## MatheMUSEments

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

## June 25, 2007

### The Beauty of the Bag

The humble brown-paper bag that you use to carry groceries is actually a technological masterpiece that solves many practical problems. Unlike a plastic bag, it can stand upright by itself; you don't need an extra hand to hold it open while you fill it. Yet it folds flat for easy storage.

A standard brown-paper bag can stand upright, as shown above.

It took inventors years to come up with a design that would behave this way. Early paper bags had "envelope bottoms" and wouldn't stand up at all. Then in 1867 Margaret Knight invented a machine that could make the standard bag's rectangular "satchel bottom" in a series of three folds. Another inventor added the accordion folds on the sides of the bag.

The folded form of a standard brown-paper bag lies flat (above).

A company near Savannah, Georgia, now makes 35 million paper bags per day, or about 9 billion per year. That's more than 100 bags for every family in the United States.

A paper bag can be easily folded and unfolded because it is creased. But these creases are more peculiar than you'd think. Suppose you made a bag from a very stiff material, such as steel, and that each crease was some sort of hinge. If the steel bag started out flat, you wouldn't be able to open it, and if it started out open, you couldn't fold it flat. The bag could open or close only if the material could bend, and steel doesn't bend.

The red lines in the diagram (above) show where a paper bag is creased.

Robotics expert Devin Balkcom of Dartmouth College got interested in grocery bags when he was designing a robot that could do "rigid origami," origami folding where the paper stays flat. You can watch his robot fold a paper airplane at www.cs.dartmouth.edu/~robotics/origami.html. The film clip will make you laugh—such a big machine concentrating so hard on such a simple task.

Balkcomm and computer scientists Erik and Martin Demaine of the Massachusetts Institute of Technology independently observed that rigid grocery bags wouldn't fold and worked together on finding some mathematical solutions to this problem. One is to make the bag very short, so it looks more like a collapsible department-store gift box. That gets rid of the place on the grocery bag where several creases meet, blocking collapse. The researchers also figured out how to fold a tall bag without bending the material, but their method requires a whole bunch of new creases and doesn't look much like a grocery bag.

A 'short' bag gets rid of the place on the grocery bag where several creases meet, blocking collapse.

So the next time you use a paper bag, take a closer look at this wonderful invention. There's a lot to ponder in its simple folds.

Muse, January 2006, p. 19.