Evelyn Lamb: Hello and welcome to My Favorite Theorem. I’m your host Evelyn Lamb, and I’m a freelance math and science writer in Salt Lake City, Utah. And this is your other host.
Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. How are you doing, Evelyn?
EL: I’m all right. I had a lovely walk today. And there are, there’s a family of quail that is living in our bushes outside and they were parading around today, and I think they're going to have babies soon. And that's very wonderful.
KK: Speaking of babies, today is my son's birthday.
EL: Who’s not a baby anymore.
KK: He’s 19. Yeah, so still not the fun birthday, right? That's that's another two years out.
EL: Yes, in this country.
KK: In this country, yes. But our guest, however, doesn't understand this, right?
EL: Yes. Today we are very happy to have Katie Steckles from Manchester, England, United Kingdom. So hi, Katie. Can you tell us a little about yourself?
Katie Steckles: Hi. Well yeah I'm a mathematician, I guess. So I did a PhD in maths and I finished about seven years ago. And now my job is to work in public engagement. So I do events and do talks about maths and do workshops and talk about maths on YouTube and on the TV and on the radio and basically anywhere.
KK: That sounds awesome.
EL: Yeah, you’re all over the place.
KK: Yeah, that sounds like great fun, like no grading papers, right?
KS: A minimal amount of, yeah, I don’t think I’ve had to grade anything, no.
EL: Yeah, and you have some great YouTube videos. We’ll probably talk more about some of them later. Yeah. And and I have stayed at your apartment a few years ago, or your flat, in Manchester. Quite lovely. And yeah, it's great to have you on here and to talk with you again. So what is your favorite theorem?
KS: Okay, my favorite theorem is what's called the fold and cut theorem, which is a really, really nice piece of maths which, like the best bits of maths, is named exactly what it is. So it's about folding bits of paper and cutting them. So I first encountered this a couple years ago when I was trying to cut out a square. And I realize that's not a very difficult task, but I had a square drawn on a piece of paper and I needed to cut out just the square, and I also needed the outside bit paper to still be intact as well. So I realized I wasn't going to be able to just cut in from the edge. So I realized that if I folded up the bit of paper I could cut the square out without kind of cutting in from the side, and then I realized that if I folded it enough I could do that in one cut, one straight line would cut out the whole square. And I thought, “That’s kind of cool. I like that, that’s a nice little bit of maths.” And I showed this to another friend who’s also a mathematician, and he was basically like, “Isn't there a theorem about this?” I thought, “Ooh, maybe there is,” and I looked and the fold and cut theorem basically says that for any figure with straight line edges, you can always fold a piece of paper with that figure drawn on it so that you can cut out the whole thing with one cut, even if it's got more than one bit to it or a hole in it or anything like that. It's always possible with one cut, in theory.
EL: Yeah. So you discovered a special case of this theorem before even knowing this was a thing to mathematically investigate.
KS: Yeah, well, I was I was cutting out a square for math reasons, because that's everything I do. But I was actually trying to make a flexagon at the time, which as I'm sure you've all been there, but it was just because I needed this square hole. And I thought it was such a satisfying thing to see that it was possible in one cut. And my maths brain just suddenly went, “How can I extend this? Can I generalize this to other shapes?
KS: And it was just a nice kind of extension of that.
EL: Yeah. So I have a question for you. Did you, was your approach to go for the, like diagonal folds, or the folds that are parallel to the sides?
KS: Yeah, this is the thing. There are actually kind of two ways to do a square. So you can do, like, a vertical and a horizontal fold, and then you get something that needs two cuts, and then you can make one diagonal fold and just end up with the thing that you can do in one cut, but you can actually do it in two folds if you do two diagonal folds, but it's along the cut. I don't know what the payoff is there. It depends on how much time you want to spend cutting, I don't know.
KK: I was thinking as you're doing this, I've never—I know about this theorem, but I've never actually done it in practice, but never really tried, but I was as soon as you said the square, I started thinking, “Okay, what would I do here?” You know, and I immediately thought to sort of fold along the diagonals. But so in general, though, so you have some, you know, 75-sided figure, is there an algorithm for this?
