 So, I'd like to bring up a gentleman named Robert Kreese. He is the chairman of the philosophy department at Stony Brook University, and also a dear friend, someone I've known for many years. He's also a good friend and supporter and fan of our speaker tonight, and so I will let Bob come up and introduce our speaker this evening. I'm sure you'll be interested in hearing from both of them. Thank you, Bob. Thanks, Indy. There's a short story by the science fiction writer Stanislav Lem about a crazy tailor, and this tailor knows nothing about people or animals or plants, but he makes clothes anyway, and he puts holes and tubes in random places and then puts these clothes in a warehouse. And people come along who have strange animals like octopuses or giraffes or something, and they usually find something in the warehouse and then take them away. And Stanislav Lem comments that this is sort of what mathematics is all about. People dream up these strange structures, and they don't care if they fit the world, but they dream them up anyway, and that's a wonderful image, I think, for what mathematics is all about. It's also a particularly good image to introduce our speaker tonight, because for one thing, she thinks that mathematics is basically fun. Her adage, she told me, was that a professor's office can never have too many toys, but it's also a particularly appropriate because she actually makes clothes, and she'll show you some of them this evening. Now she went to Amherst College, graduated from Amherst College, then went to Harvard, and she had an interesting experience at Harvard as a grad student where she was telling people about the discovery of hexaflexagons, which were discovered by Arthur H. Stone. But in the course of the talk, she forgot his name. But in the talk included someone who made a lot of interesting comments, who to her embarrassment turned out to be Arthur H. Stone himself. So a little bit later, after a few post-docs, she wound up at St. Mary's College, and she wound up being interested in beaded crochet art. But for a while, she couldn't bead crochet herself, so she was in the embarrassing position of being the leading expert on bead crochet art, but without being able to bead crochet. But she finally got up to speed. She and an artist and computer engineer named Ellie Baker wrote a book together, which is actually right over here called Crafting Conundrums, which came out in 2014. It's got a lot of interesting creations in it. She'll show you some of them. Let me just tell you about one mathematical creation that she's going to show you, because it's really interesting. There's an online magazine called Nitty that is sort of for kind of people interested in nerdy knitting designs. But it's very demanding. To get in it, the design has to be original. It can't have appeared in Facebook or on Instagram or anything. And so it's a real coup to get in it. It's sort of like winning the Oscar. So it's, if you're interested in nerdy knitting patterns. So Susan Goldstein tried in 2015, couldn't do it, but in 2016 she succeeded. So when she shows you this, make sure to be really impressed because it's very hard to do. So as I said, this is, she's the perfect MoMath speaker. She's interested in things, in very complicated things that are fascinating but that illustrate and draw you deeper into the world and the structures in it as you'll see in our talk. So let's give a warm welcome to Susan Goldstein. Thank you very much, Bob, for that lovely introduction. And thank you all for coming out. So let's see, excellent. Okay, so yes, I'm going to be talking about connections between mathematics and the fiber arts and all the things that Bob described. Very happy to have the opportunity and I'm very grateful to MoMath for inviting me here. So before we can really talk about the fiber art, there's a certain amount of math that we're going to have to explore. And so just to sort of tide you over, you will have noticed this slide that's been sort of sitting up here coily this whole time. So the thing that's photographed in the slide is this. So this is one of the things that I will be showing you later. In fact, I'll pass it around the audience but not yet because I sort of want to show you a little bit of the math first or else you won't really know what you're looking at. What I want to point out here is that this is a triangular shawl and it has a bunch of lace designs going up the right hand side. In fact, if you want to be specific, it has seven of them. And those seven designs have, well, something to say about symmetry. But we're going to have a certain amount of math that we need to discuss before we do that. So there will still be lots of pictures but not so much of created objects. So to set this up so that we can actually understand the fabric of symmetry and how symmetry connects to this shawl and some of the other things that I'm going to be showing you, we first have to address the most important question here, which is what is symmetry? And of course, there are lots of answers to this question, depending on who you ask. I'm going to give you a mathematician's answer. And the first stage in giving you a mathematician's answer is to say that's not exactly the right question. Because we don't view symmetry as a monolithic entity. To us, there are objects that are symmetric in different ways. And so they have different symmetries. So the real question is what are symmetries? And to a mathematician, what symmetries are is that a symmetry is a way of transforming an object or system that preserves its essential structure, which is deliberately worded in a very general way. Because in fact, there are many different ideas of symmetry at large in the world. Today, we're going to be talking about a very specific type of geometric symmetry. But just as an example of how this can generalize, in the world of physics, one of the things that people are often interested in is which processes are time symmetric or time asymmetric, right? So there are some things like billiard balls colliding, where if you reverse the flow of time, the laws that govern what happens are still the same. There are other things like pouring ink into water and stirring it up, where that's very much not the case, right? But in this case, we're going to be talking about geometric objects and geometric transformations of them, objects like this. So they're going to be designs in the plane, right? So what we have here, we have three patterns. And each of them has some symmetries attached, right? And so what I want to do now, so that we can all sort of explore this together, is I'm going to ask you questions about what the transformations are here. In this case, the kind of transformations we're talking about are, we're treating the plane as though it were made out of paper, right? So you can think of these objects as, you know, being made out of paper, and then you can, you know, swirl them around and pick them up and rearrange them and put them back down. And if after you put them back down, it looks the same as before you started, then that transformation, that motion of the plane, is going to be a symmetry, right? So I want us to figure all of that together. And I know that what will often happen in a talk like this is that a question like this will come up, and a few people will think about it really hard and raise their hands and everyone else will sit there quietly and wait for someone else to answer, right? Please don't do that, right? You know, obviously not everyone's going to answer. There are a lot of you here. Again, thank you so much for coming out on a rainy, rainy night. But I really want all of you to think about this, because this is really something that you can kind of see the motions and when you get a little bit of practice in, they start to be a little bit easier to see, right? So for example, on the upper left here, we have this shape that's kind of like a sort of woven star. Can anyone see what kind of motion you could do to the plane that would make that look the same? Yeah? Six meter rotation. Six, oh, this is beautiful. Be completely precise. So 60 degree rotation is the answer that was given, right? Because of course, what often happens is rotation, and then I go, oh, through what angle? But you got it perfectly, yeah. Yeah, absolutely. A rotation by 60 degrees. Actually, are there any other angles that would work? Yeah, 180 would work, right? 