And tonight we have Judy Holdner with us, and I can tell you from personal experience having been sitting out in the audience a few months ago and hearing Judy speak about her mathematical artwork, how captivating I'm sure you will find this talk. To give you a little bit of information about Judy, she received her bachelor's in mathematics from Kent State and then went on to get her PhD in math from the University of Illinois in Urbana, Illinois. And since that time she's taught in a wide variety of places including the U.S. Air Force Academy, Harvard, Carnegie Mellon, and where she is now, Kenyon College. I guess we have some Kenyon folks in the audience, thank you for coming. Her primary research area is algebra and number theory, but she's also collaborated with students in areas such as mathematical biology and dynamical systems. I know you are going to enjoy an unusual and creative and wonderful presentation today, so please welcome Judy Holdner. Okay, thank you very much, Cindy. It's a pleasure to be here. Ever since this museum opened about five years ago, I've wanted to get a chance to come and check out the place, and I'm delighted to have the opportunity today. I saw the exhibits, they're absolutely wonderful. This is an extremely fun place to be. I'm sure it's a fun place to work. So I thank you for the invitation. And I'm also delighted to have this opportunity to speak about my artwork, pictured here. I created it in Adobe Illustrator just over a year ago. I also used computer algebra systems, Maple, and Mathematica to produce some of the patterns that you see, and Photoshop to manipulate some of the images to get the effects that I wanted. Several different programs went into play here. I titled the piece Immersion, and for the next hour or so, we're all going to be immersed in it. Okay, so in this piece, what I do is I tackle the definition of immersion from two different points of view, the vernacular point of view, and the formal mathematical point of view. So in the vernacular sense of the word, what I do in the piece is I have a lot of patterns that reflect my own immersion in mathematics, in the mathematical world. An immersion that arises as a faculty member at Kenyon College and working with students, the research I do, the teaching that I do, and it's a beautiful world. So I'm going to talk about some of the patterns that reflect the work that I do as a mathematician. That's my own immersion. In the second definition, the formal mathematical definition, what you'll find is that it's that definition that defines the composition of my piece. So in the second part of my talk, I'm going to tackle that. So I sort of break up my talk into two parts. And for the vernacular definition, we're all familiar with it, but I think that the Cleveland Cavaliers sort of sum it up very well. And as a native Ohioan who grew up 20 minutes from Akron, Ohio, which is the birthplace of LeBron James, I'm also a huge fan of their two words slogan, all in, because I think it captures this notion of immersion very, very well. And I also think that LeBron's all-in mentality is what's helped to propel him to the top of his game. So I like this intensity that the notion of immersion portrays, I guess. What's interesting about LeBron's immersion, and I guess the immersion of other sports figures, just how visible it is. So we can all get a sense of what LeBron does, the hard work that he does, the fruition of his efforts are on the news or you can watch it on TV. But for a mathematician, our immersion typically happens in our head. And even when we speak about what we do, we're often speaking a language that most people can't understand. So much of the work that we do remains mysterious to the general public. And it's my hope with my artwork to change that, to make the mathematical world, my world, more visible to a general public. And also in my talks, I hope that I can reflect the nature of what it is that mathematicians do. Okay, so as a professor at a small liberal arts college, I teach at Kenyon College, student learning is my number one priority. So much of the patterns that you see reflect topics that arise in coursework that I teach. So multivariable calculus is one of my favorite classes to teach. I probably teach this every year. I just taught it last semester. And you'll see that some of the patterns that appear, so the vectors and the contour diagrams, and you have these normal vectors shooting out, these are all images that reflect coursework in the third semester of calculus. And if you watch the Weather Channel, then you're probably quite familiar already with vector fields, although maybe you didn't know it was called that. Basically, it's a slew of vectors. The length of the vector will indicate how fast the wind is blowing, and then the direction of the vector is telling you the direction of that wind. Pictured here is the Great New England Hurricane of 1938, which the speed, I guess, of the wind hitting the shore was clocked at 47 miles per hour or something like that. So this was a very deadly and costly earthquake. I have to be very dramatic in my talk. I also teach a course called abstract algebra. And in this course, you learn the tools that you can use to rigorously quantify symmetry. So if you look at my patterns and my piece, you'll see there's lots of symmetry going on. The one that's highlighted here is a pattern known as a wallpaper pattern. And what that means is that it has translational symmetry in two different directions. So let me say a little bit about what that means. So you want to imagine that the pattern extends indefinitely in all different directions. And I want you to hone in on the hexagon that I've highlighted there. Notice that if I were to shift the entire pattern in the direction of the red arrow, then that hexagon would then move to the second highlighted hexagon. And in fact, the image as a whole would remain unchanged to the viewer. This is reflecting what we call translational symmetry in that particular direction. And I could shift the whole pattern again and the same would hold true. Similarly, there's a second direction that I could shift the pattern in. And that would make the hexagon that I've highlighted move to another hexagon and also the surrounding pattern would also move to the surrounding pattern of the second hexagon. And what you would see again is that the resulting image would remain unchanged to the viewer. So patterns that have this property of two distinct directions of translational symmetry are known as wallpaper patterns, which might not be so surprising because you think about what wallpaper does and you have this repeating pattern to cover the wall in some nice way. Now if I take these two directions and extend lines through them, I have the lines here dotted in red and in blue, you'll find that they break up the plane into diamond shaped tiles. I'm going to highlight one here to emphasize the fact that that single tile can then be used to generate the entire pattern. That tile has a special name. It's called the fundamental region of the wallpaper pattern. It's fundamental in the sense that it generates the entire pattern. So if you wanted to produce this tiling pattern on your bathroom floor, for example, you could use that diamond tile and then continue... Well, you'd need a bunch of them, right, and you would just tie your floor in this way. Now teachers, how many teachers we have in the audience? A good number. You will understand this idea quite well.  What procrastigrating is, it's really any activity you might get involved in to avoid grading.  