The Fold and Cut Problem

Imagine that you have a sheet of paper in front of you, and that you draw a polygon on the paper. It can be as simple or as complex as you like. Now... is it possible to fold the paper in such a way that you can make a single, straight-line cut with a pair of scissors, in such a way that when you unfold the paper, you have exactly cut out the polygon that you previously drew?

What about if you draw a number of separate polygons on your sheet of paper? Can you still fold the paper so that with a single, straight-line cut with a pair of scissors, you can cut out ALL the polygons that you have drawn?

For instance, consider this drawing of a Jack-o'-Lantern. There are two triangles to represent the eyes, one for the nose and one for the mouth. Assuming this was drawn on an A4 sheet of paper, could you fold the paper so that with a single cut you could cut out the eyes, nose and mouth of the Jack-o'-Lantern?

Amazingly enough, the answer is yes. In fact, there are references to this type of folding and cutting as far back as the 18th century.

More recently, Eric Demaine of MIT has published work on this area. See his article and also a number of examples that you can download and try for yourself.

Another article that talks about this topic is called 'The Origami Polygon Cutting Theorem' by Eric Biunno.

While this may all seem quite theoretical, albeit fun for students of advanced origami, one application of the underlying mathematics turns up in the science of fitting roofs onto irregularly shaped buildings. See the article by David Bélanger of McGill University.

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