The Hidden Maths of Knitting - EMF 2026

[applause] Thank you so much. Uh hello everybody. Uh I'm going to start with a little bit of maths. I'm going to start with my clicker not working. Uh yes, I'm going to start with a ven diagram. So hopefully you are all familiar with uh ven diagrams. Uh three areas on here. The first area uh I've illustrated with Tom Dailyaly because he is famously someone who loves knitting. Um I found out after I made the slides for this talk that he has a level maths. So it might be that he's in the wrong section. Uh we've also got here uh Hannah Fry who very famously loves maths. Um I don't know. She might also like knitting though. So I might have got this ven diagram completely wrong. The one part of this ven diagram I am absolutely sure about though is the middle because that's me and I love maths and I love knitting. And I just want a quick show of hands to get a sense of who my audience is today. So if you would put yourself in region one of this ven diagram. Put your hand up now. Okay. If you would put yourself in region two. So the loving maths but maybe not loving the knitting. Okay. Anyone who's in section three with me? Lovely. And if you're in section four, it's not too late to sort of uh sneak out. This might not be the talk for you. Um, so yeah, I wanted to talk to you a little bit about uh maths and knitting, two of my passions. And I thought I would better start off with just a little bit of an explanation of what knitting is for the people who weren't in that section of the ven diagram. So this is a very badly drawn uh picture of several rows of knitting. It's kind of hard to see from that how the stitches are formed. Uh so I've got a couple of videos. The first one is um showing uh yes. So you sort of put your needle into the stitch, you wrap the yarn round, and then that comes off to create a new stitch. And you do this all the way to the end of the row. And then you keep on doing that. And it builds up rows and rows of this fabric of knitting. And so if we look again at that, you can now see the needle in there. I didn't want to put the needle in the first picture because it's a bit of a spoiler. Um, but the excellently drawn ball of yarn as well. So as you complete each row, everything sort of moves down once. So each stitch is looped in to the row below it. and you kind of get this pattern which is sort of like little um curly shapes or Vshapes depending on where you're looking at with the knitting and the types of stitch you do. Uh there's a lot more complicated things you can do with knitting, but that's sort of like the basic first introduction that everybody tends to learn. Now, I want to contrast this with weaving. So, weaving is um older in terms of how long humans have been doing fiber arts. We know about weaving because we found um the uh sort of the things that we use the tools that we use for weaving dating back a really long time. We've got documentary evidence of knitting and knitting related stuff um sort of over the past more like a thousand years. But weaving is going back sort of orders of magnitude almost as long as humans have wanted to wear clothes really. Uh, and for weaving, you've got these two threads going um one across the way and then one sort of weaving over and under uh the warp and the weft. Somebody gave me a good way of remembering them and I can never remember which way round it is. But I'm not really going to be talking about weaving today. I just wanted to draw your attention to the two uh sort of different uh types of ways of making fabric because if you happen to have a uh microscope uh you can buy little USB microscopes. Um you can look at fabrics that you own under the microscope and they look something like this. So on the left is a t-shirt fabric and hopefully you can see these sort of like curly V-shaped things that were the same as what was appearing in the knitting. Whereas on the right, it looks much more like a grid structure. And that would be a woven fabric that was from an old pillowcase. And so when we choose whether to knit or to weave, we're looking at the properties that the fabric will have to decide how to use it. [snorts] Okay, I want to talk a little bit about some of the maths then. So topology is the maths of how objects behave when they're deformed. And so I've stolen this little animation to illustrate this because one of the classic maths jokes is about the topologist who confuses his coffee cup with his donut. Because topologically, if you just look at what stays the same and what changes, the number of holes in a donut or the number of holes in a coffee cup is the same. And topology is looking at the shape of objects when they're deformed without worrying too much about uh the geometry of it. And it's important that we think a little bit about topology because I want to talk also about machine knitting. Now I first heard about machine knitting um not that long after I'd learned how to hand knit. There was a um distant family relative who made these machinek knitted jumpers for the family every Christmas. And so I asked my mom, "How come she can make so many uh jumpers every year?" And my mom said, "Oh, she's got a knitting machine." And I'd recently learned to hand knit. I had a good mental model of that. And so I extrapolated from it. And I decided that she must have in her living room some big robot arms that operated two needles like this automatically while she went off and and did the housework. Turned