180 would just be like if you did 63 times, right? For that matter, you could do 62 times, right? So you could get 120. Yeah, so there are a bunch of rotations that work, but the smallest angle here is going to be 60 degrees. And so that's the one I'm marking, and then the rest of them will come out of that, right? Yeah, absolutely. That's a rotation. By the way, for future reference, too, I will also accept my answers. So sometimes I get like that. So if you're not entirely sure, but you can sort of, yeah, please feel free, especially for this next one. Although I will say I'm getting a sense that I have some ringers in the audience. So it's possible that this will, yeah. So secondly, on the upper right, we've got this object that's sort of like a little cluster of sort of like flowers and leaves. What's the symmetry there? How do you transform the plane? Yeah? A flip horizontally? Yeah, it's going to be a flip, right? And this one is a little bit tricky to see, because if you want to actually do it like with a piece of paper, you sort of have to lift it up out of the plane and turn it around and put it back, right? But yeah. Now it's interesting. You said flip it horizontally. So what is that? Can you show me with your hand what that means to you? Yeah, OK, so like this, right? And I should say, too, there was before that answer, there was an answer up front that was just that. Yeah, so for what it's worth, I tend to think of that. And I thought that was what you were thinking. Like the direction of the movement is horizontal, but I tend to think of it a little bit more in turn. And when I say I, sort of like mathematician, in terms of where the axis is, and I say that because sometimes I will tend to slip and just sort of say, oh, it's a vertical reflection referring to the axis. But that is one of these ambiguities that, like, there's the motion and there's the axis. But yeah, that's exactly what's going on here. And so here we have the axis marked, right? So there's this vertical axis down the center. And if you take this and flip it across that axis, right, then the object will look the same as before you flipped it. You'll notice I also have, in parentheses, mirror up there. Of course, this is exactly sort of what we're used to seeing from looking at ourselves in the mirror every day. But, you know, if you look in the mirror every day. But part of the reason that I put that up there is as we go on, there will be bits of notation for some of these mathematical objects that aren't terribly important. And I'm not going to fully explain them. But sometimes you'll see little cues in them. And one of the cues is, in general, when you see the letter it's telling you that there is a reflection in there somewhere. OK, so now, fun one. The pattern at the bottom, which, before anyone answers, I do want to make this clear, I'm assuming that you have this sense. But what this is meant to convey is that this is a pattern that we're assuming goes on forever, right? So I just sort of let it run off the slide. And the question is, what motion do we have there? Yeah, in the blue. Yes, yeah, you. Is it translational symmetry? There's a translational symmetry, right? So translation is just, and thank you for miming that, right? That's just if you move it without, yeah, absolutely. You can just translate this so that each hook lands on top of the next hook. That, by the way, I think in some ways that's a little bit harder to sort of recognize as a symmetry because I think usually in common parlance you don't really describe just repetition and align as symmetry. But, right, it fits this, it's moving the planet, right? So that is a form of symmetry. So yeah, absolutely, there's a translation symmetry here. Anything else? There's a second one. There's a second one, yes. So can you see what the second one is? It's, yeah, that's exactly it, right? So, and this I think is the one that's a little bit trickier to see, especially if you haven't seen symmetries as much in the past. So you can almost reflect it, but if you reflect it, it doesn't quite line up correctly because the hooks are offset from each other. But if you reflect it and then slide a little bit, you can get it to overlap with the former position. So that's this motion, and because translation reflection is a little bit long to say, we describe this as a glide reflection. It just rolls off the tongue a little bit more easily, right? And so now the neat thing is that it turns out that if you're just talking about symmetries in the plane, every symmetry of the plane ends up being one of these four types. It's a rotation, a reflection, a translation, or a glide reflection, right? Which I'll point out is not 100% obvious, right? Because there's this thing that's basically a translation reflection, and so you might say, well, why isn't there a rotation reflection? And the answer is there totally is, but then when you look at it more closely, it just ends up being a reflection through a different axis. And the question of which axis, there's some fun geometry there, but we will get on to bigger and better things. So I wanna come back to the rotations too, because we have a rotation, and as we said, in this particular shape, we've got 60 degrees and multiples of 60 degrees, but for different shapes that have rotational symmetries, we may have different angles, right? And was there a question in the back? I'm sorry, can you a little bit louder? You can actually, so the question was, can you relate it to an isometry? And the answer is you absolutely can, in fact, all of these things are isometries. So this particular type of symmetry is also sometimes called an isometry of the plane. And for those of you who recognize Greek roots, that's from iso, same and metric measure, right? So the idea is that all measurements are preserved. So in fact, not just that one, but all of them count as isometries, yeah, absolutely. So coming back to the question of reflections, I wanna talk a little bit about recognizing these angles and sort of vocabulary here. So here are some objects that have rotational symmetry. And you may notice the one on the upper right, although it's different or upper left. It's different than what we were seeing before, but it's kind of similar to that star that we were looking at before. In particular, it also has a rotation by 60 degrees. But really I think the thing that is both easier to see, and once you get used to it easier to describe, is the fact that 60 degrees is a sixth of the way around the circle, right? Or to put it another way, if you do that smallest rotation, and you do it six times, you end up back where you started, right? And you can see that pretty clearly from the six equally spaced points of the star. So a couple ways to say this, one is we'll sometimes describe that star as having six-fold rotational symmetry. But a slightly more mathy way of saying it is that it has a rotational symmetry of order six, or an order six symmetry, right? And so just to clean this up, so that's order six. And once you get used to it, that's I think actually much easier to think of orders than angles, right? So again, just to sort of get warmed up and make sure we're all on the same page here. So how about this star here? What's the order of symmetry there? Five. That's gonna be five, right? Fairly easy to see with the five points. Now that is my star case for why order is a little bit easier. Because if I said, ah, you can rotate it through a 72 degree angle, I don't think that's as intuitive for most people. And I don't know about you, but I have to think about the fact that that's a fifth of a circle, right? Okay, good. So it sounds like we're kind of all on the same page. So hopefully we can do the rest of these together. What is the order of symmetry of the upper right? Three. A little more enthusiasm. Here. Lower left. Two. Yeah, that one's a little bit tricky. I know, I suckered you into that. That one is a little bit trickier, right? Because you're right, they're four points, but you notice how they're not the same four points. So you actually have to go halfway around for this one. That one was very mean of me. That one's actually order two. Okay, but look, we're good on this one, right? What's this? Order four. Order four, excellent. Okay, okay, so bonus round. Right, so ooh, yes, bonus round. So you'll notice that some of these shapes also have reflection symmetries, and some of them don't. Right? Okay, so now that I've had a long pause so that you guys can think about this a little bit, does anyone want to say something about which ones of these have reflections and which ones don't? Yes, in the black jacket, yes. Yeah, I think the green star, the blue star, pink diamond, and the blue semi, I don't know, swastik sometimes. I worked very hard to avoid swastika appearance on that. I just want to make this perfectly clear. Listen, I'm going to say order four chiral symmetry to get all chemical on this. It is really hard to avoid very, very angry at the. Why did they take that from us? Okay, so we have, basically, so you're saying everything but order three. Yes. Right, okay, so conflicting view, what do you think? Okay, so we have a contested notion on order four. Okay, well how about if we, well yeah, sure, let's do a vote, come on, we're a friendly crowd. How many of you think that, I think we're a friendly crowd, how many of you think that this order four object has reflection symmetry? Okay, and how many of you think that it doesn't have reflection symmetry? Okay, so