KS: It’s pretty horrible, depending on how horrible the thing is. Like simple things are nice, symmetrical things are really nice, because youjust fold the whole thing off and then use, you know, just do the half of it. And so there are algorithms. So the proof is done by Eric Demaine and Martin Demaine. And they've essentially got, I think, at least two different algorithms for generating the full pattern given a particular shape. So I think one of them is based around what they call the straight skeleton, which is if you can imagine, you can shrink the shape in a very sort of linear way, so you shrink all of the edges down but keep them parallel to where they originally were, you’ll eventually get to kind of a skeleton shape in the middle of the shape, and that's sort of the basis of constructing all the fold lines. And it is sort of seems quite intuitive because if you think about, for example, the square, all your folds are going to need to either be bisecting an angle or perpendicular to a straight edge. Because if it bisects the angle, it puts one side of the shape on top of the other one. And if you go perpendicular to the edge, it’s going to put the edge straight on top of the edge. And I always kind of think about in terms of putting lines on top of where the lines are, because that's essentially what you're doing if you've got a thin enough bit of paper and a thick enough line, you can actually physically see it happening. So it's beautiful. And then the other method they have involves disks in each corner of the shape, I think, and you expand the disks until they're as big as they can be and touch the other disks. And that then gives you a structure to generate a fold pattern. But they have got algorithms. I haven't yet managed to find a simple enough implementation that you can just upload the picture to a website and it will tell you the whole pattern, which is a shame because I've come across some really difficult shapes that I would really like to be able to fold but haven't quite been able to do it by hand. I'm just going, “Ah, I could just put some maths on this and throw it in the computer program!” But I actually asked Eric Demaine because I was in email contact with him about this. And then the thing that happened was, there’s a TV show in the UK called Blue Peter. Their logo it's like a giant boat that’s called the Blue Peter. It's a big ship with about 20 sails on it. And they said we could talk about this nice piece of maths, and you could even maybe try and cut out our logo with one cut. And I said to myself, “Goodness me!” Because it's all curves as well, so I’d have to approximate it all by straight lines and then work out how to cut this whole thing, and I emailed Eric Demaine and I sent him the picture and asked him, “Like, do you have a program that you can use to just you know, take a figure, even if I send the shape the edge or whatever?” And in his reply, he was like, “Wow, well, that looks, no.”
I just love the fact that they asked me to do something that not even the mathematician that proved that it's possible for any shape was prepared to admit would be easy. And so yeah, I'm not sure if there is kind of a, I mean, I would love it. I’m not enough of a coder to be able to implement that kind of thing myself. I would love it if there was a way to, you know, put in a shape or word or picture and come up with a fold pattern. Yeah, no, I don't know if anyone's done that yet.
KK: Well, this is how mathematicians are, right? We just proved that a solution exists, you know, and then we walk away.
EL: And so I seem to remember you've done a video about this theorem. And one of the things you did in it was make a whole alphabet, making all of those out of one-cut shapes.
KS: Yeah, well, this was, I guess this is my kind of Everest in terms of this theorem. This is one of the reasons why I love it so much, because I put so much time into this as a thing. So essentially in the paper that Demaine and Demaine have written about this, they've got a little intro bit where they talk about applications at this theorem and times when it's been used. So I think it's maybe Harry Houdini used to do a five-pointed star with one car as part of his actual magic show. It was really impressive. And people watch me do it. And they go, “Wow, how do you do that?” Such a lovely little demo. They also mentioned in there that they heard of someone who could cut out any letter of the alphabet, and I saw that and thought, “Wow, that would be a really nice thing to be able to do!” You know, that would impress people because it's kind of like if you can do any shape, then the proof of that should be whatever shape you tell me, I can do. And of course, a mathematician would know that 26 things is not infinity things, but it's still quite a lot of things. It's an impressive demo. So I thought I would try and work that out. And I literally had to sit down and kind of draw out the shapes and kind of work out where all the bits went and how to fold them. And some are easy, some are nice ones to start off with, like I and C and L. As long as you’ve got a square sort of version of it, they're pretty easy to imagine what you’d do. And then they get more difficult. So S is horrible, because there’s no reflection symmetry at all. It's just rotation symmetry and you can't make any use of that at all. R is quite difficult, but not if you know how to do P, and P is quite difficult, but not if you know how to do F. And so it all kind of kind of builds gradually. And I worked out all of these patterns and and in fact, it was one of the reasons I was in communication with Eric Demaine. Because he'd seen the video and he said, “As well as being mathematicians, we collect fonts, like we just love different fonts, type faces, and we wondered if you could send us your fold patterns for your letters so that we can make a font out of them.”