out that knitting machines are actually way cooler than that. So, uh the box on here is um one of the knitting machines I brought to EMF uh this year. It is uh on here and you'll see it a little bit later on. Um but this is the knitting machine that I should have had to teach me about knitting machines when I was first asking those questions. This came out in the um in the 1980s. I think we dated this one to the Argos catalog 1987 or so. Um, and I never had one of these as a child, but I found one of these on eBay as an adult, and I absolutely love it. But the thing about machine knitting is that although the mechanism to create the stitches looks different, topologically the stitches that you get out of it are the same. So, let's look at how a knitting machine creates the stitches. So this is uh this machine uh set up and you can see that instead of being along a needle to work one at a time, the stitches are all uh sort of hanging on needles like these are individual hooks that are coming out at right angles to the knitting. So rather than one needle with all the stitches, you've got lots of needles each holding one of the stitches. And then when the carriage moves across, it brings the yarn with it and much quicker than I could knit individually, it makes a stitch by pulling the old stitch through the new yarn, just like I would do with my needles. Now, it's a bit hard to see what's going on with the carriage in the way. So, what I've got here is just manually doing what the carriage does for each knitting stitch. So, there's a little latch on that hook and the hook catches the yarn and then the latch closes over it to pull the yarn through the previous stitch and you can get all the way along to the end of the row and it is much quicker. It's much slower doing it by hand this way. But with the carriage, it's much quicker uh to knit things. Okay. So I want to talk a little bit now having introduced the idea of knitting and the idea of uh topology. I want to think about socks because socks is one of the things that when you're a beginning knitter you might aspire to make someday. And you might think well they're quite small so it won't take me long to knit socks. Now, there's a whole lot of stuff. I could probably do an entire hour on why that's a a mistake to make. Um, but when you're thinking about how to knit a sock, you've got some choices to make about the construction. And so, I want to just uh switch uh underneath the visualizer for a moment to show you some of the things that you can do. Now, the visualizer is playing up a little bit, but hopefully you can all see my hand there. Um, so yeah, we can decide about how if you want to knit a sock on a knitting machine, you're going to have to turn it from like a 3D tubular object into something flat and then sew it back together. So, one thing that you could do [snorts] is you could make a really, really long thin thing that will turn into a sock. So, let me explain how this works. This bit here is going to be the ankle of my sock. And then this bit that I've done in blue as a contrast, I've sort of done a trapezium shape by knitting fewer stitches on each row and then increasing back again, but keeping the rows the same so that you don't get a gap just there. And that gives you something that can then become when you turn it into three dimensions, the heel of a sock. And when I first learned about how to do heels on socks, I was like, that's so clever. Um, with this method of doing socks, I got more even more excited when I found out that you can knit along the bottom of the foot and then put another heel there. Because if instead of having your heel at right angles, you fold your heel flat, it turns from a heel into a toe. And then you just have to knit the top of the foot and then sew up all your seams and you have a tiny sock. So that's one way of knitting a sock flat. Instead of doing that though, you could do it as a very short and fat sock. And it's a very similar construction. I've only done the heel on uh half of this width because this bit is going to be the front of the ankle. And then this bit is going to be the uh top of the foot. But then again, I've done the same thing. I've done a heel here that folds over to become a toe. And then the yellow bit of yarn here is just uh waste to to take it off the machine. So if I wanted to sew it all up and turn it into a sock, I wouldn't drop all of my life stitches. So that's two ways of knitting a sock flat. Um you can of course knit socks using something called circular knitting. And you can do this with knitting machines or by hand. But because I'm a showoff, I wanted to show off the knitting I'm doing at the moment, which is a pair of socks. I'll put these under the visualizer. I'll hold them up as well and then hopefully everyone can see one way or the other. But I've got a really really long needle um a circular needle with uh needles at each end and a big tangly mess which allows me to knit along the stitches and then back the other way by pulling the needle through and knitting continuously. And because I'm showing off, I'm actually knitting both socks at the same time here. Um, so you knit along the first sock and then you knit the front of the second sock, the back of the second sock, and then the back of the first sock. And that's a complete row. Um, somebody told me about this method that even though it's harder than just standard uh circular sock knitting techniques, uh, it avoids second sock