since math works by democracy, well yeah, so the room seems to be in favor of no reflection symmetry, and I have to say I do agree with them, but this is a good sign of the fact that it is, like it takes some practice to sort of see what's going on here. If you reflect this, like for example this way, then the part that used to be on the upper right is gonna be on the upper left, so it's gonna kind of spiral around the other way. And this question of things that aren't mirror symmetric, that really is something that with practice, it gets a little bit easier, but yeah, that one's subtle, I would say. I think the order three, one, it's a little bit clearer that it doesn't have reflection symmetry. So yeah, indeed, that's exactly the way that it breaks down. The order five, six, and two have reflection symmetry, the order three and four don't. Okay, good, so now we're ready to up our game, because what I'm really interested in is not so much these isolated objects as repeating patterns in the plane, patterns like this, right? So what I have here are three patterns in the plane, and for all of these, I'm just showing you a little piece of this, but kind of like that pattern that stretched on the bottom of the original symmetry slide, you wanna think of these as patterns that really fill an infinite plane, right? And I hope that even just glancing at them, you already get the sense of like, yeah, there's a lot of symmetry going on there, right? But here's the major point that I wanna make. These three patterns, superficially, they look rather different, right? Different shapes, one of them's a little bit more decorative, the others are purely geometric, but I claim that they have the same symmetry structure. So in other words, they have the same types of rotational symmetry with the same orders, they have the same reflections, and the same glide reflections, and the same translations, and the same relationships to each other, right? Now when I say the same, I don't mean literally the same, I mean structurally the same, and this probably won't be quite clear until we go through this, so that's part of what the slide is about, but so to get us started, right, I'm claiming you can see all the same types of symmetries here. I wanna do a little bit more in the way of symmetry hunting, and so let me start off with, there are some rotations here, right? You can sort of see there's some rotational symmetry. What orders of rotational symmetry can you see here? So I think I heard a six, yeah, I'm hearing six, okay, and I should add too, part of it, right, I'm claiming these have the same symmetries, you may find that some symmetries are easier to see in some patterns than others, right? So a bunch of people have said six, yeah, we see order six. In any one of these three patterns, can you tell me where I would have to point to get the center of an order six rotation? Yeah, he's saying in the leftmost pattern in the center of a hexagon, if you rotate around that, right, one sixth of the way around, and the hexagons will line up to themselves. Here it's at the corners where six triangles come together, and in fact it is true that you could also see there are a bunch of hexagons in the triangle pattern kind of hiding in there, so that sort of shows you, and then these stars, and that's a little bit of a giveaway because that really is the same star from the previous slide. If it looked kind of familiar, that's where I stole it from, right? Okay, excellent, so we have order six rotations. Any other orders? Yeah? Order three. Order three, we have some order three rotations. Where? At a point between the three hexagons. And then it could be between any of the point, the vertices between the triangles, and the vertices would do for the second one. Well, weren't the vertices order six? But they're also order three. They are also order three, but they're, like it's order three just because it's order six. What you've described here is a little bit different. That's order three and that's the smallest order. So if I'm not lying about the fact that these have the same symmetries, in the triangle somewhere, there should be a place where we can point where you have to go a third of the way around. Yeah? The center of the triangle? Yeah, center of the triangle, right? So the role's kind of reversed. Here it's corner, center, before it was center, corner, right? And over in this one, you can kind of see right in the middle of each of the circles, right? There are three stars around it. So you have to rotate a third of the way around, right? Okay, any other orders of rotational symmetry? Way in the, oh, actually there are a couple of people way in the back, way in the back corner. Yeah? Order two, I think, in the triangle one, a diamond. Order two in the triangle one, where did you say the center was? The diamond. Diamond, oh yeah, there are these diamonds that are like two triangles right in the middle there. Oh, how about that? I like that, by the way, that's a really nice way of seeing that order two symmetry. I usually think of it as being more like you've got the edge and the edge goes to itself, but yeah, you can see it either in terms of that edge of the triangle or this big diamond shape if you rotate exactly halfway around, right? And you can't do a smaller angle there. So yeah, there's some order two there. That's the really sneaky one. So all of you who are like, oh yeah, I see it. This is good, these are advanced moves. How about, can you see where the center is in the hexagons? Yeah, same idea, right in the middle of each of the lines, right? And here there's actually also an edge that gets bisected so if you go on the edges of these sort of pieces in between the circles and the stars, right, you see it there. Okay, great, so in each of these we see order six, order three and order two symmetry, yeah? I was just asking to generalize it, is that the case just for any tessellation or does the tessellation have to be at a certain angle or it's just generally any tessellation will exhibit plain symmetry? These are wonderful questions, hang on. Those are exactly the questions you should be thinking about, that's fantastic, right? So is there something special about these tessellations or does this happen in general? Oh, what a good question. Okay, so anyway, we've got all of these rotations. We're so getting there, right? Reflections too, can you see reflections, right? So, and there's so many, I sort of want to go where are the axes, but there are just so many, there are horizontal axes for each of these where you can reflect, there are vertical axes for each of them, and then there are a bunch of slanted axes as well, right? So, I have now, I'm gonna show you a labeling of the symmetries here, but just some of them, because if you actually label all of the symmetries it completely obscures the diagram, so I just have here a label of like one symmetry of each type, right? So these are the symmetries labeled, so the centers of rotation are shown by these little dots, so red is order six, blue is order three, green is order two, the orange axes are axes of reflection, and the really subtle ones are there are actually some glide reflections here that are not themselves reflections, they're super subtle, but they're there as well. So here's the fundamental point, if we strip the designs away and just look at the symmetries, we're actually looking at the same structure, right? I mean, you can see the scales different and one of them's rotated compared to the other two, but you have the same types of rotations and they're connected in the same ways to the reflections, right? Like for example, each one of the centers of order six rotation has this cluster of reflection axes coming out of it, so that's what I mean when I say these things have the same symmetry structure, right? And so we actually have a name for this underlying symmetry structure, these are all patterns of a type that are called wallpaper patterns, and so what makes them wallpaper patterns is that they repeat in two different directions, in fact, many different directions, but two is enough, and so they're kind of like wallpaper and that you can take a little piece of the pattern and stamp it over the plane over and over again and get the entire plane pattern, right? Which for wallpaper is sort of a manufacturing necessity. And so these have the same symmetry group, which is the symmetry structure and the notation for the symmetry structure, there are a couple of different schemes. The one I'm showing you here is the international crystallographic union notation, and so this note happens, a group happens to be called P6M, and again, I'm not gonna fully explain this notation, but just to point out the six, as you might expect, that's because the order six rotation, and as I mentioned before, the M is for mirror, so it indicates that there are reflections here, right? Okay, so these have the same symmetry structure. By contrast, question in the back, by contrast, here we have two diagrams that look very similar, right? They're both based on the same triangular grid, but the fact that the one on the right is shaded in is going to change the symmetry structure because there are some moves of the grid on the left that if we do it to the grid on the right, it will change the appearance. Can anyone give me an example of such a move? Yeah? Uh, 50 degrees. Absolutely, right? Like if you take this whole thing, so there are a bunch of ways that you can flip it by 180 degrees, so like here, for example, if we just go up one of the corners, we can flip it 180 degrees, and the unshaded grid, it doesn't make any difference. In the shaded grid, suddenly the gray triangles would all be pointing down instead of up, right? So it's going to change the appearance, yeah? 270 degrees? 