Origami at its finest!
In this video, Katie Steckles, a Manchester-based mathematician, shows how she can cut any straight-edged shape with a single cut. She begins by cutting a square and a triangle, but quickly moves on to cutting out the entire alphabet.
She was inspired by the cut-and-fold theorem that states that any shape with straight edges can be created with a single cut if you fold a piece of paper in the correct manner. It can also be thought of as stating that, for any straight-edged shape, there is a way to fold it so that all of the edges will end up along a single line.
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Houdini, the famous escape artist and magician, described how to make the five-pointed star in his book Paper Magic, published in 1922, so the concept is not new. In fact, mathematicians have been looking at the problem since 1721.
The set of lines that indicate where to fold the paper is called a crease pattern or plane graph. You may remember this type of graph from our article on the Seven Bridges of Königsberg. In this type of graph, pairs of edges must intersect at a shared vertex. In other words, you cannot have a line segment that ends without intersecting with another. Lines that have to be folded inwards with reference to the top of the page are called valley edges whereas those that are folded outwards are called mountain edges.
In this paper folding, as with all origami, some rules have to apply. Every crease must line up with another when folded — the paper cannot be stretched to make this happen — and the folding must not result in any crossing.
If you want to try some of your own paper folding and cutting, there are some fun images here with the folds pre-traced to help you out.