So, you know, it might be that you're answering email or cleaning toilets, whatever it is. If you're trying to avoid grading, it's procrastigrating. And in this particular case, I was trying to avoid a stack of linear algebra exams, and so I decided that it was absolutely necessary that I figure out how to visualize sources and sinks in vector fields. And I'll say what that means in a minute. My exploration led to these three patterns, which are the ones that I had highlighted in the previous slide, and all of them turn out to be wallpaper patterns. Okay, so I'll need a definition. The divergence of a vector field is a number, and it's a number that measures the tendency of the vectors to emanate outward from a point. So you can compute the divergence at a point, and if the number that you get is positive, then that indicates that the vectors have a tendency to point outward. And if the number is negative, then there's a tendency for those vectors to point inward towards that point. If you know some calculus, it's very easy to compute this divergence quantity. If you have the vector field described by two coordinate functions, say p and q, what you can do is compute the partial derivative of p with respect to x and then q with respect to y, sum them up, evaluate the point, and then you sort of say, okay, is my number positive or negative? If it's positive, I have the situation where the vectors are pointing more in an outward direction. A typical picture of this situation where the divergence is positive is given here on the left. I like to think of vector fields when I talk about divergence as representing the velocity of fluid. And if you get a divergence that's positive, you should think of the point as sort of a source of new fluid. So in fact, if the divergence is positive, we call those point sources. You might think of a spring or something if you're talking about water flow. And in fact, if you computed the divergence for this vector field, it's clear that when looking at the picture that the origin is a source, but in fact all points in the planar source is because the divergence turns out to be a constant value of two everywhere, which means that you have a source everywhere. And to make sense of this geometrically, what you can do is take a small circle around your point in question and look at what happens to the flow out of the circle versus in. If you look at the flow out, can you see that there are more vectors that are longer sort of coming out versus in? So the net gain would be sort of in an outward direction. That would indicate that you must have like a spring of fluid inside to produce that extra flow. So in fact, every point in this particular vector field is a source. And similarly, if I were to multiply the vector field by negative and swap the direction of those arrows, they all point inward now, you would find that all points are syncs of old fluid. These are easy situations to visualize or easy vector fields to visualize, but if you get a more complicated one like the one here and you try to visualize where are the sources and syncs, it's not so easy. So this is the question I was thinking about when I was trying to avoid migrating. And if you look at the origin, you could say, okay, well, the vectors emanate outward there. It looks like there would be a source there. And maybe in the corners, it looks like there are sources. But in other places, the vectors are changing direction and length, and it's really hard to tell. So I got this idea that what I could do is for a given point in the plane, I could plot that point white if the point is a source, meaning the divergence is positive. And I can plot the point black if the divergence is negative. And I did that with various vector fields, and some of them produced some really delightful symmetric patterns, like the one that's shown here. So the vector field on the left produces this pattern, which has some nice rotational symmetries and some reflective symmetries. And if you zoom out, you'll find that it's a wallpaper pattern. So in fact, you get nice translational symmetries as well. So I was delighted to find this. And so I started playing some more. And all these three patterns here, and then some others as well, they all come from this particular vector field here that has this form. So what I quickly honed in on in trying to find interesting patterns were vector fields that had this form. And if you look each term, so the first term is corresponding to describing the vector in the horizontal or x direction. And the second term is describing the vector in the vertical direction. So what I found was vector fields of this form produce interesting results, and I took, you know, so the first quarter function is a product of a sine and a cosine, and similarly for the y-coordinate. And investigating this further, it's not hard to see why this produces the interesting patterns that it does. So if you compute the divergence, again, you have to compute those partial derivatives. You have to remember your chain rule, right? Derivative sine is cosine, apply the chain rule, multiply by the constant a, and do that with both terms. One thing to notice is that all the terms involve all cosines, right? Each term has a cosine of x involved and a cosine of y. And if you remember that cosine is a periodic function, what that means is that you're going to get this periodicity in both the x direction and the y direction, which means you have translational symmetries in both the x and y directions. So this particular divergence is, oh, this is going to give me a wallpaper pattern, which is what I set out to obtain. And the different parameters for each of the three patterns pictured here are given below. So if you put in these different numbers up here, you'll get these various patterns. These are the fundamental regions of each of the three patterns. So you can see that those regions actually generate the entire pattern in the plane. There's even more in the way of symmetries here because the cosine is an even function. So what you can do is replace x with negative x and y with negative y, and the divergence will remain unchanged. And what that means geometrically is that we're going to get both horizontal and vertical reflections in the patterns, right? So it means more symmetries, which often means more beautiful patterns. So that's nice. Additionally, you could replace both x and y with negative x and negative y, and the divergence would remain the same. And what that tells us is that we're also going to always have 180-degree rotational symmetry in the patterns that I'm creating here. And if you apply or restrict yourself further and impose more conditions on the parameters a, b, c, and d, then in fact you can get even more symmetries. So for example, if I set a equal to c and b equal to d, or if I say that a has to equal b and c has to equal d, then you also will get 90-degree rotational symmetry like what's pictured here. Okay, so this one satisfies one of those two. I don't remember which one. Here's a sampling of some of the patterns that I produced using this method. These are the three that I've talked about already, but if you vary different parameters, you get these different patterns. I find them to be incredibly beautiful. They remind me of the black and white patterns that you see in African textiles and weavings and baskets and such. I just love them. I think they would look great on pillows, maybe fabrics for skirts. This would be a wonderful thing to use, right, to produce some beautiful patterns in culture. Okay, so I'm going to sort of change directions a little bit. I want to talk about three more patterns that appear in the piece and these hold special meaning to me because they stem from some undergraduate research that I've done at Kenyon with some students there over a span of about 10 years. If we get a better look at these patterns, you might recognize them. Well, you might recognize the one on the left. That one's a famous curve or it's a famous iterate of a... It's an iterate of a famous curve known as the Cox-Snowflake or the Von Cock curve. And then the other two images are related to a very famous sequence known as the 2e Morse sequence. So I'm going to talk about those famous objects in turn, starting with the Von Cock curve. So the Cox-Snowflake is a fractal object. It's defined iteratively starting with the equilateral triangle. And with each iteration, we define a replacement rule that we apply to each edge. And the replacement rule is such that you take the edge and remove the middle third and replace that middle third with two more edges that form two sides of the right triangle that have base equal to the edge that you removed. It's kind of a hard thing to say. But there you have it. It's easier to see. And you apply that same replacement rule then to every single edge of the new figure. So with each iteration. So I'm going to do it here, and I do it again. And the Cox-Snowflake is an infinite object. So it's what you get if you were to do this infinitely many times, which of course you can. This is a theoretical object. But you notice that after a couple more times, when you get to about the sixth iteration, the resolution doesn't allow us to see any change anyway. So we might as well just stop there. Think of it as an approximation