270 degrees? 270 degrees, how many is 270? So 270 would be minus 30, or minus 60, right? Yeah, so that's going to be... Oh, minus 2. Oh, minus 90, yeah, see, the problem is that's actually going to change the one on the left as well, right? So yeah, if you rotate by 90 degrees, then these lines that are horizontal will become vertical, and there'll be no other lines that will become horizontal, right? So that's not quite going to work. You're going to need to have multiples of 60, right? But there are actually some multiples of 60 that do this, yeah? So 360 degrees actually gets you back to where you started, right? So that's going to work for every pattern. In fact, by the rules of math, we actually don't even count that as a move, because you just go by the end position. So if you rotate by 360, it's like, oh, you didn't move it. Shh, it's our little secret that it all ended up back where it started. But for example, 60 degrees will also mess this up, right? If you just rotate it one sixth of the way around the circle, you're going to be swapping gray and white. And there are also some reflections that will mess up the pattern on the right, like the reflections through horizontal axes. That's not going to work because it's going to flip all the gray triangles. Vertical axes are still fine, right? So if we mark the symmetries here, the pattern on the right has fewer symmetries than the pattern on the left, right? It's missing some of the reflections and some of the glide reflections and it only has rotations of order three. And so these have inherently different symmetry structures. So in fact, these are wallpaper patterns with different symmetry groups. Or for what it's worth, well, sometimes if we're talking about the symmetry group of a wallpaper pattern, we'll just say wallpaper groups, right? And so these two groups happen to be P6M and P3M1. This question of shading, though, does actually create an interesting side issue, which is that depending on what you're doing, you might want to look at shading or you might want to ignore it, right? And in fact, a really good example of this is the MoMath logo, right? Or the Meta logo, to be more precise, right? The MoMath logo is really like a rule for taking a symbol and then there's some coloring and some symmetry involved. And so here we have the MoMath logo made out of lowercase e's. And so there are two ways you can look at this. You can say, okay, I don't really care about the color. I'm just gonna look at this as a shape, right, against a white background. And so if you're not counting colors, if you don't care that yellow lands on yellow, what's the order of rotational symmetry here? That's gonna be eight, right? Because there are eight points. The colors help a little bit because you can see that there are four colors and two points of each color, right? So it's order eight if you don't count colors. How about if colors do matter? What's the order then? It's just gonna be two, right? Because yellow has to land on yellow and so you have to go halfway around, right? So if you discount color or if you allow the colors to change places, then we have order eight symmetry. Counting color, it's only order two symmetry. Incidentally, so you will notice because you can use any symbol here. I did make this from lowercase e's. Does anyone here know why I chose this particular symbol? Yes, because today is e day. Oh my gosh. Yeah, I should put it in the states. It's e day in the states. So if you were in any of the most of the countries in the world that do it the other way around, this doesn't happen until July 2nd. Or to put it another way, you have another chance to celebrate. But other than that, this is it for the century, right? Yeah, it's e day. So I'll mention other than some uses of e's, which you may also have noticed if you looked at the clipboards you were given, there's not that much mathematical e in this talk. So in interest of observing the day properly, I'll point out a lot of people have written some very nice things about what you can actually do with e day. And one that just recently came out is in slate. There's a really nice write up by Evelyn Lamb, who is a wonderful popular mathematical writer. If you haven't seen her writing, I definitely recommend tracking her down. And so she has this thing on what you can do for e day. It's a little late in the day. So either you need to do it quickly after you leave here. Or like I said, July 2nd, right? You got another go. Anyway, so back to this question of wallpaper groups, right? So we've seen now two different wallpaper groups and a natural question is okay. So there are two of them, how many are there? Right? And this is a question that depending on how much you've looked through the materials we've given you and counted, you may already know the answer to, even if you didn't know it before this talk. Probably not though, because you probably didn't go through and count, I would guess. The answer's actually been known for a while. It was found in 1891 by Evgra Fetteroff. And it turns out that there are exactly 17 different possible wallpaper patterns. And one of the nice things is that not only do we have this result, but there are also a lot of nice schemes for taking a given wallpaper pattern and identifying which wallpaper group it corresponds to. And that's the handout that I've given you. I also have it on a slide up here with the understanding that you can't read that. In fact, to be perfectly honest, you can probably barely read the one on the handout that I gave you. It's really hard to fit on a page. But I gave this to you partly because it may be a little bit helpful in later parts of the talk, but more because I'm hoping that you'll take it with you out of this talk and start looking at things and identifying what groups you're seeing. Because once you start looking at this, you'll see these everywhere. But the basic way that the flowchart works is that you start down this main line by looking for rotations and what rotation orders you see. And in the flowchart, there's this added piece of information which is a fairly deep fact about wallpaper groups. And that is that the only orders of rotation that are compatible with wallpaper groups are six, four, three, and two. Which is kind of a deep fact. That turns out to be part of this theorem of Fedorov. Later rediscovered by Polya. So you start by picking out the highest order of rotation and then there are branches for each of the orders. And then after that, you ask, are there any reflections? And then there are more subtle breakdowns for some of these orders. So I'll also mention this, sort of describes following the paradigm of Dorothy Washburn and Donald Crow, that paradigm is fascinating. If we had three hours, I'd tell you all about it. But I will say that on the handout, one of the sources that I give, one is the book that Bob alluded to that Ellie Baker and I wrote on Beed Crochet. That's where this diagram is modified from. There's a larger, more readable version in that that spans two pages. But I also have symmetries of culture by Washburn and Crow. And it's a fascinating book. Just something that you might be interested in looking at. So these are the wallpaper groups. These are the symmetries that you get when you have patterns that repeat in two directions. But in the design world, there are also lots of interesting patterns that repeat in only one direction. And so we also have a classification for those. Those are called because of the corresponding art form freeze groups. Because freezes are pieces of art that are linear and repeat in one direction. And so here is a photo array of all seven freeze groups. This actually, this is quite fun. So St. Mary's College of Maryland, from where I hail, is a lovely school. If you ever happen to sort of be in the D.C. area and then want to drive like two hours south. I do welcome you to come and visit us. It's a beautiful campus. It's also sort of a deceptive name. We're actually part of the Maryland Public System, which is fairly confusing to people. It's the place was named by the Catholic settlers who went there and so. But part