You Can Cut Any Straight-Sided Shape From One Sheet of Paper in One Cut

In kindergarten, you probably folded up construction paper and used those frustratingly dull craft scissors to slice up some snowflakes. If you remember doing a lot of chopping before your final result, you were definitely taking the long way: You could've done the whole thing with a single cut.

Magical Math

The fold-and-cut theorem states that you can cut any shape with straight sides from one sheet of paper with a single cut. Every polygonal shape. Even shapes with diagonals and shapes with holes in the middle. Anything. If it sounds more like magic than math, a man named Harry Houdini might agree with you: he cut a five-point star out of paper with a single cut as part of one of his acts. In his 1922 book "Paper Magic,"* he explains how it's done. Legend has it, Betsy Ross had this theorem in mind when suggesting a five-point star for the U.S. flag instead of the six-point stars originally requested by George Washington. This tale was published in a 1873 article of Harper's New Monthly Magazine.

But the theorem itself goes back even further than these historical figures. The first published reference we know of is from a Japanese puzzle book, "Wakoku Chiyekurabe" (Mathematical Contests), from 1721 by Kan Chu Sen. It wasn't until relatively recently that the theorem was given a formal space in mathematics, however.

The fold-and-cut problem is the corresponding problem to the fold-and-cut theorem, which asks: What shapes can be obtained by the so-called fold-and-cut method? Mathematicians Erik Demaine, Martin Demaine, and Anna Lubiw formulated a proof of the fold-and-cut theorem in 1999. The answer, as previously stated, is any and all straight-sided shapes. Their solution was found using the "straight skeleton" structure, which focuses on capturing the symmetries of shapes. A second solution and algorithm for the problem is the newer "disk-packing" approach, which includes placing disks on top of paper. This method is more practical, seeing as the desired cuts are the union of radii of disks. If you're getting lost, the video below can illustrate.

One and Done

Since you're probably not going to go through all the trouble of either measuring out the symmetries in a polygon or laying down some disks, there are plenty of resources that offer folding templates. Using the fold-and-cut theorem, you can cut a swan, angelfish, butterfly, and any other straight-sided concoction you have the patience to dream up and carve out. Click here to see how to make the creatures we just mentioned out of paper with only one magical cut.

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Fold-and-Cut Theorem

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Written by Joanie Faletto February 2, 2018