to the actual object. But for all practical purposes, from our point of view, here we have the Cox-Snowflake. A close inspection of my artwork will reveal that it's the fourth iterate that appears in the pattern there, and I've repeated it a second time just to make it look more interesting. To convince you that this really is a popular object, this Cox-Snowflake, here's a rendition of it in the snow produced by snow artist Simon Beck, who works on canvas the size of, say, three soccer fields, gets his snowshoes and tramples it all out. Isn't this great? It's not going to last for long, so you have to get a good picture and put it on the Internet, which is what he does. It's beautiful, right? It's beautiful. He does lots of different geometric designs, so you should check them out online. Here's another rendition that he did on a beach, which, again, this is going to last for long. He's got to do it all in like a six- to eight-hour span or something, right? All right, so for the two-emorse tiling, I want to introduce the idea of the two-emorse sequence by way of this tiling, and I'm going to do so by posing a question to all of you. If you have a tiling pattern, you might think of it as a decorative border that might go around a room, maybe a coffee lounge or something. A question I want to pose is, what's the color? What will the color be of the next two tiles? So let me give you a little while to think about that. You see some pattern here. What will the colors be for those last two tiles there? Anybody want to venture a guess? Black and white. Black, white. People do so well. Yeah. Good. So it turns out that this tiling pattern was constructed by contrarians, and I like to think of this group of contrarians as, let's say, a faculty committee, because we can be pretty contrary. And the idea is that the first faculty member puts the tile down, a white tile, and the second faculty member comes along and says, no, no, no, that's all wrong. It can't be white, it's supposed to be black. Third faculty member comes along and says, white, black, that's wrong. It's way wrong. It's not white, black, it's black, white. The third faculty member does the same thing, and you can see what the next four tiles are going to be. It's the contrary version of the first four, so it's black, white, white, black. And you continue in this manner, and if you do so, you'll see those last two tiles are black, white. Okay? Suppose this same faculty committee wants to tile a floor. First faculty member puts down a white square. Second faculty member comes along and says, oh, no, no, no, that's all wrong. It should be black, put black one down next to it. Of course, if that's the case, and the lower right-hand corner should then be white. What's interesting is you can take those four tiles and think of them as one big tile and play the same game with that one big tile. So I'm going to put that one big tile down, and the next person says, oh, no, no, no, that's not right. They put the contrary tiles next to it, and you get a tile that looks like this, and now treat that as your tile and apply this contrary construction to produce the next iteration, and if you continue to do that, you get this tiling of the plane that's known as the two-emorse tiling. Okay? And that appears in my artwork. I've warped it a bit. I've warped it to sort of create a landscape. It's my hope that this landscaping effect will sort of invite the viewer into my mathematical world. All right, so more formally, the two-emorse sequence is typically defined by using sequences of zeros and ones, so iterating them, and you start with the initial string x sub zero equals zero, and then we're going to define each new iteration based on the previous one by way of concatenation. So let me explain what that means. So what I'm going to do is I'm going to say the n plus first iteration is defined by taking the previous iteration, which I'm defining to be x sub n, and I concatenate, which means I put it just next to x sub n bar. x sub n bar is the contrary map. So what x sub n bar is, it takes x sub n, and it changes all the zeros in that string to ones, and all the ones to zeros. So for example, suppose you wanted to compute x one. x sub one is the concatenation of the previous iteration x sub zero with x sub zero bar, which will be the contrary of zero, which is one, so you would get zero one. Similarly, x two is x one, x one bar, so you get zero one concatenated with one zero, and you continue this indefinitely. Once again, the two amor sequences is an infinite object. You have to continue this process indefinitely, so it exists in theory, but if you were to do that indefinitely, you get what's known as the two amor sequence. Like the number pi, the two amor sequences are ubiquitous mathematical objects. It pops up in unexpected places, and it pops up a lot. So just to give you a sense of this, I'm going to talk about some of the properties that it has. One of the properties is that it's cube free, and what this means is that there's no substring anywhere within the sequence of the form w w w, where w is constructed out of zeros and ones. So you're constructing that word from the alphabet that you have, which is zeros and ones. It turns out that there are plenty of squares. Just to convince you of that, let's give you a few examples. Here's a bigger square right there. You'll never find a cube anywhere in this. The sequence is not periodic, meaning that you don't have some word that repeats. Nonetheless, there's a lot of repetition within the sequence. So for example, if you take the first 28 symbols of the sequence and you look at just the first four symbols, so just think of the word zero one one zero, that word repeats itself five times in the first 28 symbols. So lots of repetition there. In fact, the sequence is what we say is recurrent. And what that means is that if you pick any string anywhere within the sequence, that string is guaranteed to show up in some window of some size. So there's going to exist some length, which we call l sub w. And if you take a window within the sequence that long, you're guaranteed to capture that word within it. And the longer the word, the longer the length that you're going to need in order to make this happen. So a very simple example is what you get if you take the string just w equals one. We know that the two or more sequences is cube free, because I just told you that. So what that means is that if any window of length three is guaranteed, contain a one in it, because if it didn't contain a one, it would have to be three zeroes, and then you would have a cube. So the fact that it's cube free means that l sub w would be three. If I take the word one one zero one, it's a bit more complicated. So what that means is that the length is going to have to be greater than 14, because if you look, I have two occurrences of the word here, and I look at a window of length 14, and inside of that window there is no copy of that word, and so that's saying, oh, this length isn't long enough. I need to get bigger. So we really do have a lot of repetition within the sequence. The coolest property of them all, I love this property, is the self-similarity that it exhibits. And what I mean by that is I take the sequence, break it up into blocks of size two. So there you have it. There are my blocks. And now I'm going to define a replacement rule. The replacement rule is that I replace every occurrence of zero one with a zero, and every occurrence of one zero with a one. So let's do that. Zero one becomes zero, one zero one, one zero one, zero one zero, et cetera. We keep going. Smush those symbols together. And what do you get? You get that two more sequence right back. Interesting. It gets better. So now I'm going to take the sequence and break it up into blocks of size four. Define a replacement rule where zero one one zero, every occurrence of that is replaced with zero with zero, and every occurrence of one zero zero one is replaced with one. And we do that. You get zero for the first one, one for the second one, one for the second one. Can you see what's happening? Smush the symbols together, and there you have it, the two more sequence again. In fact, you can take the sequence and you can break it up into blocks of size two to any power you like. So it has to be a power of two. And define the replacement rule in a similar way, and you'll find that the sequence has self-similarity at that scale. It's