of our mission is outreach and that includes an interest in global studies and international programs. And so one of our philosophy professors, Michael Taber, regularly leads tours to Greece. And I was lucky enough to be the second faculty member to come along on this tour in 2015. And when I was there, just because why not, I decided to go on a freeze hunt. So from that summer, I've got like not kidding, hundreds of photographs of freezes that I saw in various places. And so this is just me picking out one from each of the seven groups. One of the sort of interesting things, and this is very much connected to the work of Washburn and Crow, is that some of the symmetries are more common than others. And so you'll notice that most of these are legitimately free symmetries as I described them before. There are patterns in the plane that repeat in a straight line, except for this last one. So here the pattern is bent around an urn. And the reason that that was what I chose is because out of my hundreds of photographs, I had exactly two that had this group on it and neither of them were flat. Right? But it is a good illustration of the fact that very often in the sort of in the natural world, freezes will be wrapped around curved surfaces, right? So like for example, the edges of urns are actually a fairly common place to see freezes out in the wild. Okay, so the thing that you accidentally got the sneak preview of, this brings us to the reason why you have sheets of stickers and paper with squares on it on your clipboards, because this is the point at which you get to play the game. So this is make your own free symmetries. And just so you know, because I have been asked this before, I was asked this when I tested this out with some of our students, you may well ask, why didn't you just give us this? Why are you making us make them? And the answer is because I mean, you can let me know if I'm right about this, I genuinely think that taking these stickers and positioning them is going to give you more of a sense of how the designs are moving and what the symmetries actually are. So in honor of E Day, you will notice that the design is basically a stylized E. I'll also point out, I guess for the sake of the video, this is sort of a computer sketch. The stickers that you have are actually photographs of a needed piece that I made, because you know, fiber arts, it seemed appropriate. And so we have pieces that have E's on them, and then we have pieces that have the mirror reverse flipped E's on them. Couple of observations I'll make. The first one is that this first diagram is all E's and the second one is all flipped E's. And that's important to get the counts right for the rest of the stickers. The other designs, it so happens, use equal numbers of E's and flipped E's. So that's observation number one. Observation number two is don't worry about laying the stickers down too precisely in the squares, it's all good. You'll see the patterns anyway. And observation number three is I'm gonna suggest, especially after you get through the first few, that you don't complete these patterns, but instead you just do the left half of the stickers. And the reason for that is because this takes a little while. And so we'll have a nice pause for you guys to work on this, but then I'm gonna wanna sort of pick up the discussion a little bit and we'll talk about these. If you have the left half, the right half just literally repeats the left half. And while we're doing this, I may circulate around and see how you're doing and see if there are questions. I see. So if I can cut in for a moment here. Oh, here we are. Find my little clicker. Okay, so the pictures are gonna be up for a little while longer here, just a little bit harder to see, because what I wanna do now that you've had the opportunity to play stickers for a little bit is talk about what the actual symmetries are that you're seeing here, right? So we can just sort of quickly go through this. And I'll start off by pointing out this first one, that's sort of clearly kind of the baseline pattern. It's just a bunch of repeated non-flipped E's. And the thing about a lowercase e, besides the fact that it represents the constant that is this day, is that it doesn't have any symmetries of its own, right? You can't rotate the E's and not notice that you've rotated them and you can't reflect them. So this one is just going to have the translations in one direction that make it a freeze pattern, right? Whereas the one right beneath it, this is the one that you made entirely out of flipped E's, that's got something else. Can you see what else it has besides translations? Yeah, there's some rotations there, right? There are, I will point out, no reflections or glide reflections. And you can see that from the fact that you made it entirely out of flipped E's, right? If there were a reflection, then some of the flipped E's would flip to become regular E's. So this one just has translations and rotations. You'll notice those sort of two different types of rotation centers. There's one that connects the tails of two E's and there's one that connects the backs of two E's, right? Okay, so those are our first two groups. We've got only translations and for the rest of these I'm gonna leave off the translations parts because they all have translations. Again, that's what makes it a freeze group is that it repeats in one direction. So the next one is rotations. How about this third one? What does that have? Remember, I will take mine, yes? Yeah, it's got reflections. Which way do the axes go? Yeah, they're vertical, right? So yeah, this has vertical reflection axes. How about this one down at the bottom of the left-hand column? What have we got there? Actually, no, because then forward pointing E's would become backward pointing E's, right? So that's not, you're not gonna be able to rotate that. Yeah, that's what it is. You've got a reflection with a horizontal axis, right? Okay, now for the second column I actually wanna start with the bottom one because I think that one's a little bit easier than the other two. What's going on with the bottom pattern down here? Yeah, that one's gonna have a glide reflection, right? And the big giveaway is that's the one that I made you set up in this weird way with a half step. That's to get the glide reflection going, right? And I think that's one of the places too where the red corners really help. You can really see the glide reflection in the red corners. Okay, so how about the top of this column? What have we got here? Do we have rotations? Yeah, we've got some rotations, right? Again, there's sort of two different types of centers. You can see here, there's the red corner centers and the orange corner centers. Reflections, yeah, yeah, you've got reflections both ways, right? Here you've got both horizontal and vertical reflections. So it's sort of like all the reflections that you could have. Can we both have a glide reflection? Yes, and this is by the way a detail that I sort of glossed over in the rest of these. Very often when you have a reflection, there's also a glide reflection. You know, the actual word that she used was inadvertently. I like that way of wording it. It's sort of incidental because you happen to have a reflection and then a glide separately. And so if you combine those, you get a glide reflection. That's the reason why I didn't put glide reflections up here. And for what it's worth to, no one asked about this, but on the flow chart in a bunch of places, I have the phrase indecomposable glide reflection. Indecomposable means not by accident. It means that you can't just do the translation and you can't just do the reflection. You have to do both of them, right? OK, so how about this last one, the middle of the second column? Does it have rotations? We have a vote here for yes, if extended it would. Do you have a center in mind? That's the tricky part, right? It's a little bit harder to see in this one because the rotation centers are actually in the middles of the sides, very sneaky. I think the red corners help a little bit. Yeah, you see that one's kind of sneaky. So there are actually rotations there, right? And there are reflections with vertical axes. And then there is a glide reflection down the middle. That one's a little bit more subtle. And so yeah, for what it's worth, I had made this mention in passing. And again, this is a lot of what the work of Washburn and Crow is about, that you will often find in various parts of the world and various art forms that some symmetries appear more often than others. I actually learned literally just last month in a talk by Dara Shavi, who is one of Donald Crow's students, that in many cultures, this group that's got the rotation and the reflections in the glide reflection is actually much more common. Because there, the glide reflection kind of arises from the way that you've arranged the simpler rotations and reflections. This is often, in many settings, much