syndrome, which is when you finished a sock and then you go, "Oh god, I've got to knit an entire other sock before I can wear them." Um, so, uh, the other annoying thing though about this is, uh, I've done it in a sort of yarn to get this stripe pattern. Instead of changing color every few rows to get stripes, you can get this wonderful invention called self striping yarn where all of the different colors are on the ball and they're dyed in a repeating sequence. Which means that if you find the same point in the sequence and start both socks from that point, then in theory, if you can keep your tension really really consistent, each sock will end up exactly the same. And I'm quite pleased that for the most part on these, this is my second pair of uh no, my third pair of two at a time socks. Uh and I've managed to keep mostly the color patterns working there. Um so that's a couple of different ways of making socks. If you want to make socks using uh knitting machine and you don't want to knit it flat, there are techniques you can use. You can uh use something called a ribbing attachment for a knitting machine, which you gives you a second set of needles at right angles. And then that sort of allows you to do the same thing that circular needles are, cuz you can knit one way along the top and then the other way um along the back. But you can also get something, and this is something that's on my wish list to buy called a circular sock machine where it just knits tubes and you just crank a handle and it it magically uh knits tubes. It's a a really lovely thing, but they are quite pricey. So, uh that's something I'm saving up for. I believe that circular sock machines were one of the first knitting machines that were um that were created and they've actually been around longer than flatbed knitting machines. Uh okay. So that was deconstructing socks. Then I want to talk a little bit about crochet now because um like I my journey to knitting. I started learning to knit when I was I don't know about six or seven. Uh I didn't start to learn crochet until uh I was in my late 20s and then crochet just captured my uh imagination. It's a different technique than knitting cuz you only ever have one stitch at a time. So it means that if you drop your stitch, you can just pick it back up again. Uh but it does give you a slightly different pattern with things. And sometimes crochet is the best medium for making some things. sometimes knitting is most talented people in one art or the other can think of ways to make most things using their preferred one. Um, but yeah, what you can see on the screen at the moment is uh this hat, which is something I crocheted for myself last summer because I needed a bucket hat and I wanted one that would actually fit my head. And this is another type of knitting with circles because the socks were to do with knitting with circles cuz I was knitting in the round. I was using circular knitting. But for crochet I can actually uh crochet a circle to be the top of my hat. And when we look at what's happening mathematically uh you can start to see some interesting things happening. So the inner ring of my hat, the thing that I started off with was just doing six stitches. And then by increasing the next round ended up with 12 stitches in. So I increased into every stitch on the first round. Then I increased into every other stitch. So I ended up with 18. And then I increased into every third stitch and I ended up with 24. And hopefully you can see that a pattern is starting to emerge here. we're going up in what mathematicians would call a linear sequence where the sequence is changing by the same amount every time. So I can then predict that uh if I got up to round 10 of this I would have 60 stitches. I can then count and all the knitts in the room and all the crocheters in the room will know that when you count sometimes you don't have the right number of stitches and it makes you very sad but at least you know how many you should have. And so because we are increasing in this linear fashion with this particular stitch, we're getting something that is flat and it is approximately coming out as a circle. It's actually going to be more like a hexagon. And you can do things to change where in the rows the increases are to make it look a little bit less hexagonal, a little bit more circular. You could do uh stitch patterns where it was instead of coming out as a hexagon, you can do uh larger things that approximate better to a circle, but it was good enough for the purposes of my hat. And so the type of geometry that we are in with this kind of crochet is uklidian geometry. And so the picture that you can see on the right shows what happens if you put equilateral triangles together in uklidian geometry. The picture on the left is just a picture of a bag that I crocheted cuz I'm quite proud of it and I thought it was nice. But you could make the Thank you. Uh you could make the bag using triangles instead of hexagons because if you look on the right, six hexagons will go together to sorry, six triangles will go together to make a hexagon. And so triangles and hexagons both have this property that uh on a flat uh plane in normal space they are going to tile it without leaving any gaps. And you can think about why that happens by thinking about the angles involved because equilateral triangles have three 60° angles. So when you put six of them together you're going to get 360° round a point. But if you're