fascinating. A little history about the sequence. In fact, it was first discovered by French mathematician Eugene Pruitt, even though the sequence is named after two mathematicians, Morse and Tui. Pruitt discovered the sequence while trying to solve a problem in number theory, and that problem now bears his name, the Pruitt-Terry-Escott problem. But Pruitt didn't actually define the sequence explicitly. What he did was sort of use it along the way, and that's probably why the sequence isn't named after him. It was Axel Tui who first defined it explicitly, and he was interested in avoidable patterns in the sequences of symbols, so he was interested in the sequence for that reason and particularly proved that the Tui Morse sequence is overlap-free. And what that means is that you cannot find a substring of the form W, U, W, U, W, where W and U are words constructed from 0s and 1s anywhere within the sequence. So just to give you an example of a word that does have overlap in the English language, the word alfalfa does have this form, right, because if you break it up in such a way and define the W and U appropriately, you'll see that it has that form. But if you look within the Tui Morse sequence, you're not going to be able to find it, that there's no overlap in that sequence. The sequence was... Marston Morse was the first person to bring the sequence to sort of worldwide attention. And when I say worldwide attention, I don't mean worldwide attention in the context of, say, one sort of attention, right?  And Marston Morse, he did... he stumbled upon this sequence while working in the area of differential geometry. He was interested in apiridicity, and he sort of discovered the sequence in that way, and it had significance in his work. Finally, there's a Dutch mathematician and world chess champion who stumbled across the sequence while trying to design infinitely long chess games, right, under certain draw rules. I believe it was the draw rules that were in existence at the time. Max Houve had a PhD in mathematics, and he wrote this really long paper analyzing chess, and the Tui Morse sequence makes its way into it, so that's kind of interesting, too. So I was reading about this sequence and all these interesting properties and the self-similarity and so on, and I just thought, wow, I wish I could see this sequence. Like, what does it look like? I can see the symbols, but I want to actually visualize it. And since the sequence is a two-symbol sequence and just a sequence of letters, I thought, how can I transfer that into a geometric object? And I immediately thought of turtle graphics, right? So I was in school in the 80s, and turtle graphics hit it big. Turtle graphics were used as a way of teaching programming to young children because it made programming very accessible. The idea is that you have a turtle with a pen and you give the turtle commands, and the symbols are the commands, and in doing so, the turtle draws some figure in the plane. So if you wanted to draw a square, for example, you tell the turtle, you know, move forward, and when he moves forward, he creates an edge, then turn left 90 degrees, move forward again, turn left 90 degrees, move forward again. So basically, if you do the FL four times, it's a sequence, well, it's eight symbols, FL, FL, FL, there you have a set of turtle commands that actually defines a square, right? So I thought, okay, I have two symbols, I'm on two turtle commands. Two turtle commands are F for forward motion, and L I define to be counterclockwise rotation by 60 degrees, and then I take my two emorse sequence instead of using zeros and ones, I create two emorse turtle programs, starting with this alphabet and applying the contrary map like we've talked about. So the first initial string is just a forward motion and F, and then I get FL, FLLF, et cetera. What I want you to notice is that the even iterates are all palindromes, because I only have a few even iterates here, but if you were to continue, this will always be the case, you'll always get that those are palindromes, and that has some relevance in a moment. All right, so let's try to visualize this, and we'll just look at the fourth two emorse turtle program and see what happens when we tell the turtle to move in this way. So I'm going to say that the turtle starts with its heading upward, which is depicted by that blue arrow, and we're first going to look at the first four symbols, FLLF, so the turtle moves forward one unit, rotates 60 degrees twice, which is depicted by the two blue arrows, and then lays down another edge, and then you see the final heading there. Next four rotates 60 degrees, now it's pointing downwards, moves forward two steps, creates two edges, and then rotates twice, and that's reflected by those two arrows. So the first eight symbols produce this polygonal curve, and then what we notice is that because I have a palindrome, you can take the second chunk of this turtle command, the turtle program, and that's going to actually draw a bilateral sort of reflective version of this. So what it means is that my turtle trajectory is going to exhibit bilateral symmetry. So what I can just do is draw the rest of my polygon, it looks like a pointy heart of sorts. So there's a picture of just that fourth iterate. Not too interesting, so let's continue. Let's look at iteration four, iteration five, iteration five, that's an odd iterate, so we lose the bilateral symmetry. TM six, now we have an even iterate, bilateral symmetry is back. TM seven, let's look at TM ten. This hopefully looks familiar, that's the pattern that appears in my artwork. Now let's take it a little further, take a little further and go to TM fourteen. Does that curve look familiar to anyone? That's one-third of that Cox-No-Flake. So I was not expecting to see this, but when I saw it, I had goosebumps. I thought, wow, here is this very famous sequence, and when I visualize it by way of a turtle program, out pops this other famous object that I had no idea was related to the sequence. That's really, really cool, more evidence of the sequence sort of being ubiquitous in mathematics. Of course I can look at this, and I can say this is one-third of the Cox-No-Flake, but that's not a proof. In mathematics, we need to prove things. You can't say, oh, it looks like this. You have to actually have sequence of steps that lead to definitive truth. Truth is beauty. So I enlisted the help of an undergraduate at Kenyon, and here's my collaborator's name is Jun Ma. He's standing in front of his poster, which reflects our work at a national math, meaning is the American Mathematical Society joint, and AMS, and also the MA, Mathematical Association of America. They have joint meetings every winter. In fact, I'm leaving tomorrow to go to Atlanta to hit GMM. And here he is with his poster. He then went on and got a Ph.D. in financial engineering at Princeton. All right, so that completes the first portion of my talk, which is on, again, the vernacular sense of the word for immersion. Now I want to focus our attention and what you see here are some interesting mathematical objects highlighted. Those all reflect the notion of immersion in mathematics. Formally, there's a very technical definition. I don't want to get into that because we really don't have time to handle technical details in very, in any real depth, so that you would get any understanding of this. So I'm going to be very intuitive in my approach here, which means I'm not going to really tell you everything. I'm going to give you a sense of what's going on. But let me start by saying that the notion of immersion resides inside of a subfield of mathematics known as topology. How many people here have had a topology class? Few people. Okay, well, don't worry because those of you who have not have designed a special course for you. It's called MoMath 999 Crash Course in Topology. So I was asked, when I was asked to give this talk, I was asked to also incorporate a fun activity. And to me, there's nothing more fun than a math class. Right, nothing more fun than a math class. And in fact, what I really like are math exams. So at the end of our fun math class, we're going to have a really fun math exam. Sound good? Does that sound like fun? Don't worry, I won't grade it because as we've already established, I don't like to grade. And in fact, what they really are are fun puzzlers, and you're going to be encouraged to work with a neighbor. So this really