One-Cut Angelfish

By Ivars Peterson

Muse, July/August 2003, p. 27.

You probably know how to make a lacy snowflake, a chain of identical spruce trees, or a line of paper people by folding paper and cutting some notches out of the folded wad. Ah, but do you know the one-cut angelfish? It's a paper cutout that will amaze and astound your friends.

Intrigued by paper cutting, computer scientists Erik and Martin Demaine and Anna Lubiw wondered what sorts of shapes it would be possible to make by folding and just cutting once. To simplify things a bit, they assumed that the shapes would have straight edges. A mathematical figure with straight edges is called a polygon. It can be as simple as a triangle or as complicated as a lacy star.

Remarkably, the researchers proved that after just the right set of folds, any straight-line drawing, or polygonal shape, can be cut out of one sheet of paper by a single straight cut, no matter how complicated the shape may be.

The hard part, however, is figuring out how to fold the paper properly and then knowing exactly where to cut it to get the design you want. Demaine and his coworkers have come up with a procedure for converting a design into fold-and-cut instructions.

Although the procedure can get pretty messy, with all sorts of tricky folds, it works, and the researchers have invented many new fold-and-cut designs, including beautiful angelfish, swans, butterflies, turtles, and fancy stars. There's even a way to do your own name or initials in block letters!

Angelfish Instructions: For easier folding, lay the pattern over another piece of paper or on a sheet of cardboard. Using a ruler and ballpoint pen, trace the pattern to score the paper. The dotted lines are "valley" folds and should be folded toward you. The dashed lines are "mountain" folds and should be folded away from you. Once you've made all the folds, you have to "collapse" the completely folded form into a compact wad. This is the frustrating step. Then one snap of the scissors should suffice to cut all the bold lines at once.

For other one-cut cutouts, go to http://theory.lcs.mit.edu/~edemaine/foldcut/examples/.

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