EL: Oh wow.
KS: And I thought that was really nice, so they've got a list on their website of different fonts, and they’ve now got a fold-and-cut font which I’m credited for as well.
KK: Oh nice.
KS: So yeah, the video I did with Brady was for his channel Numberphile, which is as I understand it a hugely popular maths channel. I've done about five or six videos on there, and I've genuinely been recognized in the street.
EL: Oh wow. That’s amazing.
KS: I walked into a shop and the guy was like, “Are you Katie Steckles?” I said, “Yes?” Like, the customer service has gone way up in this place. And he said, “No, I’ve just been watching your video on YouTube.” It’s like, Oh, okay. So that was nice. So he asked me to come and do a few videos, and that was one of the things I want us to talk about. I said, “What do you want me to do? I mean, do you want me to spell out Numberphile or your name or whatever? Brady, who’s Australian, said, “No do, the whole alphabet.” His exact words were, “If you're going to be a bear, be a grizzly.” A very Australian thing to say, he was basically saying let's do the whole alphabet, it will be great. I think at that point it was early enough I wasn't 100 percent sure I would get them all right, but his kind of thing that he has about his videos is that they always write maths down on brown paper, so he had this big pile of brown paper there, and he cut it all into pieces for me, one for each letter. And it was such a wonderful kind of way to nod to that tradition of using brown paper. But I just sat there folding them all, and he filmed the whole thing, and he put it in as a time lapse, and then I cut each one, one cut on each bit of paper, and open them all up, and they all worked, so it was good. But it was this very long day crouched over a little table cutting out all of these letters. But people genuinely come and ask me about it because of that video, so that's quite nice.
EL: Yeah, well I think after I watched that video, I tried to do—I didn’t. H was was my kryptonite. I was trying to fold that, and I just at some point gave up. Like I kept having these long spindles coming out of the middle bar that I couldn't seem to get rid of.
KS: I think somewhere I have a photograph of all of my early attempts at the S. It’s just ridiculous. Like it's just a Frankenstein's monster parade of villains, just horrific shapes that don't even look like an S, and like how did I get this?
But it kind of gave me a learning process, and I think it was maybe just a few weeks of solidly playing around with things. I think I had one night in a hotel room while I was away working so that no one else around. I just spent the whole evening folding bits of paper. I don't know what the maid who cleaned the room the next day thought. The bin was full of bits of cut up paper. I've got like a big stacks of scrap paper at home that's like old printouts and things I don't need that I use for practicing the alphabet because I go through a lot of paper when I’m practicing.
KK: This is a really fun theorem. So you know, another thing we like to do on this podcast is ask our guests to pair their theorem with something. So what have we chosen to pair the fold-and-cut theorem with?
KS: Wow. So I know that you sometimes often pair things with foodstuffs, so I'm going to suggest that I would pair this with my husband's chili and cheddar waffles.
KS: And I’ll tell you why, so my reasoning is that I kind of feel like this is a really nice example, as a theorem, about kind of the way that maths works and the way the theorems work. So my husband's chili is a recipe that he's been working on for years. He comes from a family where they do a lot of cooking, and it was natural for him when he moved out to just have his own kind of recipes. His chili recipe is so good that we've taken his chili to parties and people have asked for the recipe. And I'm just like, there isn't one. It's not written down anywhere. It's just in his head. He has this recipe. And he's obviously worked really hard on on it and achieved this brilliant thing. And kind of the ability to do the alphabet, the ability to kind of make things using this theorem for me is my equivalent of that. It's my special skill I can show off to people with. Because, you know, I've put in that time and I've solved the problem. And one of my favorite things about maths is that it gives you that problem solving kind of brain, in that you will just keep working at something, you keep practicing until you get there. And then the reason why I’ve paired it with cheddar waffles is (a) because that is a delicious combo.