is a fun exam. All right, I want to start our course in topology by posing very famous problem in mathematics known as the Bridges of Konigsberg. It was first posed to Swiss mathematician Euler in 1736. And the question is, well, first let me say that there's a river in the city of Konigsberg, which was in Russia, it's modern-day Russia. And there are seven bridges that connect two islands with two banks of this river. And the question is, can you come up with a walking tour of the city that will visit each of these different land masses crossing each bridge precisely once? Right, so you're going to come up with a walking tour where you visit both islands, both banks, but you don't want to go over any given bridge more than one time. One possible attempt, a failed attempt, would be this one, right? You say, okay, I'm crossing over these bridges, but I can't get back over to cross the bridge that's connecting the two islands. Let me give you a moment to just think about this. See if you can come up with a way of creating such a walking tour. And perhaps you'll agree that the shape of the islands don't really matter here, right? It doesn't matter if the islands were shaped like a kidney bean, or rectangular, or what have you, right? That doesn't matter. So let's take a look at the pictures with a more basic picture. So let me just keep it here for now while you think about it. And let me see if somebody has a tour, a walking tour that works. Let me know. Nobody's raising their hand. This is a pretty big audience, so I would imagine that if one existed, somebody would find it by now, right? And there's lots of smart people in here. In fact, it's not possible, right? This is not a possible thing to do. And Euler figured this out. And his solution, what he did was he said, okay, what really matters here is not the shape of the islands, not how thick the bridges are, or not the shape of the bridges. So he simplified the picture into a graph, right? So a graph is a set of edges and nodes. The nodes you can think of here as representing land masses. So you might want to think about the land masses as being like shrunken down to a point. And the bridges are then compressed into these edges, right? What's interesting about this problem, I mean, his solution was interesting. He created some interesting results about graphs in order to solve it. But what's particularly interesting for my purpose today is his approach. His approach in transforming the problem into a simpler one without changing the important information, right? The connections are what really matter. People think of his solution as the birth of topology. And the reason for that is that topology, also known as sort of rubber sheet geometry, is the study of those properties of geometric objects that remain unchanged under certain transformations, right? And I'll talk about what those transformations are in just a moment. But we're going to say that if you perform these transformations on an object to change it into another object, as long as you use these valid transformations, you're not changing the inherent nature of this object topologically. They're considered the same. They're considered to be topologically equivalent. All right, so valid transformations include compressing, distorting, bending, stretching, twisting, right? And here's an example of this. So this is my son Chase, 15 years old. He likes to use Snapchat, it's the latest craze in high school. They like to take pictures, mess them up. He's like, you know, expanded his eyes, compressed his nose. And this is the sort of thing that we might try to accomplish way back in my day using mirrors and funhouses, right? And they're using this app on their iPhone. It's kind of fun, right? I like it because it's mathematical. It's going to give them a sense. You know, they hear about the notion of distortion and this kind of thing. Oh, yeah, they don't know exactly what this means. So that's the beauty of technology. It's bringing in some of these ideas right into children's hands. Which changes are invalid? Well, you can't rip or tear a hole into an object and change the inherent topological makeup of the object. And it's illegal to glue pieces of the object together. So we can't do that. That's illegal. With those rules in place, the question now next for this example is can I transform the mug here, that I picture on the left, into a donut on the right? This is a very famous first example in topology. And it leads to a very famous joke. The joke being the definition of a topologist and a person who can't tell the difference between a coffee mug and a donut, right? Because both have a hole in it. So it turns out that if yes, you can transform one into the other. And here's one way you might try to do this. So imagine you take the mug and sort of stretch it downward and continue to do that until you have a slab with a handle. And then imagine that you take that slab and you shrink it inward until you get something that looks like this. And finally, you can sort of shrink the slab all the way in to get your donut, right? So these things are topologically the same. And the key observation here is that they both have a single hole. All right, so now we're ready. We're ready for our fun math exam. Hopefully you all have it here. Let me start by having you work on the first page, Problems 1, 2, 4, which I have pictured here. And for all four of those, you're given two objects, Object A and Object B. And the question is, can you transform Object A into Object B using the valid transformation? You can't rip or tear any holes in the surface, but you can distort, twist, compress, stretch, et cetera. So I'll take some time to let you think about this, and I'll be circulating if anybody has questions. The stem of the wine glass in the first problem is solid, right? So you don't have a hollow stem. Are we ready to go over the first page? Should we do that? So let's see. So let's talk about number one. So can this object be transformed in Object A into Object B? The one on the left, no hole, right? The one on the right, hole? So the answer is that it's not possible, right? Somehow you'd have to rip a hole and then to get to the other one. So you say not possible for that first question. How about question number two? Now we have two objects, both have some holes in them, so maybe this is possible. The first one, the one on the left, appears to have some sort of linkage. What do you think? Can I transform that object A into Object B? Lots of no's. And the answer is yes, you can. I love it when things are counter-intuitive. In fact, you can. And I have a little arrow here that's going to help tell us how to do this, so I want to move that arm upward and get to there. And then I have an arrow moving downward there, meaning I want to imagine putting your finger through the hole and sort of moving it to make the hole bigger. And now I want to take the bottom of that hole and move it all the way up until it reaches the other link, or the other circular region, I guess. So it looks like this. The next thing you want to think about, so I can get my pointer to work here. Oh, it's very light. See this arm right here? The top part that's going over. Take that and pull it up out of the hole. So I'm going to take this upper arm and just pull it out, and I get this. Can you see that? And now you can flatten it out to see that. In fact, these two are topologically the same. It's not easy to see, is it? It's very cool. It's very cool. Great puzzler. Now maybe you want to go back and change your answers to the previous problems. OK, how about this one? Can I transform the one on the left into object B on the right? Now we don't want to answer, do we? I mean, why answer this question, right? I'm like, this is mean. Cruel and unusual punishment here. OK, the answer is yes. In fact, they are topologically equivalent. And let's see how this works. OK, so I have these arrows indicating that you want to, again, make the hole bigger and sort of move it downward, much like I did before, and until the arms of the horse, the hole sort of meet all the way down at the circular region. OK? And then here's the part that might be hard to see, but can you see where this arm here attaches on the top? Migrate that upwards. Can you see that? I'm trying to show it with my pointer. So take this piece right here that's connected to the circular region, migrate it upwards, and then you can see that you're home free. You can then pull it out. So they are topologically equivalent. All right, how about this one here? I transform the one on the left into, like, no, this better not be the case.  