rarer. And that was that pattern that I only found two of in Greece. But that varies from place to place. OK, great. So now we're ready to look at the actual fiber arts that show these different patterns. And so before I get into that, I would be really remiss if I did not point out the great source of all things math art related. I cannot recommend this site enough. It is just fantastic. Bridgesmathart.org is the site run by the Bridges organization, which was founded by Reza Sarhangi, who we have here. Reza, unfortunately, we lost a couple of years ago, which was a really great loss to the mathematical art community. But we take some comfort in the fact that there is still this thriving community of people that he built. And so one of the big things that the Bridges organization does is that every summer, there is a math art conference, which is the biggest math art conference in the world. This summer, it's going to be in Stockholm. So Stockholm late July, if you are interested and up for international travel. But the real reason why you want to look at the website is twofold. The first is that the Bridges organization sponsors both at the Bridges conferences and also at the joint mathematics meetings, which are the big national meetings every January, and an exhibition of mathematical art. In fact, I'm going to have pictures of fiber arts here. A lot of those pictures come from the catalogs from those exhibits. And all of the catalogs are on the Bridges website. So that's just hours of looking at amazing, beautiful things. And then at the Bridges conference, the people who speak are speaking on papers that appear on the conference proceedings. And those proceedings are all open access. So right now, you can go to the bridgesmathart.org website and find all the papers that have been printed in the Bridges proceedings. And there are papers not only on things like this. In fact, some of them are papers of mine, but papers on math and the visual arts, math and music, math and literature, math and culture, math and architecture. It's just an amazing collection of things. So I strongly recommend that you peruse that. So what I'm also going to do, although we're not quite getting to these yet, in the interest of people having time to look at these, now that you've thought about the groups a little bit, I'm going to pass these two things around. So this is the shawl from the opening slide. And this is a scarf. And so hopefully you guys can have a chance to look at that. And then some of you may be lucky enough to have them when they come up on the slides. OK. So the first example of the kind of artwork that got me really interested in this connection between symmetries and the fiber arts, at least the first example that I know of, is this piece of embroidery. It's a countered cross stitch by Mary Shepard. And it actually appeared in an exhibit at a different conference, a math fest in 2007, called Wall Papers and Cross Stitch. And this is the earliest example that I know of, of an art form that I and other people have come to call a symmetry sampler. So it has something to say about symmetries, and we'll get to that in a moment. But it's also a sampler piece. And those of you who actually do fiber arts, I already know that there are at least some of you in the audience, you may be familiar with this idea. A sampler piece is traditionally a piece that assembles a bunch of different patterns into a single piece. And they're usually designed to let you practice different techniques or show off, yes. There's a sample, do you want to stand up? Actually, you should stand up and come up here. So this is absolutely perfect. Here we are. This would be a sampler piece, right? So this is a sweater that has a bunch of different examples of whether they're all cable work. No, it's all different. Oh, it's all different things, yes. So there's some cable work. Oh, that's fantastic. Did you make that? Yes, so here we are. This is, what is your name? Margaret. Margaret? Yes, so Margaret. Thank you, perfect, right? So that's what a sampler piece looks like, right? You cannot plan these things. It's great. Fantastic. Lovely sweater, by the way. I've been sort of admiring it from the front row. So this is a sampler piece, but in this case, the patterns have been assembled for a specific mathematical purpose. Before I go into the details there, though, again, and the things that you should know about, I should also mention that this is actually one of 10 chapters in this really lovely book, Making Mathematics with Needlework, 10 Papers and 10 Projects, which is a collection of papers that was edited by Sarah Marie-Rail Castro and Carolyn Yackel, who are sort of the founding mothers of the current mathematical fiber art trend, in my opinion. They have been running a knitting circle at the joint math meetings for many years, and that turned into a special session with talks about mathematics and the fiber arts, and that turned into this book, which features all sorts of different crafts. And if there were, for example, any, I don't know, teachers in the audience, there are also, in each of the chapters, sort of indications about how you could conceivably use them in the classroom. So it's a great book. I definitely recommend checking it out. And actually, while we're at it too, so Carolyn Yackel, both Carolyn and Sarah Marie, make a lot of interesting artwork, but Carolyn Yackel made her own, I'm gonna say sampler piece. You might quibble, because it's not one piece, but they're spheres. So what are you gonna do, right? So these are examples of tamari, which is a Japanese art form where you embroider on the surface of a sphere. And so this is a collection of tamari spheres showing spherical symmetries that Carolyn put together. And again, I'll get to the details of that in a moment, but I would be remiss if I didn't mention that this is actually featured in the sequel to making mathematics with needlework, which is crafting by concepts. I mean, especially because they're on the cover, right? You can't not show the book when Carolyn's tamari balls are on the cover. But coming back to this sampler that she made. So this is an example of what I would call a complete symmetry sampler, right? Now these are symmetries that we didn't look at because we were talking about symmetries in the plane. These are symmetries on the surface of a sphere, but you can categorize them as well. There turn out to be 14 types. And so what this is, is 14 tamari balls, one for each symmetry type. That's what constitutes a complete symmetry sampler. And they are, as I've indicated here, in what I would consider to be, in this context, an unconstrained art form. And what I mean by that is that according to the rules of tamari, you're allowed to put your needle at any point on the surface of the sphere, right? So there's nothing in the craft itself that is going to stop you from making any of the symmetry types. There were 14 symmetry types on a sphere. She has all of them because tamari does not impede that in any way, right? Whereas, if you take a look at Mary Shepard's counted cross-ditch sampler, and if you were looking at this before, you might have had a moment where you went, wait, okay, that's weird, because it says wallpapers, and we learned that there are 17 different wallpaper patterns, but she only has 12. What's going on there? And the answer is, it's because this is a constrained art form, and the constraints won't allow all of the symmetries, right? In particular, counted cross-stitch is done on a square grid. That's gonna mess with some of the rotations. Can you see what orders of rotational symmetry you can't get here? What can't you get? Yeah, six and three are gonna be out, right? Because squares have four-fold symmetry, right? You rotate a square by 60 degrees. Suddenly, horizontal and vertical don't work anymore, right? So if you take a look at the flow chart, and even if you don't, I will just tell you, right? You have to cross off their two groups with order six symmetry. There are three groups with order three symmetry, so that's five you had to cross out, and 17 minus five is, I think, still 12. So there we are. These are the 12 groups that are left, and Mary Shepard Sampler is basically a proof that you can get all 12 of those groups. Okay, so Bob had mentioned this work that Ellie Baker and I did in Bead Crochet. And by the way, I should also mention, so the book is by Ellie Baker and myself. In fact, some of the original work on Bead Crochet was also done by Ellie Baker's daughter, Sophie Summer. And so Ellie and Sophie were trying to work out some Bead Crochet design issues that they were happening, and ended up getting in touch with me, and then amazing things happened. But this is an example of what Bead Crochet looks like, if it looks kind of familiar, because it's the set that I'm wearing now. So if you wanna see it closer, you can after the talk. And the thing that's tricky about Bead Crochet is