. The new version aroused his interest, and he carefully compared it with the one that was already familiar to him.

Long found that the alternative recycling symbol was based on a different surface -- a one-sided band formed by gluing together the two ends of a long strip of paper after giving one end three half-twists instead of just one.

Cliff Long made a Mobius band the basis of his wood carving "Bug on a Band."Photo by David Hampshire, Bowling Green State University

If you were to lay a string along the strip's edge until the string's ends met and pulled the string tight, you would end up with a trefoil knot in the string. If you did this with a standard Mobius band, you wouldn't get a knot.

In general, all bands with an odd number of half-twists (and their mirror images) are, roughly speaking, one-sided surfaces. Bands with no twist or an even number of twists have two sides. Topologists generally apply the term "Mobius band" not only to the standard form (one half-twist) but also to the symmetric version (three half-twists) and anything else "homeomorphic" to the standard form.

All this made me curious about the origin of the recycling emblem. It started with a contest sponsored by the Chicago-based Container Corporation of America (CCA) as a special event for the original Earth Day in 1970. Art and design students were invited to create a symbol to represent paper recycling. The winning logo, selected from more than 500 entries, was submitted by Gary Anderson, then an art student at the University of Southern California.

"The figure was designed as a Mobius strip to symbolize continuity within a finite entity," Anderson recounted in an interview published in the May 1999 issue of the trade magazine Resource Recycling. "I used the [logo's] arrows to give directionality to the symbol. I envisioned it with the small edge or the point of the triangle at the bottom. I wanted to suggest both the dynamic (things are changing) and the static (it's a static equilibrium, a permanent kind of thing). The arrows, as broad as they are, draw back to the static side."

Anderson's original design was then refined by Bill Lloyd, CCA's public relations department manager. He sharpened the lines and rotated the symbol so that the stylized outline of a tree can be seen in its center.

Initially, CCA licensed the design to trade associations for a nominal fee. The company later dropped its application to register the logo as a service mark, leaving it in the public domain. In the 1970s, the American Paper Institute and the American Forest and Paper Association started promoting use of this symbol to describe recyclable and recycled paper products. Its use spread rapidly and expanded to many other items.

Long discovered that many people are unaware that two different forms of the recycling symbol are now actually in use. Where did the second version come from?

Perhaps it was introduced accidentally, when someone failed to notice that the direction of the twists in the arrows makes a difference. One possibility is that an illustrator drew just one bent, twisted arrow, made two copies of it, and put the arrows in a triangle pattern, never realizing that the original symbol was meant to conform to the shape of a standard half-twist Mobius band.

What's fascinating about the entire recycling-symbol episode is how a geometric shape that came out of pure mathematical research, done in the 19th century by August Ferdinand Mobius (1790-1868), has become a modern cultural icon.

Recycling Topology