Yeah, this one's impossible. So the one on the left is known as a link, and you just can't unlink it. There are, I guess, two examples before that. So question number two, you had an arm that you could sort of work with, right? And this one you don't have that arm. Not possible. OK, so let's just pause for a little bit more to think about these two, and then we'll come back together and talk about a solution. So I don't know if you can see this. Both objects are like a double bagel of sorts. The one on the left has a curve on the surface, and the one on the right, or the one on, I guess, your left here, the one on the right does two, but the second one sort of encloses both holes. So the question is, can you transform it so that the circle enclosing only one now encloses two? Is it possible? You can't rip the circle. No, good. That would be illegal, right? You can't rip the hole on the surface.  Anybody think it's possible? Yeah, you have a solution? I think so. You want to describe it? So you take the left hole, stretch it out, and then take the right segment and sort of flip it underneath into a basic one. Yes, all right. Awesome.  That's great. Yeah, so the answer is yes, you can do this, right? They're a topologically equivalent. And we're going to stretch that second hole out sort of like what you said, right? And then in a similar way, I'm going to migrate those arms around and then pull it down underneath, which is what he was saying, and then push it up in between, right? That's pretty much what you described, correct? Yeah. Fun puzzlers. These would be great for the teachers in the audience. Great puzzlers to share with your students. So pass it on, right? All right, how about this one? You have this punctured inner tube for, say, a bicycle tire, and you want to be able to turn it inside out. Can we do that? I see some nos. I see some yeses. What do you think? The answer is yes, we can, right? So the idea is, when the arrows are indicated, you take your hole and you stretch it in the vertical direction. It looks something like this. And now take that hole and stretch it in a horizontal direction. Okay, it starts to look like that. Continue. And what you notice at this point is that there's a nice symmetry in this object between the white side and the black side, right? Which means that I can sort of undo what I just did, but with the white on the outside instead of the black. All right, so let's illustrate that. So now the white's on the outside, I bring it all the way back around, and we have the black side of the inner tube is now on the inside. Isn't that fun? Isn't this a fun math test? Fun. All right, last problem, last problem is to take these household items and to sort them by way of topological equivalents. So you're going to take these objects and the ones that are topologically equivalent. You put them together in the same set. And then of course, if they're not topologically equivalent, then they should be in separate sets. All right, and my goal here is that this makes you think differently about your world. You go home and now you're going to start thinking about topological properties of objects in your household. And who knows? Maybe it'll even affect the way you buy objects because you want something that's more topologically interesting. So let me just leave it at that and give you some time to try to sort these objects and see if you can do it. And I don't know if you can see the spoon, which is quite silly, has two eyes and a smiley face spoon. Those are holes in the spoon. And you can buy that. You can buy such spoons. Oh, no, no, no. Some sets could be bigger than other sets. They're not, yeah. Yeah, some sets could be bigger than other sets. Okay, how about if we touch base on this? Does that sound good? Okay, so let's look. So how many different sets do you get? Three, you're saying, yeah? The three corresponding to the number of holes, right? So it turns out you get three different sets. Here they are. You have objects that have only one hole in them. You have objects that have two holes in them. And then objects that have three holes in them. Well, yeah, people were making great progress. I think the hardest thing to visualize are those black stoles, right? You have like, I mean, it's interesting because you have two three-legged black stoles that are very different from a topological point of view, right? Even though you might think of them as the same. They're both black, they both have three legs. They're very, very different, right? If you have a hard time visualizing it, this is how you might want to think about it. Migrate that one stick over, flatten it out, and you can see then the number of those holes. The triangle is right here, right?  You have this hole, this hole, this hole. Imagine this is made out of pretzel, right? It's the same. You see what I mean? Yeah. So it turns out to have three holes in it. If we pulled the middle of the pretzel up, then it would, you'd have a little triangle under you. Exactly, exactly. Okay, so hopefully now we're starting to think like topologists and you're primed to think about how immersion plays out in a formal mathematical way in my artwork. Okay, so it turns out that topologists really don't study these household items that I just had you looking at. Instead, they study these very abstract spaces like I have pictured here. And both of these, or I guess all of these things that I've highlighted represent the notion of immersion. I'm going to start by focusing on the closed curve because that's a simpler object. It's simpler because it's lower dimension, right? The curve has dimension one, surfaces have dimension two. The higher dimension just makes things more complicated. So I'm going to focus on that closed circle. And the way I want you to think about the closed circle is it's a function mapping a circle into the plane, right? So if you take a circle and you map it into the plane, there's various different ways to do this. And this is how I'm going to think about it here. It's a closed curve. Let's look at some other examples you might imagine of closed curves in the plane that could be thought of as what you would get if you were to map a circle into the plane. So on the upper left is the curve that shows up in my artwork. Here I have a rubber band depicting a closed curve. And I like this thinking about it in terms of a rubber band because one way to map a circle into the plane is to take a rubber band and just drop it into the plane. You can imagine it maybe crossing over itself, maybe not. I also like the notion of the rubber band because it's going to encourage you to think like a topologist. The rubber band, you can stretch and distort so you can get all sorts of interesting closed curve shapes in the plane like some of these other ones that we have here. So it might be hard to accomplish with a rubber band, but you can think of at least theoretically that you could stretch and get these different types of shapes. It turns out only three of them are topologically equivalent to a circle, the ones on the right, because they don't have cross-intersections. So the two on the left are not going to be topologically equivalent to the circle in the plane because of the two self-intersections in the top example and then there's three in the bottom. But I'm not really interested in that as much as I'm interested in immersions because that's what my artwork is about. It turns out that four of the five reflect immersions of the circle in the plane. The heart is not an immersion. Does anybody have an idea of what the problem is with this heart? Yeah? Those pointy areas precisely. So the heart turns out to not be an immersion because of those pointy areas. And let me give you a sense of why that is. So you can think of, at least in the context of the circle mapped into the plane, that you're going to get an immersion when you have this nice property where you have well-defined tangent lines at every point. So mathematicians love to talk about tangent lines. Of course it's the focus of calculus. But what it represents is like a linear approximation to your object. So in this case, if you're at this point up here, you take that small little line segment. That's a tangent line. It's going to give you a linear approximation