Fold and Cut Theorem – Numberphile

Fold and Cut Theorem – Numberphile

I have a pair of scissors ’cause I want to talk about a piece of maths called the Fold and Cut Theorem. It’s one of those things in maths where they just name it what it is. This first came to my attention when I was trying to cut a square hole in a piece of paper. So, like, my first instinct – as a person who cuts holes in paper – was to jab a hole and then cut round the edge of the square. But I thought, ok maybe there’s an easier way to do this, and kind of my mathematical brain kicked in, and I realised that if I fold this in half like that, I can just cut along those three edges. Ok, I’ve roughly approximated the hole I was originally trying to draw. There’s a square hole, and I guess like, the mathematician in me was like, “Can I do that more easily? Can I make this more efficient?” So what I thought then was, if I fold this twice, so I make the first fold the same, and then fold that at 90 degrees to that, I now have this bit of line, and I can now cut two cuts like that and, I’ll end up with a square hole! Yeah, it’s two cuts it’s less than three cuts it’s more efficient but of course you want to optimise this so your mathematically efficient way of doing this with just a square is to start off with one fold in half, a second fold like that, and then your third fold, is a diagonal fold through that corner. So as long as you get all of those kind of folds exactly right, and the lines that you’re cutting along are all lying on top of each other, you can now cut out the entire square, the whole shape, with just one straight cut, like that. And this is the moment of truth, the bit where you don’t really believe what you’ve done has worked, but it turns out a square is pretty much the easiest one to do! It is efficient and it’s one straight cut And this kind of idea like I went to the pub as you do when you’ve discovered a nice thing, and I was like I wonder if this is a thing you can do in general, I wonder what shapes you can cut out with one cut and a friend of mine who was in the pub said isn’t that a theorem? As all mathematicians will tend to do I went and looked up that It is a theorem, it’s called the Fold and Cut Theorem and it says that any shape whose edges are all made up of straight lines, can be cut out with one cut, if you’re prepared to fold a bit of paper. which is just insane! and I love this it was proved by a guy called Erik Demain and a team of other researchers, they’ve proved that it is possible to cut out any straight line edge shape, even if it’s got more than one bit to it, even if it’s got a hole in the middle, you can make one cut and cut out the whole thing! There are various examples of this throughout history So Harry Houdini, the magician, used to apparently do a star with one cut as part of his magic show and it is sufficiently impressive to go in a Houdini magic show, it turns out So I’ve looked that up and I’ve figured out how to do that So it’s in half first, It’s… in half that way and in half that way To get a cross shape And then the fold is there And in theory that should give you 108 degrees-ish You can then fold that one in half Fold that back and fold that And you can tell you’ve done it right ’cause that’s the same size as that, ish And there’s actually a nice little story behind this because apparently when America was founded in the 1770s they needed a flag and they went to see a seamstress called Betsy Ross I don’t know if it’s a true story, could be one of these made-up stories that George Washington walks into a fabric shop and was like “Gonna need a lot of stars cutting out” and she had a piece of fabric and she just grabbed it and folded it up like this and she did the one cut, so it’s from there, kind of up that way and she unfolded this and showed George Washington that she cut out a perfect five-pointed star with just one cut and this, this story goes that he was like “oh well, you can make all the flags then.” And she then had like work for life making all of the American flags and at that stage there were only like a few stars on the flag they kept updating it and adding more stars so she had to remake all the flags so it was quite a decent amount of work just ’cause she happened to know this little trick and I love that, I loved it The most, like the most impressive thing for me is that not only have you got a star, you’ve also got a piece of paper with a star-shaped hole in it That’s a five-pointed star. That’s my favorite bit, is that you get the opposite thing as well ’cause it is just that line that’s cut, it’s nothing else The next thing in the paper that they list in, kind of, things that they’ve seen was that someone had once seen an example of someone who could cut out any letter of the alphabet on demand and I looked this up and tried to find it and I couldn’t find anyone who did this so I was like “right, well that’s… I’m gonna do that, I’m gonna figure out how to do every letter of the alphabet.” And it turns out it’s not even just a cool thing to be able to do it’s actually a really fun thing to work out So you start with the simple letters like L ’cause they’re quite easy, that’s just one, kind of, diagonal fold and then you need to make the ends You can do things like O as long as you don’t mind it being quite square. And then things gradually get more and more difficult and the easier ones tend to be quite symmetrical. So T is quite easy. S is not easy! S is a bit of a nightmare, but it is possible. So I’ve worked out a fold pattern for every letter of the alphabet. Having done this, I then got an email from someone saying, “I’ve just found this pdf with a fold pattern for every letter of the alphabet.” I was like, what!? That already existed. Oh, well, never mind. But, you know, the system that I’ve worked out I’m happy with. It’s slightly different in places. But you know, there are these things out there, but it is more fun to work it out yourself. Well, we’ve got A B C D E F G H I J and K And then, L M N O P Q R S T U V W X Y and Z That’s an A B That’s C Here’s D E F G … that’s the closest I can get to G without any curves H There’s I J my initial … K That is completely different from J … that’s L That one is M there’s no way I’ll get this the right way round … N O P … it’s good enough it’s tiny, but it’s a Q That’s R Brady: “nice!” A bit of a nightmare that one. Brady: “well done!” S Slightly funky T U Slightly ’70s looking V W X … it’s quite a fun one to work out Very happy … that’s a Y It’s looking good from there, isn’t it? So that’s Z Like, working out how to fold that is such a mathematical thing to do. Like, it’s the type of thinking that mathematicians use can be applied to so many situations, and this is exactly one of them. But this is an example of something which also mathematicians have properly used, you know, mathematical analysis on, and they’ve proved a thing about it, which is, you know, for me, that is what maths is about. It’s not about calculating things, it’s about proving things. And it’s about understanding how the universe works. And it turns out that in this universe, one cut suffices. We’d like to thank for supporting this video. Now Lynda’s great. They have a huge range of video courses. I was actually going through the catalog earlier, and came across some great ones about how to use typefaces and typography. This is something I find really interesting. This one here is called The 33 Laws of Typography, and it’s gone right to the top of my watch list. I’ll be having a look at that one soon, I think. But whatever it is you want to learn, whether it’s something creative, techy, maybe just getting better at business, Lynda will have something for you. There’s over three thousand video courses, and one hundred thousand tutorials in the vault. You can sign up now for a free ten-day trial. That’s unlimited access to all those videos by going to or clicking on the link in the description. That’s Lynda, L-Y-N-D-A dot com slash numberphile Give it a try. And our thanks to Lynda for supporting numberphile. I did a show at Cheltenham a couple years ago, and one of the things that I did in it was, as the challenge, like, to pick a person in the audience, ask them what their name was, and then cut out that word while explaining the maths behind this. It was brilliant. I asked for a four-letter word. I said, has anyone got a four-letter name? And this guy shouted his name, and I can’t remember what it was, but it was five letters. And everyone was like, mate, that’s five letters. And it was just a gift, it was hilarious.