a mathematician, then when you're putting uh things together around a point, you ask the question, well, that's what happens if I put together uh six triangles around a point. In fact, here we are. Here are six triangles together around a point. You might ask the question, what happens if I put together five triangles around a point? And I'm glad you asked that question because this happens. So this is five equilateral triangles put together around a point. And of course if each of these is a 60° angle 60° times by five gives you 300. So it's not enough to form a full circle. And that's why a little bit tricky to see here but you are getting some curvature here. And curvature is one of the things that I needed to use to make my hat because you'll notice that my hat comes down the side of my head as well as uh having a top of my head. So the top of my head's quite flat but then it needs to curve round. So I can exploit uh geometry. The linear process gave me the um uh flat surface. But if instead of doing something linear, I increase by less, then I can start getting this um spherical geometry. So this is a little guy that I crocheted. Uh he is a little octopus. Uh I particularly like him because although you can't see it in this picture, uh one of the yarns that I used for one of his sides is a glow-in-the-dark yarn, which means that this octopus uh does some sort of bioluminescence uh at nighttime. But the way that it was made was starting off with the same linear increasing for the first few rows, but then once I got up to 30, I just continued straight down. So 30 on every row. And that gave a cylinder because at that point I'm not keeping the flat pattern to uh linearly increase. I'm increasing by a less than a linear amount. So it is now just going straight down. I could have done something in between the two and that's what I did for the hat because my head doesn't come straight down. It sort of slopes down. So, by deciding which rows you want to do the increases in, you can decide on a rate of curvature. And you can see that the curvature again at the uh bottom of the hat uh for the brim it goes more flat again, but not completely flat because you want it tilted slightly down to keep the sun out of your eyes. Um, that's what happens then if you uh put fewer than six uh triangles together around a point. Some of you may want to know the answer to the question, what if I try to put more than six triangles together around a point? And some of you might think, well, that's a silly question. They won't fit. They won't fit if I just put them together like that. But if I were to glue these two edges together so that I had seven triangles, I get something that won't lie flat on the 2D plane. It sticks up out no matter how I try and lie it flat. Without squashing my paper, it won't lie flat. And we are now in the world of hyperbolic geometry. So in hyperbolic geometry, what I can do with my crochet is I [snorts] can start off with six stitches again. I can increase into every stitch to get 12 stitches for the second round. But then instead of increasing into every other, what if I increase into every stitch again? So it doubles. So I get 24. And then on the fourth round, I get 48 stitches. I've doubled again. And by keeping on doubling and doubling, I get rows that are too long and I get really bored. So I don't normally go past about five rows. But the last time I gave this talk, uh, somebody in the audience was doing some hyperbolic crochet uh, while we were there and she made this. I think she'd already started it and then she gifted it to me at the end of the talk, which was really rather lovely of her. Um, and she does them sort of as as uh, fidgets. um it's like to keep her hands busy and then she ends up with lots of these at the end and doesn't quite know what to do with them, which is why she was happy to give one to me to use as a demo model. Uh but yes, if you keep going, you get this lovely thing where it will not lay flat and it goes into these almost organic looking uh structures as a representation of the hyperbolic plane. And people have used these in art projects. They've used them to uh there was a crochet coral reef that someone did because it uh very closely resembles some coral structures. And I think in the the pure mathematical context, the rather lovely thing is [snorts] that this was actually used as a teaching aid. So back in 1997, uh Dana Taimina used crochet to make models of the hyperbolic plane to use in undergraduates teaching because making a physical hyperbolic plane that you can pick up and examine to understand the maths of it was quite hard to do with most manufacturing methods. But crochet turned out to be uh the best way to visualize what was going on with hyperbolic geometry. Okay, I want to talk now a little bit now then about knitting machines. Um I've talked a little bit about the little knitting machine. I started off with no knitting machines, which I think is the default state for most humans. Um, I have had the joke made about me recently that the um the one about the guy with all the spiders who drags up the mean number of spiders that I am becoming that for knitting machines because the average number of knitting machines owned by people in my friend circle is a quite reasonable number. But I think I'm the only one of my immediate close friends who actually owns a knitting machine. Um, I wanted to get an electronic knitting machine because I saw them and I saw they were