You probably remember from your kindergarten school-going days when you use to fold up construction paper and slice ‘em up into some shapes. That awkward moment when you had to keep on cutting and slicing your paper over and over again just to get the right shape. Maybe spoiler: you failed. Did you know that you could’ve done the whole thing with a single cut?
Related media: What Is The Fold And Cut Theorem?
The ‘Mathemagical’ Cut
The fold-and-cut theorem states that you can cut any shape with straight sides from one sheet of paper with a single cut. Yes — every polygon, and even diagonals and shapes with holes — like any shape whatsoever. Anything! That sounds pretty much like magic than math, right?
The legendary magician Harry Houdini might agree with you if you thought as much; he once cut a five-point star out of paper with just a single cut in one of his stunts.
He later explained how its done in his 1922 book ‘Paper Magic.’ Legend says, Betsy Ross once knew of this theorem when he suggested a five-point star for the United States flag, instead of the six-point stars George Washington originally requested. This tale was published in an 1873 article in Harper’s New Monthly Magazine.
However, the theorem dates back even further than these historical figures. The first account of the theorem was from a Japanese puzzle book from 1721 by Kan Chu Sen, “Wakoku Chiyekurabe,” which literally translates as “Mathematical Contests.” No surprise. But it wasn’t until relatively recent times that the theorem was grounded in formal mathematics.
It’s Just A Cut
The fold-and-cut theorem has led to the corresponding problem of the fold-and-cut problem, which asks: What shapes could be cut by this so-called fold-and-cut method?
In 1999, mathematicians Erik Demaine, Martin Demaine, and Anna Lubiw, came up with a formulated proof of the fold-and-cut theorem. The answer? As aforementioned, is to cut any straight-sided shape with just a single cut.
Their final solution was based on the “straight skeleton” structure, which captures the symmetrical structure of a shape. Another solution and algorithm, and a more recent method is the “disk-packing” approach — which involves placing disks on top of the paper. This is a more practical method by cutting the union of radii of disks.
The video below can give you a better illustration.
One ‘N Done
You’re probably wondering: “How on Earth in the complex but simple name of math would I be able to measure out the symmetries of polygons just to cut any shape with just a single cut?” That’s one heck of a question, but don’t worry.
There are plenty of resources that offer folding templates. With the fold-and-cut theorem, you can cut an angelfish, butterfly, crane, and any straight-sided figure out of paper. To learn how to cut the creatures we just mentioned, click here. It’s just a cut away.
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Written by: Nana Kwadwo, Thu, Apr 18, 2019.