EL: That sounds amazing.
KS: Yeah. As soon as we got a waffle maker, that was our first go at it, was “What can we put with this chili that will make it even better?” And I just found the recipe for cheddar waffles on the internet, because we don't have that, you know, we don't do that many waffles. We don’t really know how to make them. And but the fact that you can go online and just find a recipe for something, is a really nice kind of aspect of modern life.
This is one of the things about maths I appreciate is that once you prove the theory that kind of goes into a toolbox, and other people can then you know, look at that theorem and use it in whenever they're doing, and you kind of building your maths out of bits of things that other people have proved, and bits of things that you're proving, and it's sort of a nice analogy for that, I guess. So those are those are the two things about it. Now that we've got the fold-and-cut theorem, nobody needs to prove it again, and anyone can use it.
EL: Yeah. And I guess if it were a perfect analogy, in some ways, maybe the chili recipe is sort of like these algorithms for making them, they’re really—well maybe that’s not good because the algorithms seem really complicated and difficult. Here, it's more that the recipe is hidden in your husband's brain.
KK: Well, a lot of algorithms feel that way.
KS: It really is quite complex. So you get some more things out of the cupboards that I've never seen before and they all go back in again afterwards. There’s a lot to it that people don’t realize.
KK: It’s a black box. my chili recipe is a black box, too. I can't tell you what's in it I mean it’s probably not as good as your husband’s, though.
KS: It’s got roasted vegetables in it. Yeah, it's that's that's one of the main secrets if anyone's trying to recreate it. But then just a whole lot of other spices that only he can tell me
EL: My husband doesn't like soups with tomatoes in them very much. I mean sometimes he does. But I don't do chili very much. So yeah, I don't have a good chili recipe we have a friend who's allergic to onions and that's a nice excercise in, can you cook or modify your recipe and still have it taste like what it’s supposed to be? because without us yeah a lot of things that don't work and she must have a nightmare with it. Because like a lot of packaged foods, they've got it.
KS: They’ve got onion powder or stuff.
EL: Every restaurant.
KS: We made chili without, and it kind of works. It kind of works without onions. It was great. I think there was a bit more aubergine that went in and some new spices, just to give it a bit more oniony flavor, but it still works.
EL: Oh, nice. Yeah, cooking without onions is tough. Does it extend to to garlic—does it generalize to other things in the allium family?
KS: Yeah, it’s all alliums, so she can’t really have garlic either she can get away with a little bit of garlic, but not any reasonable amount. Yeah, it must be completely horrible. Actually it kind of reminds me of Eugenia Cheng, her first book was about maths and baking. But one of the really nice points that she makes about the analogy between recipes and maths, which we have apparently stumbled into is that, you know, understanding something in a maths sense means that you can take bits of it out and replace it with other things. You've got a particular problem and you go, “Okay, well, do we need to make this assumption, do we need this particular constraint? What happens if we relax this and then put something else in?” And that's how you explore kind of where you go with things. And if you relax a constraint and then find the solution, that maybe tell us something about the solution to the constraint problem, and things like that. So, you know, tweaking a recipe helps you to understand the recipe a bit more. And as long as you know roughly what goes in there and you've got something that is, you know, recognizably a chili, then, you know, it doesn't matter what you've changed, I guess.
KK: Yeah, so we also give our guests a chance to plug anything you're working on. You want to plug videos, websites, anything?