to the curve. Lines are much easier to work with. Linear functions are much easier to work with than objects that have higher degree or curves in them. So we want to rely on what we know about lines and we use that. So what I want you to do is imagine now that you're driving around that curve, that closed curve on the left, as you drive around, the tangent line that you would use to approximate your path would change. But you would have a nice well-defined tangent line at each point along the curve. No problems there. Now if you played the same game with the heart, you start here, you sort of travel around that heart, you can see what happens when you get to the bottom. When you get to the bottom, do you see that on the right, you're going to have one tangent line. But on the left, it's going to be an entirely different one. And in fact, if this were really a road, it might look nice from above in an airplane, but it wouldn't be a very nice road to drive on because all of a sudden, right down there, you have to change your direction instantaneously and that's just not going to be good for our necks. So to have this property, to have a curve that's actually an immersion is a nice property. And that's what we have with this closed curve here. It represents the immersion of a circle into the Euclidean plane, which you can just think of as the plane. All right, let me turn our attention now to the closed surfaces floating in space. It turns out that these are immersions of a very famous space known as the real projective plane into Euclidean 3 space, which is the three-dimensional space that we live in. So just imagine that somehow you're immersing them into this space. And of course, you can't really understand what this means until you know more about the real projective plane. But let me say for now that all five of these surfaces here are topologically equivalent. They're all representing the surface, which is an immersion of the real projective plane into Euclidean space. The surface is now known as the Boy surface, named after a mathematician by the name of Warner Boy, who discovered the surface in 1901. He was asked by his thesis advisor, who is David Hilbert, a very famous mathematician, to prove that it's impossible to immerse the real projective plane into Euclidean space. And so Warner Boy went off and said, oh, it actually isn't impossible. Here's an example. This surface doesn't, right? And so that surface became the focus of his PhD thesis. So even David Hilbert can be wrong. That's great. Okay, so what is the real projective plane? It's a very interesting space in which all of the points in the space are actually lines. Lines that run through the origin in three-dimensional space. Okay, so what I want you to think about is that sphere here is centered at the origin, the lines all go through the origin, and each line represents a point in your space. Okay? It's a weird space, but maybe not so weird. It has importance in art and in particular in linear perspective. Because if you think about the lines of sight of an artist, they all go through the artist's eye. Think of this as a one-eyed artist. And the artist's eye is at the origin. So imagine that the artist's eye is placed at the origin. So the lines of sight are lines in space, and you could think of those as points in projective space, and where the line of sight intersects the canvas, that's a point that's representing projective space. So the artist's canvas here is representing a portion of projective space. It's only a portion. It's not all of projective space because the canvas only captures some of the lines. It's only going to capture the subset of lines of sight. So for example, there would be one going straight up and the artist isn't reflecting that on the canvas. So it's only getting a portion. The portion that reflects the intersection of those lines of sight in that canvas. If you wanted to capture all of projective space, then you have to do a little more work and try to understand this in a way. So right now the points are lines. It's hard to picture that. So what we want to do is we want to come up with a model in which points in projective space are points. How do we do that? Well, we start with the sphere and we notice that each line intersects the sphere in two different diametrically posed points, which we call intipital points. So we don't want the point represented twice on the sphere because that would be redundant and inefficient. We want to come up with a model that represents projective space where a single point in your space reflects a single line in projective space. It represents a single line. So what we do is we take one point that a line intersects the sphere with and the posing point, and we identify. We'd say, oh, these are the same point. And if we do that, then we have, again, this sphere is representing projective space in a redundant way. It's represented twice. So what we can do is throw out half the sphere. Now you could throw out the left half of the right half, but I'm going to throw out the bottom half and use the upper hemisphere to represent my model. So now what I have is every point on my upper hemisphere representing a single line that goes through the origin except along the equator. I need to include points on the equator, but I really do have to identify those opposing ones to make sure that you get one point representing one line. So I have the two points labeled by P. Those are going to be glued together if I want to construct a space that represents projective space in three dimensions. I'm going to try to glue those together. It's not an easy thing to do because it's not just those two points I have to glue together, all opposing points on that equator. And how can you do that? Try to envision this. It's not an easy thing to do. There's going to have to be a self-intersection of the surface. So let me show you how we might do it. So this is all rendered in Mathematica here. So you start with the hemisphere and I warp the equator into a triangular shape. And then what I'm going to do is identify opposing points while maintaining three-fold symmetry along the way. So I'm going to start making these opposing points meet up. And it might be hard to see here, but I'm trying to... Well, I'm keeping a three-fold symmetry. So what's happening on this side is happening in the other two sides. And it's hard to see what's happening once the surface starts to intersect itself and it's crossing over itself. It's hard to see. But you can start to see what's going to happen. If I rotate it a little bit, you can see that that's the surface that appeared in the bottom right-hand corner of my artwork. So there it is. If I want to talk about why this is an immersion, this is an example of an immersion, it's because we have this well-defined local linearization. You know, it was a tangent line when we talked about the closed curve, but in this situation we have a surface and so it's going to be a well-defined tangent plane. Tangent plane is going to be what we need when we're talking about a local linearization at a point. So if any given point, you're going to have this nice well-defined tangent plane. You don't have any kinks or corners in the surface. It might look like you have a corner in the center, but you actually don't because these, remember the surface is intersecting itself, so it goes right through itself and as it goes through you're going to have a well-defined tangent plane there. Surface floating on the upper left, that's bigger. That's actually the reverse side of the one on the lower right. So if you were to sort of get the view from the backside, you would get this. And the other two are also renditions of the boy's surface and I won't really get into that because we really don't have much time. Tangent plane is a really important space largely because it provides an example, a very simple, one of the simplest examples of a closed surface that has only one side. Right, and there are some examples of one-sided surfaces right here in the museum, namely the Mobius Strip. There's this wonderful Mobius Strip exhibit upstairs where you can sort of drive on it, which is really great. What I like to think about when I think about the Mobius Strip is sort of think about it together with the cylinder. In fact, you can create the Mobius Strip from the cylinder. The cylinder is not a closed surface because it has boundary. It has those two closed curves on the boundary