cool. And I found one on Facebook Marketplace, but it was part of a huge job lot of stuff. And the woman selling it said, "Uh, I'm not splitting it. If you want the uh, incredibly exciting electronic knitting machine, you've got to take the what I thought then was slightly less interesting punch card knitting machine." And so this is the first punch card knitting machine that I acquired. It's a a knitmaster machine. And because the electronic one didn't work cuz it was from the uh early 1990s and the capacitors had done the thing that capacitors do until we could get around to ordering the correct parts and getting it up and running again and checking power supplies and stuff. I thought, well, I'll test this one and see if this one works. And so, this is actually uh the machine that I learned machine knitting on. And the punch card actually turned out to be a thing of joy and wonder because it felt like this old technology that I'd heard about for stuff. I mean, my parents had told me stories of computers that ran on punch cards. Uh but yeah, I was just a little bit too young uh to remember any of that. And so this idea that you could use ones and knots, you could use binary to make the knitting machine do cool pictures just felt very exciting. So, uh, this is a punch card in the knitting machine. It has 24 ones or zeros, holes or filledin bits on each row. And so, if we think about that in terms of bytes, that is three bytes of information on each row. Uh, we've got about 60 rows on a longest punch cards that you tend to get. So, yeah, that means 180 bytes of data on one of these punch cards. Not bad, eh? And uh yeah, the the way that it works then on the back of the knitting machine carriage, you've got this little uh wheel and when it passes in front of the punch card reader, it is setting the state ready for the next row that you're going to knit. So you've got one on each side and so as you go back and forth, it picks up the pattern for the next row whilst depositing the pattern for the previous row. So, as the punch card feeds through the machine, you then get uh each row of the punch card being knitted in uh you can do two different uh stitch styles or you can do two different colors. I use it for color work because I think it makes it a little bit more obvious uh what's going on. So, here it is in action. So, the knitting machine goes Oh, the knitting machine should go, if the video plays, in front of the punch card reader. And you can't really see what's happening with this one, but as it goes in front, it is picking up now because you can't really see what's happening on this. This was the one and only reason that I chose to buy Oh, that is uh the sort of patterns that come out of it then. So you can see if you have a checkerboard pattern on the punch card, you get a checkerboard pattern on the knitting. But yeah, this is one of the reasons that I decided to buy this knitting machine, which is the other knitting machine that I brought with me to AMF. And if you want to come up and see it in the maths village, uh I should be around for most of this afternoon and various other times over the course of the weekend. But uh this machine has a slightly different mechanism. So instead of uh storing the uh row on the little drums on the carriage, it's got this uh timing belt mechanism and lots of stuff inside the machine that although I have had this one apart. I only got this machine a week ago and it's already been apart trying to fix it to get the punch card working to make this video to show in this demo and also to have up in the math village if anyone wants to see it. Um, but yes, the way that this one works is slightly different. So, it sets the needles ready for the next row as it's knitted the previous one. So, I've got a little bit of uh video that was shot uh this morning in my tent uh by uh my very friendly uh videographer and partner Sam. And so what you can see as it comes along, oh, I didn't turn the sound off on this one. Okay. Um, you can see that it is moving some of the needles forward and some of the needles backwards. And the punch card advances after each row as the carriage turns around. So different needles are forward and different ones are backwards. So if you've got two colors, then it means that the front color, front needle can be one color and the back needle can be the other. And I'm hoping to have this hooked up and actually knitting some color work a little bit later on when it's maybe not quite so warm. Um, so yeah, that is uh one of the Brother knitting machines and it uses very similar ideas of needles forwards or backwards, just different ways of uh setting where they are. Okay. Um, I want to talk about a little obsession that I ended up with then to to finish off today, which was asking myself once I had punch cards, once I had uh the ability to do color work without having to do it by hand, how would I knit a circle? And I asked myself this question because I thought, well, if I can figure out how to knit a circle, that's going to help me to understand how to design patterns and pictures uh for a lot of other things as well. And so the first thing I did was I went to the idea of um just naively coming up with if I had a square grid, drew a circle on it, and then colored in the points that the circle passed through, what would that look like? And I looked at and I thought, well, it's sort of circularish. And I thought it reminds me a bit of computer graphics. So I talked to