KS: Oh, I’m always working on a million different things. I guess probably the nicest thing for people to have a look at would be the Aperiodical, which is a website where I blog with two of my colleagues so we write— it's kind of a maths blog but aimed at the people who are already interested in maths, so it's one of the few things I do that is not an outreach project. Which is essentially it’s aimed at people who already are interested and want to find out what's going on, so we sometimes right like opinion pieces about things or, like, “Here’s a nice bit of maths I found,” and then sometimes we just write news. And there’s a surprising amount of maths news, it turns out. It's not just “They’ve discovered a new Mersenne prime again.” There are various other maths news stories that come up as well, so we write those up, and bits of competitions and puzzles and things as well and it's at aperiodical.com. And we get submissions. So if anyone else wants to write an article and have it go out on a blog that’s seen by, you know, a couple of thousand people a day or whatever, they’re welcome to send us stuff, and we’ll have a look at it.
EL: Yeah, it's a lovely blog, and you also organize and host the math blog carnival that is, like, every month a round-up of math blog posts and stuff like that.
KS: We sort of inherited that from whoever was running it before, the Carnival of Mathematics. Every month someone who has a maths blog takes it in turn to write a post, which is essentially just a bunch of blog posts that went out this month. And we have the submissions form and all the kind of machinery behind it is now hosted at Aperiodical and has been for a few years, so if you have a maths blog elsewhere, and you want to get an opportunity to put a post on your site that will be seen by a bunch of people because there's a bunch of people who just read it every month, then get in touch because we're always looking for hosts for future months. And essentially we just forward to your email address all the submissions that people put in during the month, and you can then write it up in kind of the first week of the next month.
EL: Yeah. And I always see something cool on there that I had missed during the month. So it's a nice resource.
KS: So one of the other non-outreach, I guess, maths things that I'm involved with is a thing called Maths Jam. Or in the U.S. the equivalent would be Math Jam. We do have both websites, basically. So I coordinate all the Math Jams in the world. So it's essentially a pub night for people who want to go and do maths in a pub with people. It's aimed at adults because a lot of kids already get a chance to go to math club at school and do maths puzzles and things in their classrooms, but adults who have finished school, finished university, don't often get that chance. So we basically go to the pub once a month or to a bar or restaurant somewhere that will allow us to sit around and drink do maths. And there are now I think, getting on for a hundred Maths Jams in the world. So we've got about 30 or 40 in the UK. And then they’re popping up all over. We just picked up one in Brazil, we’ve got three in Italy now, three in Belgium, and there are a few in the U.S. But what I'm going to say is that I’m very sad that we don't have more because I feel like it would be really nice if we had a whole load of U.S. jams. I think we've got more in Canada that we have in the USA, which interesting given the population sizes, or relative sizes.
KS: I think Washington DC has just gone on hiatus because not enough people are coming along. So the organizer said, “I'm getting fed up of sitting in the pub on my own. No one else is coming. I'm just going to put it on hold for now.” And so if you live somewhere in the U.S. and you want to go meet with the people and do maths in an evening, essentially to start when you just need a couple of people you that know you can drag along with you to sit around the case no one else turns up. And we send out a sheet with some ideas of puzzles and things to do. And you can play games, chat about maths, and do whatever. People can bring stuff along. And all you need to do to organize it is choose a bar and send the email once a month. And those are the only requirements. And go to the pub once a month, but I think that's probably not a big ask if that's the kind of thing you're into. So if anyone is interested, you can email to email@example.com and I can send you all the details of what's involved. You can have a look on the website, mathsjam.com, or math-jam.com, if you want to have a look at what there is already, what’s near you.
EL: Yeah, it'd be nice to have more in the U.S.
KS: Yeah, well, I get a lot out of it. Even though it's kind of sort of my job, but also I always meet people and chat through things and share ideas and people always go, “Oh, that reminds me of this other thing I saw,” and they show me something I've not seen before. And it's such a nice way to share things. But also just to know that everyone else in the room is totally sympathetic to maths and will be quite happy for you to chat on about some theorem or whatever and not think you’re weird. It’s quite nice.
EL: Well thanks a lot for joining us. I enjoyed talking about the fold-and-cut theorem. It makes me want to go back and pick up that alphabet again and try to conquer Mount H, that felled me the last time.
KS: I can send you send you a picture of my fold pattern for each, but I’m sure you would much rather work it out for yourself. It’s such a lovely puzzle. It's a really nice little challenge.
EL: Yeah, it’s fun.