there. But if I were to cut it and then flip one side of the strip of paper and tape it together, I get my Mobius Strip, right? So I go on the left, the cylinder has an inside and outside. It's two-sided. But if I do this twist and tape, I get something that's one-sided. Similarly, at the objects on the left, like the torus and the sphere, those are closed surfaces. They have a well-defined volume, right? So for example, you learn with the sphere that the volume of a sphere is four-third pi r cubed, where r is the radius of the sphere. It would hold that much liquid, right? On the right, if I try to think about either the boy's surface or the Klein bottle holding liquid, well, it doesn't make sense because you put it in and it would just come out. There's just one side to this object. You can't talk about a volume. There's only one side. On the final slide of my talk, I want to end with a surprising twist. I think that's always a good way to end a talk is to have a surprising twist. And to do that, I need to explain a second model that exists, a very common model. In fact, it's the model I first learned when I learned about the projective plane. And that model starts with the square instead of a hemisphere to create the projective plane. So let me just show you how these two models are actually going to be the same. So imagine that you take your semi-spherical or hemispherical model, I guess, for a projective space and think like a topologist. Flatten it out. If you were to flatten it out and look at it from above, it would look like this circle. What I'm going to do next is mark the two boundaries, the circular boundaries with an additional point, these two blue points.  So I'm not really changing the topological surface. I'm just saying, oh, instead of using one edge, I'm just going to use two. And now, again, think like a topologist. Warp the circle by taking those two blue points and making them the corners of a square and similarly with the point P. And you'll get this, what's known as, again, so it's a projective plane, but it's the square model. It starts with this square in the plane. And then in order to actually create or realize a projective plane in space, you have to glue opposing edges with a twist. So in other words, I'm going to take the bottom edge. Imagine this is made out of paper, or maybe rubber, it'll be maybe easier to do it with a rubber sheet. And you twist the bottom edge so it lines up the two arrows with the top edge and glue it. And then do the same thing with the left and right. And by doing so, you're identifying those intipital points and creating the projective plane. The artwork actually lives on the projective plane in this way. So if you were to examine, we'll examine the curves that are exiting the bottom of my picture here. And imagine the curve and then consider the curves that are exiting the top. If you were to twist the top and glue it to the bottom, then they would line up perfectly, which I'll illustrate here. So I just took the top image, flipped it, and lined them up and it's the same. I'm going to do the same thing here, that curve. And I can do the same thing with left and right. So the left, you have that surface that's floating off the left-hand side. But when you flip it and then identify the right-hand side, it's actually connected to this piece here. So this piece is the same as that piece. It's all one, right? So there you have it. We actually have two closed curves living on the real projective plane defining the composition of my piece. And then the floating surfaces are sort of other renditions of the real projective plane immersed in three-dimensional space. So I think I'll stop there. All right.  And we have some time for some questions. So if you could raise your hand. Any questions for Judy? We can take them now. Yes. So when you were doing the turtle command you came up with this 60 degrees that you were going to turn. What made you choose 60 degrees? A great question. I didn't at first. First I tried 90 degrees and it looked ugly. So I experimented and then I threw in 60 degrees and of course you have to take it out of ways to see it. And I was not expecting to get that coxonoflake. All of a sudden it just appeared. And I'm like, oh, wow, this is bizarre. This is really great. But I didn't. It wasn't the first thing I tried. After we proved the theorems that the turtle programs or turtle trajectories converged to the coxonoflake once you see why it works theoretically then of course it's 60 degrees. But when I was experimenting I didn't know. I was just throwing it in there and it worked. Ultimately it did. Yeah, great question. Any other questions? Yeah, in the back. What artists and musicians have used the two more sequence? I'm not sure about the two more sequence. I do know of some artwork that features the boy's surface. There's this beautiful steel sculpture in front of the research institute in Oberwaffe and it sort of graces the ground so as you enter the building and it's a very beautiful sculpture and it also sort of illustrates the mathematics that creates the surface because you can look it up online and it's a piece of steel and those strips are the image of a polar grid that's used in the parametrization of the surface so the mathematics are actually built into the surface itself which is kind of cool. But I don't really know of any artists that's actually using the two more sequence to create art. The reason I used it here was that I was trying to create a piece that represented a sampler of sorts as a professor and I worked with the two more sequence in a research project and so that had to show up in one of my patterns. There's other mathematical patterns here that I just don't have time to talk about. I mean one thing is if you look right in here this might be hard to see from where people are seated maybe they can come up later and look at this. This is the picture of the whole piece shrunken and stuck right inside there so you have infinity embedded in here. And then you can imagine that maybe it keeps going because there's another piece right there that would... I was pretty obsessive compulsive when creating this piece. I was just curious I think you mentioned at the very beginning that you have done or do research in number theory I was wondering what kind of research do you do in that? Oh in number theory I'm really interested in visibility problems sort of related to some of the divisors of a number and I actually did create a painting let me say that my website is completely out of date I need to actually spend time on this. But that painting is on my website so I created a painting where I have a spiral of let's just say kind of like a one dimensional tiling where we're interested in tiling to the border for the two amorph sequence but it spirals out and each color that I choose my palette is defined by way of this number-theoretic function called the sigma function which is the sum of divisors of an integer. So as I work my way out I go through the integers I compute the sum of the divisors of each of them and then I have a coloring associated with it you get this beautiful pattern at least I think it's beautiful but I've done some work in number theory of odd perfect numbers and identifying certain types of numbers that can't possibly be perfect and also identifying the form of what you might have to have in order to make there exist an odd perfect number I don't know if you know anything about this but divisibility problems related to integers is a major thing I'm concerned or interested in. Alright then let's give another hand to Judy Haldiner Thank you


The Scottish Parliament hosted a picnic lunch today. The great get-together was in memory of the late MP Joe Cox who was murdered a year ago. It brought together MSPs, Parliament staff and members of the Holyrood community. Well as you can imagine, as parliamentarians we were acutely sensitive to the death of Joe Cox, somebody who believed that there is more that we have in common than that which divides us. But you know it's very easy in politics to have interpret language and to move to the political extremes. What we've done here today is celebrate our sense of community, celebrate that what we have in common with our neighbours in the nearby primary school in the Whiteford Armed Forces residence, a community sharing a common space. Great get-together events are taking place throughout the UK over the course of the weekend.