actually tied to the thing that's easy about Bead Crochet too. So part of the reason that Bead Crochet is really popular, it's actually sort of gain some popularity, is that, so you have to string all the beads in advance, and you have to make sure that they're in the right order, and so that takes some attention. But once you've done that and started the bracelet, it's the same stitch over and over and over and over again with no change. So it's very meditative, you know, you can do it in waiting rooms, it's awesome, right? Here's where you pay for that. When you are designing Bead Crochet, it's a lot harder than it looks. And here's an illustration of that, right? So this is an example of what a typical chart for a Bead Crochet bracelet would look like. This is telling you what order to string the colors in. And so it's basically like if you could imagine slitting the tube of the Bead Crochet and laying it flat, and it didn't fall apart, which is what would actually happen if you did that, right? So that's what we have here. And so if you look at this and you're not familiar with Bead Crochet, you look at it and you go, oh, that's lovely, it's got lots of symmetries. In fact, it's got this really nice reflection symmetry across this vertical axis here. Except, and I see some of you already know the punchline, except no, because here's what that bracelet looks like from the other side. And that's exactly the thing that kept happening when Ellie and Sophie were trying to design bracelets. And so long story short, the solution that we came up with is to say, well, we're kind of focusing too small. We don't wanna look at these two little patches that constitute our bracelet. These are really both pieces of a design that fills the plane like that. And so what this is is this is a picture of this conceptual design space, which is where we now do most of our designing called the Bead Plane. So it's this infinite grid of beads, right? And then you learn the rules that allow you to wrap it around bracelets of different sizes. And if you do that, then it becomes a really flexible design tool. But part of what's sort of exciting about this is this is now a pattern in the plane. And it repeats in two directions, which means it must be a wallpaper pattern, right? Question? Maybe a little to one side on the question, but when I looked at that, it immediately reminded me of a snake. So the comment was that it immediately reminded him of a snake. You are not the first person to say that. In fact, there are some early examples of bead crochet from the Victorian era that were deliberately made to look like snakes. They're like you can look around for pictures. Yeah, yeah. And people will commonly have that reaction to them. Because they have certain symmetries too. Yeah, yeah, yeah, they do. I mean, a little more approximate, as is often the case, but yeah. Okay, so the question now is what symmetries are possible in bead crochet? And you'll notice that there is a sort of restriction kind of similar to what you see in countered cross stitch right off the bat, which is that the way the beads are packed, you can get order six symmetries and order three and order two or five, but order four doesn't work. Because these are packed in this way that sort of has a lot of hexagons and triangles in it. So right off the bat, when you're trying to figure out what groups are possible, you have to cross off the branch with order four symmetries. So that's these three up here. But then it turns out that because of the wrapping around a bead crochet bracelet, there's another group that's excluded for much more subtle reasons. So subtle that I cannot possibly explain them to you now. But you can ask me about it later if you like. And that group is P31M. And in fact, even more than that, so these four groups are impossible. There are two more groups that are technically possible, but only in really boring ways, right? Which I'll grant is maybe not a mathematically rigorous term. So let me show you what I mean. So P3M1 and P6M, literally these are the only three patterns that you can make that have those symmetries. These are your choices. You have to do three colors like this. You can pick the three colors, right? But it has to be like this. Or two colors or one color, and that's it, right? Okay, so four of the 17 wallpaper groups are impossible. Two are technically possible, but super boring. There are 11 left. And so Ellie Baker and I put together this piece for one of the joint math meetings art exhibits. By the way, the JMM that keeps coming up, that's the joint math meetings. And so this piece, you can see it's sort of designed so that we have the bracelets, and then we have little pieces of the bead plane so that you can see the pattern that generated the bracelet. So as mentioned before, these things are all in the book, in the last chapter of the book before How to Bead Crochet is actually all about the wallpaper group material. This necklace, by the way, is also one of the patterns in the book. But there's a sequel to this too, because as you may have noticed in the photograph of this necklace, this actually has multiple patterns and kind of smoothly transits between them. And I'd never tried this with wallpaper groups and had sort of thought, oh, it would be too hard. And then I thought, you know, maybe not. And so I sat down and did some experiments, and so this is actually my latest bead crochet set. This, by the way, this is a jewelry set, and this is actually one of the things that I have with me, I'm not gonna pass it around because it's a little bit fragile, but in this tin, I have this jewelry set. So if you have a chance to run up after the talk, I'm not gonna try and get it out now, you can have a look at that. So this is the necklace, which has a smooth transition between all 13 patterns. And what I did was I put the boring ones behind the neck. Right? And then the earrings and the bracelet are flat bead weaving, and they show the 11 interesting patterns. As they appear on the plane. So it turns out too, like this is sort of what you get when you do this sort of simple form of bead crochet rope. If you're a little bit creative, you can also get bead crochet to lie flat. And so this is a piece that I made when I was experimenting with this. That's another piece that I have here. So if you wanna take it and look at it after the talk, it's I think both on the slide and sitting right here, gonna be sort of hard to see the details, but you can kind of look at it a little bit closer. After I speak. Huh? It's not a tube, it's flat, right? So this is a flat disc of beads. And so you will notice that there are seven bands of colored patterns that are wrapped around this circle, which correspond to anyone? Yeah, these are gonna be the seven freeze groups. Right now, cheating a little bit because they're not in straight lines, but each one of these patterns, if you straightened it out, it would give you one of the seven freeze patterns. And so this was around the point where I started to get interested in freeze patterns. And so I was sort of noodling around with these. And then there was the joint math meetings art exhibit in 2016. And even before the conference, looking through the catalog, I was very excited to see this artwork. So this is a piece called Norwegian Freeze by an artist named Heather Ames Lewis, artist and mathematician. And this is a piece of stranded colorwork knitting that shows all seven of the freeze patterns. But what's really exciting about this one, and the reason that it's called Norwegian Freeze, is that these are all historical Norwegian patterns. So she wasn't just trying to find the seven freeze patterns in colorwork, she was wondering, do they already exist in traditional Norwegian patterns? And as was the case when I was searching for the Greek freezes, some of them were harder to find than others, but she was eventually able to find examples of all seven and she put them together into this work. And this was particularly interesting because this appeared in 2016, which means she was making it in 2015, the same year that I was working on this. And so, oh wow, like we both were trying to do freezes and knitting. And this is around the time where I realized, oh my gosh, they're like a bunch of us making symmetry samplers, that's kind of cool. So yeah, this is a sweater that I threw together. By the way, for what it's worth, you remember when Bob mentioned that there was a thing that fails to get into knitting, that's the thing. But yeah, so this is a sweater that contains all seven freeze patterns. Here are them charted. And the reason why I wanted to show you the chart is just for one thing, and that is, so these are the freeze groups in knitting. So you might go, well, what about the wallpaper groups? Right, and if you look at this, you go, oh hey, square grid. So it's just gonna be the Mary Shepherd thing again. It's gonna be counted cross stitch. But the charts lie, because as those of you who knit know, standard knit stitches are not square. So in this form of color work, you really can't get square symmetries. And in case you don't believe me, take a look at this waist chart, and you see how these flowers all look like they kind of fit in squares? That's what the waist actually looks like. They are not square. And they're not even close enough to fake it. So trust me, I tried to do all of the patterns for the Mary Shepherd thing. 