Mathematical Paper Folding Exhibit

- Introduction
- Interview with Dr Robert Lang
- Single Cut Origami
- Curved Crease Origami
- Business Card Menger Sponge
- Links to Other Sites and Resources

Single Cut Origami

Take a piece of paper and make a bunch of folds so that the final construction is laying flat. Now with one straight cut, crop across all the folds. What are the forms you can make this way?

The origami purist eschews the blade, limiting himself to the activity of creasing, but in single-cut origami that purism is augmented by a lone slice. Before Houdini turned his talents to escape, he astonished audiences by making a single-cut five-pointed star. Another magician, Gerald Loe, explored the problem in greater depth and in his book, Paper Capers, detailed how with one cut he could fold paper to produce any specific letter of the alphabet. How universal is this method? Is there a folding in which a single cut will produce, say, the form of a fish or the figure of a unicorn?

It turns out the answer is yes. In 1997, three researchers in Canada (Erik Demaine, Martin Demaine and Anna Lubiw) proved that any polygonal shape can be produced from a sheet of paper by a single cut. The more complex the form, the more folds you need, but theoretically there is no limit. We can even produce multiple shapes – it’s easy to make one star, but if you know how to fold the right way a single cut will yield two stars, or five stars, or a hundred stars. Letters of the alphabet may also be combined so that a properly folded single-cut could produce any given word or phrase – your name, the make of your car, or the entire text of King Lear.

Dr Robert Lang's single-cut model consisting of the letters IFF

In the following single-cut models, the solid black lines represent “mountain folds,” which should be creased with the peak pointing up; the dotted green lines represent “valley folds,” which should be creased with the peak pointing down. The final model should lay flat before you make the cut, which is indicated by the A----B symbol. It can be a little tricky at first to get all the creases in places, but gentle nudging and a bit of experimentation should produce the desired result.

Download pdf of a single-cut star designed by Dr Robert Lang.

Download the pdf of a single-cut model of two stars designed by Dr Robert Lang.

At a lecture for the Institute in Fall 2004, Dr Robert Lang presented a single-cut model consisting of the letters IFF.
Download the pdf of IFF single-cut model.

For Further information on the single-cut problem see
Erik Demaine’s “Folding and Unfolding” webpage

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