The Fold and Cut Theorem

Okay so let’s say you folded a paper in some arbitrary way (so that the paper is flat). What shapes could you create on the original paper by cutting a SINGLE STRAIGHT LINE into the folded paper?

For instance, a star can be made by making the following single cut. (This is how Betsy Ross cut stars for the original US flag)

Well, as it turns out, ANY shape made of straight edges can be created via this “fold and cut” method. This includes all polygons, as well as any series of nested polygons (the regions don’t have to be connected!). Any shape whatsoever that is just made of straight edges can be made by just folding a paper flat in a certain way, and making a single straight cut.

Here’s a video of making a swan in this way:

And here’s an image of the fold lines needed to make a turtle:

Interestingly, as early as the 18th century, the Japanese wrote in books how to create certain shapes by this method. But, it wasn’t until 1999 that a formal proof showed any shape with straight edges could be produced with this method. The theorem is known as the “fold and cut” theorem.

The algorithms that compute, given a specific shape, how to fold a piece of paper such that the shape can be produced with this method are fairly complex, but a basic approach is as follows:

Given a goal shape made of straight edges, fill the paper with disks, such that no disks overlap, such that the space between all disks touches either 3 or 4 disks, such that there is a disk center at each verges on the original shape, and such that all edges in the original shape can be represented by the Union of some radii of the disks. An image demonstrating this is below (blue is goal shape)

If you then add edges between centers of adjacent disks, you get a collection of triangles and quadrilaterals, which form the basis of the edges you must fold.

This method is called the “disc packing method”

There is another method called the “straight skeleton method” which can also be used.

If you’re interested, you can see a lot of online tutorials on how to create random cool shapes using the fold and cut theorem. Check it out :)