90 degree rotations just do not work. So you have to cross those three off. And so instead of the 12 groups that Mary Shepherd had, you end up with nine groups left, and that's where the pattern that did get into knit comes in. So this is the scarf that's passing around that some of you have been looking at. There are these nine groups that are possible that are left, and they're all represented in this double knit scarf. And you may notice that up here, I have crystalline, the name of the scarf, and nitty in red, and that's because if you are interested in making this and you knit, then those are probably what you're gonna wanna put into Google. The great thing about nitty is that it's free. So this pattern is available free, basically in perpetuity, if you would like to try your hand at it. And then for the art exhibits, this is a wall hanging version of it that I made on smaller needles. It's like a 20 by 20 piece. So one of the things that's really cool about sort of math art people is, so I was chatting with Heather Ains Lewis about her freezes and my freezes, and we both had the next thing that we wanted to do. Hers was actually knitting in cable work, to figure out what symmetries she should get in cable work. And I actually have to check in with her because I'm not sure if she's pursued that. But my next thing was lace. And so this, so that you can see it a little bit better, I put it sideways, it actually hangs like this. This is a piece of lace knitting that appeared in the most recent joint math meetings art exhibition that has one, two, three, four, five, six, seven freeze patterns. And there's an interesting reason why this one is at right angles to the rest of it. Ask me about it later if you want. But in addition to this, I can actually show you this is sort of like really hot off the presses. This is something that I've just done in the past month. And that is, so I would give talks about pieces like this. And when I was trying to show the symmetries, what I would usually do is show the knitting charts. So these are the charts for the four patterns that are in the upper left. And the knitting charts are sort of designed to look kind of like the knitting looks. And then I would mark the symmetries on them. So these are the reflection axes in red, glide reflection axes in yellow. They're rotation centers in dark blue and then some arrows for the translations. And I really thought I was done, like this particular pendant, lace pendant, was like the fourth one that I'd made. And I was like, okay, I'm done with this. I've worked it out. And then I had this idea which was what if I could incorporate the diagrams in the knitting itself. So these are two pieces that I made just this last month. They are a little bit hard to see in the slide. So for what it's worth, I also have the one on the right with me. In fact, I'll see if you can see it if I hold it up. But it's a little, I'm a little bit more, but yeah, so this has beads incorporated in it. And the beads are actually marking all of these axes in a color-coded way, right? So yeah, thank you. Yes, I'm very excited about these pieces too. So this is here if you'd like to take a look at it. Okay, so just a few. That's my own little secret. Now I'm not, I feel like, see part of it is I don't know that there are that many people who would actually want to make a lace pendant. I don't know, I could be wrong. Well, okay, but here's the thing, like if that's what you want, that's where this thing comes in, which is that the shawl, oh, I'm sorry, I almost forgot before I have this, I do have this slide. So this is a public service announcement for those of you who haven't done lace knitting and are knitters and are interested in trying it. There's this little secret that they don't always tell you which is that when you knit lace, it looks bad. And so you'll look at it and you'll be like, oh my gosh, it doesn't, you'll notice that my knitters here in the audience are going, huh, right? This is what the symmetries diagram in the self looked like when it came off the needles. And so I think a lot of people see this and are like, oh my God, I can't knit lace, it looks terrible. The key missing step is you wet the fabric, you stretch it out, you let it dry, right? Okay, so the shawl that I was passing around, this is actually a pattern that is available for sale. Again, the search terms that you will want are linear lace and it's available on Ravelry.com. And it's actually not free, but it's for a relatively modest price. And the proceeds go to the Association for Women in Mathematics, which seemed appropriate, given the craft element of it. Okay, so just a couple more artworks that I wanna show you. So lace is not only done in knitting, there are other forms of lace. And one of them is bobbin lace, which is, I think the short way to describe it is that it's basically extreme braiding. And so there is a computer scientist named Veronica Irvin, who has done some amazing work with bobbin lace. She'd been doing it herself because there was a family history of bobbin lace. And then she discovered that no one had come up with a mathematical model for it. So she came up with one, she figured out how to feed it into a computer to search for patterns. She ended up generating some of the first new stitches in bobbin lace in over a century. And so this is an artwork that she put together with Lenka Suchanek. And I say coming soon because very recently, Veronica Irvin and Frank Rusky have proven that you can get all 17 wallpaper groups. They just have to actually make the lace. So I'm hoping at some point we'll have a sampler. No, it's not. It really is like braiding. You sort of pass the threads over each other. And so just one more piece that I wanna show you here, just because, it's so lovely. So this is, as I say, an Uber complete symmetry sampler. I think it has the most symmetries in any sampler that I know of. It is also the pattern that you will see on the tilework in front of you there. It is the smallest, simple, perfect squared square, which is a square that is cut up into smaller squares, no two of which are the same size. And what Wing has done here is that she has managed to fit into these squares all of the freeze symmetries. They're all in the biggest square over here. All of the wallpaper symmetries, the ones that were in the counter cross-stitch embroidery, and the rosette symmetries, which are the ones that have no translation. So they're like an individual flowering and individual star. And so they are all in this one piece, which is just beautiful. And so with that, I will say thank you. All right, we have time for a couple of questions. You could raise your hand. And I can pass the mic to you. And in the back here. On the topic of the proofs, can you give us some sort of a glimmer into the techniques that are done to prove the things that you're talking about, like prove they're all 17, et cetera? I can try to give you a glimpse into that. So the proof techniques that are used to prove that there are only 17 wallpaper groups. And I should say, I'm not actually sure what the original proof technique is that was used. The one, for anyone who's actually interested in looking this up. There is a book. Oh my gosh, am I gonna have the name here? It's by, I'm blanking on the name. This is John Conway, Virgil, and Strauss Goodman, is it? Yeah, Strauss Goodman. What is the title of the book? It's, do you remember? The symmetry is, yeah, that's right. The symmetries of things. Or the symmetry of things. I can't remember which. It's this sort of more recent proof. And what they do is they kind of look at, suppose you took one of these symmetry patterns, and then everything that's the same, because you can move things around to get them to line up, you glued them together. And so instead of having a plane, you would have the small glued together object that might be like sort of a puffy triangle thing, or a puffy square thing, or whatnot. So they basically are able to classify the wallpaper symmetries that you can get by looking at these small glued objects. The formal mathematical term is an orbifold. So it's graphically done? The proof is a graphical proof? I mean, I guess, part of it is it depends on your definition of graphical. There are certainly graphical elements to it, but it's not entirely graphical. I guess would be my answer to that. All right, if anyone has any further questions, you can come up later and speak to our presenter. Let's give one more hand. Thank you.