my partner who uh knows a lot more about computers than I do and he said oh well there's algorithms for this sort of stuff and uh he pointed me in the direction of uh Breham's algorithm and this is an animation from the Wikipedia page about it and so it's using yeah a midpoint circle algorithm and it gives something that looks pretty smoothly circular when you go round. It's exploiting the symmetry of the circle to be able to work out eight points simultaneously. And so that means that you're going to then end up with something that looks nice and symmetrical at the end. So each step all it's doing is it's divi deciding do I stay at the same height or do I drop down a height or go up a height depending on which point of view you're thinking of. So, I got a circle uh template out of all of this. I filled in uh the circle because I wanted to knit the whole thing, not just the border. And then I had a go and had my first go at uh making a circle. And I was really disappointed cuz it ended up looking sort of like this. And you will notice if you are a fan of maths and you know that circles have constant radius that this does not look like a circle. It is all squishy in because I made a mistake. I forgot that knitting stitches knitting pixels are rectangular, not square. So when I measured my first attempt at a circle, it was 6.5 cm wide and 5.5 cm high, which is not a circle. So I thought a little bit about the maths behind this. I plotted the graph of uh x^2 + y^2 equ= 100 and I made the axes different on the x and the y axis and that gave me something that looked like an ellipse. So I thought well I can work backwards here. If I plotted an ellipse instead of a circle on a normal X and Y axis, then I've sort of inverted the process. So, as long as I know what the aspect ratio of my knitting is, if I can make the punch card elliptical, then it means that I can make the knitting circular. And so, if I just show you what that looked like. So, that was the uh circular um punch card giving me a It's so hard to look on the screen and realize figure out where I want to put things. There we are. Uh a circular um punch card gave me an elliptical knitting. And so when I put in the elliptical punch card, I came out with a pretty good circle that I was very happy with. Now, that does not deserve a round of applause because so those were my better circles. I then asked myself the question, what about that self- striping yarn, the yarn that I used to make those socks before? Can you make your own self striping yarn and use it to knit a picture? Because in theory, I know where every pixel is going to come from from the length of yarn. I know what my tension is because I always knit with a consistent tension or I mean fairly consistent if I use the machine. I can predict whereabouts in the yarn each bit of knitting is. So, if I had a design, I could get the yarn, color it in with some fabric paint, and then knit it. So, on the machine today, I have mostly knitted something. I'm not knitting the whole thing on stage because that would never work and it would come out really badly, but it's mostly already knitted. I'm going to do the last couple of rows for you. And I tell you what, let's switch to the visualizer. And if I point it over there, then you're not going to see. The camera is not going to want to show that. Uh I'll leave it there then to show things once it finishes. Yeah, that's me. We don't want to see that. Definitely don't want to see that. Right. Okay. Um I will do one troubleshooting and if it doesn't work then I will just do the knitting and you'll have to come and see it afterwards. Nope, that's not what I wanted to do. Do you want me to hold your laptop? >> It's all right. Uh, it's going to come back. Right. Okay. So, I'm going to do a couple more rows of knitting. I'll take it off and then I'll show you what we end up with. So, I'm going to take it off the machine and hopefully what we will get is some picture knitting. Now, it might be a little bit hard to see what this picture is, but I do have a prop to help you with that because hopefully what you can see is three regions of color where I can put Tom Daily, Hannah Fry, and me Now, [applause] I have of course lied to you because that would be a really silly way uh of making self striping yarn. I did think about this as whether you could use it as a way of sort of doing some knitting and espionage to hide a hidden message into some yarn, send it to someone else with the same knitting machine knitting at the same tension, get them to knit it. Um, and it really works uh very very badly because to make that vin diagram, what I actually did was knitted some yarn um drew the diagram on it, then pulled the knitting down again, filled in the gaps in the yarn, and then uh put it back together again. And even then, it did not come out looking like it should. So, this is not a viable way of making uh pictures in the knitting. Uh but there's much more math I could have talked to you about today that knitting and espionage is only a tiny bit of the story of how knitting has been used in espionage. Uh you can make uh objects like merbous uh scarves or klein bottle hats. Uh there are ways that you might be able to calculate pi uh using knitting. So if you want to find out about any of these things then come and find me in the math village or uh contact me on my socials. But I hope you've enjoyed uh hearing about the hidden mass of knitting today. Thank you very much. [applause]