100 thoughts on “Fold and Cut Theorem – Numberphile

  1. Does this theorem also hold in higher dimensions? Like, is it possible to turn a cube into any abritrary shape by folding it in the 4th dimension?

  2. Maybe being an American there's something I'm missing. So cutting things like letters and turtles out of paper with one cut is certainly impressive. But this idea that an adult had no idea how to cut a square out of piece of paper is unfathomable. An American kindergartner could demonstrate this.
    Do they not make paper snowflakes where your from?

  3. could u try and line up the different bits of paper correctly and cut out the whole alphabet with literally one cut

  4. So does that mean if you were willing to do all the folding that you could do one cut and get every letter in the alphabet from that one cut?

  5. You need to make up for the less number of cuts with more number of folds so essentially the "work done" is same.
    For example, in a square, no. of folds + no. of cuts = 4 always.

  6. now someone needs to get the letters better so that they are more curved like pixels. no curved lines but just make it like 8 bit letters

  7. When I was around 7 or 8 my dad had some missionaries over for dinner. After dinner they found out I was into origami so they decided to show us a trick (that of course turned into a religious lesson) and his first question to me was: "Can you fold a piece of paper, so if you make only one cut and unfold it, it'll unfold into a cross?"

    So I thought about it for a few seconds, and said "I think so" and proceeded to fold a piece of paper in half twice, and then on the diagonal (like in the video) and snip it in half. Unfolding it it was a cross and the missionary says "Wow, I've never had someone actually do it before."

  8. If you can cut out any two dimtional shape with one cut by folding in 3 demension can you cut out 3 denensional shape with one cut by folding in the 4th deminsion and so on?

  9. Francis Hopkinson (1737 – 1791) designed the first official American flag. He was an author, a composer, and one of the signers of the Declaration of Independence as a delegate from New Jersey.

  10. So we can basically create any polygon with a single straight cut. Now can we build any shape with a single curvy cut 🤔

  11. Can't you create any sort of piecewise continuous shape, given the condition that you don't have to cut a straight line, and so long as all the piecewise functions are similar? Seems a pretty trivial extension of the theorem that might allow G, S, O, and maybe Q to look nicer.

  12. it was stated that you can "one-cut" anything with straight lines. but how about a really big circle? i mean a really big collection of straight lines that it is a circle? im sorry i cant express this well in english.

  13. I've worked out for myself how to cut out all 26 letters of the alphabet in one cut! It was really fun (and really frustrating, especially for R and S)

  14. That's really impressive!
    Now do Å, Ä and Ö.
    Jah just kidding, but that's mighty interesting that a field like that exists in math! ^^

  15. Katie, you can actually improve the one cut square even further, you can do a single cut with only two folds. If you fold it in half diagonally, you get a triangle, and then you fold it in half, and cut on the line, unfold, and you have a square!

  16. Ya it is pretty cool but to make it more intuitive use jordan curve theorem you have 2 regions an inner and outer region. So when lining up the lines your essentially separating the regions into to distinct pieces when you cut.

  17. Does that mean if you could fold 3 dimentional objects through the 4th dimension, you could carve them into any shape with a single slice

  18. Umm… Can't you just fold it along the diagonal twice, then cut once? That would be mathematically more efficient, correct?

  19. wtf? for a square just fold the thing diagonally twice and then make one cut… why would you claim that 3 folds and one cut is the optimum?

  20. More efficient? That is if you assume that making one cut is easier than making one fold. I don't see how making three folds and one cut is more efficient than making one fold a three cuts. It is still four actions in all.

  21. The accurate part about Betsy Ross is that whatever seamstress(es) made those flags probably did use this easy trick to make a reliable symmetrical star, because it was a common thing known among such folk. No secret, no mystery, no advantage over seamstress competition in the next town — just an age-old reliable way to do the thing.

    It's terrible for mass production, though, and not likely used when flags were mass-produced. It would waste wayyyy too much cloth!

  22. You can cut a square with a single cut with only 2 folds. It will be rotated by 45 degrees, but still a square.

  23. Your square in three folds is not efficient. It take only two folds. Fold across the diagonals.

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