Fold and Cut Theorem - Numberphile

Key Takeaways

I have a pair of scissors um cuz I want to talk about a piece of MK called the fold and cut theorem it's one of those things in Mass where they just name it what it is they first came to my attention when I was trying to cut a square hole in a piece of paper so I like my first instinct as a person who cuts holes in paper was to jab a hole and then cut around the edge of the square but I thought okay maybe there's an easier way to do this and um kind of my mathematical brain kicked in and I realized that if I fold this in half like that I can just cut along those three edges this okay I've roughly approximated the hole I was originally trying to draw there's a square hole um 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° to that I now have this bit of line and I can now cut two cuts like that and I'll end up with my square hole two cuts it's yeah it's two cuts it's it's less than three Cuts it's more efficient um but of course you want to optimize this so you know your mathematically efficient way of doing this for 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 uh and this is the Moment of Truth a bit where you don't really believe what you've done has worked uh but it turns out square is pretty much the easiest one to do it's it is efficient and it's it's one straight cut uh and this kind of idea like I I went to the pub as you do when you've discovered a nice thing uh 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 uh and a friend of mine who was in the pub said isn't that a theorem uh as 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 the bit of paper uh which is just insane and I love this it was proved by a guy called Eric Dain and the team of 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 uh used to apparently do uh 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° is uh you can then fold that one in half fold that back and fold that and you can tell you've done it right CU that's the same size as that is and there's actually a nice little story behind this because apparently when uh America was founded in the 1770s uh needed a flag and they went to see a seamstress called Betsy Ross I don't know if it's a true story it could be one of these madeup stories that George Washington walks into a a fabric shop and was like I'm going to need a lot of stars cutting out uh 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'd cut out a perfect five-pointed star with just one cut uh and this The Story Goes that uh 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 uh just because she happened to know this little trick uh and I love that I love that 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 that's a five-pointed star uh that's that's my favorite bit is that you get the the opposite thing as well because it is just that line that's cut it's nothing else the next thing in the paper that they 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 uh and I looked this looked this up and tried to find it and I couldn't find anyone who who did this I was like right well that's I'm going to do that I'm going to figure out how to do every letter of the alphabet and it it turns out it's not even just um a cool thing to be able to do it's it's actually a really fun thing to work out uh so you start with these simple letters like L uh because they're quite easy that's just one kind of diagonal fold and then you need to make the ends um you can do things like oh as long as you don't mind it being quite square and then things gradually get more and more difficult and the 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 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 oh what that already existed oh well never mind uh but it you know the the system that I've worked out I'm happy with uh it's slightly different in places uh but you know there 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 QR s t u v WX y and z okay that's an a that is B that's C that is d e one's f g that's the closest I can get to G without any curves little H there I J so of course is my initial K that is completely different from J that's L that one is M that is there's no way I'll get this right way around is that the right way around yeah and oh it's p it's good enough it's tiny it's tiny but it's a q that's all nice bit of a nightmare that one this worked well done s yeah Sly funky T you slightly 70s looking V so that's w x it's quite a fun one to to work out it's time very happy that's a why it's looking good 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 it's the the type the type of thinking that mathematicians use can be applied to so many situations and this is exactly one of them uh 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's what mass 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 linda.com for supporting this video now Linda is 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 type faces and topography this is something I find really interesting this one here is called the 33 laws of topography and it's gone right to the top of my watch list I'll be having 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 Linda will have something for you there's over 3,000 video courses and 100,000 tutorials in the vault you can sign up now for a free 10day trial that's unlimited access to all those videos by going to linda.com number file or clicking on the link in the description that's Linda l y nda.com number file give it a try and our thanks to Linda for supporting number file um I did a show at chelham a couple years ago and um one of the the things that I did in it was as the challenge like to pick a person in the audience asked them what their name was and then cut out that word while explaining the MTH behind this it was brilliant I asked for a four-letter word um as then it was anyone got four four-letter name uh and this guy shouted his name and I can't remember what it was but it was five letters uh and everyone was like mate that's five letters and it was it was just gift it was hilarious