 And we are delighted to have Dave Masanaga here. He's actually our introducer tonight. He is an award-winning, in fact, presidential award-winning high school math teacher from Hawaii. He literally just got off the plane yesterday morning. So we're delighted that he made the trip. He's been working with MoMath for a lot of years, both here in New York and at various conferences and festivals around the country. And I know you'll enjoy Dave as well as our presenter tonight, so please join me in welcoming David Masanaga. Thank you so very much, Cindy. And aloha, hi, everybody. Many of you remember me or know me from various and other MoMath spectacles. But here, I'm here tonight to introduce a very good friend of mine and a very good friend of MoMath, one of MoMath's prominent mathematicians, Dr. Carl Schaefer, professor emeritus in mathematics from Deonza College in California. Dr. Schaefer has always been right at the crux of interpreting mathematics for the performing arts and seeing the performing arts in mathematics. And you would just be amazed tonight at the various things that he has in store for you as participants of tonight's performances as well, too. Dr. Schaefer is not only a math pro, but he is a dance pro, being co-founder and co-director of Moose Speak Spin with Eric Stern. And he has always seen the intersections of mathematics and dance. He's applied things from mathematics into dance and seen how dance can pose new problems and new ideas in mathematics as well, too. You'll be quite excited to see how Dr. Schaefer can take very simple things like a piece of string like this. And oh, there's somebody at the end of this string as well, too. So he can take a piece of string like this. And model a one-dimensional line segment like this. Turn it into a three-dimensional tetrahedron like this. Take the three-dimensional tetrahedron and turn it into its analog in the fourth dimension, the fourth dimensional tetrahedron, the hyper-tetrahedron. And with Chandler's help, we'll take the fourth dimension and Dr. Schaefer will turn it into a two-dimensional peregrine. So we're off to a game. Speeding up. That's a rhythmic tessellation. Let me show you what we're doing here. Let's see. We're going to actually try a different rhythmic tessellation. And it goes like this. This half is going to play this. Each x for your group A is a clap. Each O is a slap. So it goes like this. This group, try it with me. And, great. And Dave is going to play with group B. Great. So let's do it together. Ready, set, and don't speed up this time. Ready, set, go. Ah, what happened? OK, one more time. And, great. So that's a rhythmic tessellation. It exhibits a certain kind of symmetry and you might have noticed that each one is four beats separated from the other one. And on each beat, you hear and see a clap and you hear and see a slap. So that's an example of the kind of rhythmic mathematics we might play with. But I want to start actually. Anyway, this thing called an inversion based loosely and not very well based on the work of Scott Kim, who's created some wonderful examples of this, should look like it spells dance one way. And then after it rotates, it should look like it spells math. And it indicates our philosophy about dance and math. That is this. Sometimes, we might take a math concept, translate it to bodies moving in space on the stage in a classroom or in the dance studio and see what happens. Just see where that takes us. Sometimes, we might take a dance that we or other people have choreographed and examine it for the mathematics embedded or embodied in that dance, just to see what's there. But we have the most fun in our dance company playing with dance and math in ways that show that they really are so connected like this inversion that they can't be pulled apart, one from the other. Let me skip this. And I want to just say a little bit about those of us who created this. Up there on the right is Eric Stern. And Dave called me Stern. Once Eric called me Eric in rehearsal. And so we gave up using first names. But he teaches at Weber State in Ogden, Utah. But we started working together in Santa Cruz, California. That is me when I had more here and could jump that high. And that's Scott Kim, who worked with us a lot in the 1990s and who's done a lot of things with MoMath. Also, in fact, in the reflection gallery, you can see some of his work upstairs. So I'm going to try and show you a number of ways in which math and dance are related. Like I said, sometimes we go from math and dance, sometimes from dance to math. We also play movement games or invent movement activities like that rhythm tessellation we just played around with. But we also like to use the mathematics, mathematical ideas as part of a kind of choreographic palette with which we paint or create dances. And the dances aren't necessarily about, say, the math curriculum or anything like that. They might be about history, might involve storytelling, or might involve social issues. But the mathematical elements are probably more prominent there because we're paying attention to them. But sometimes the dance work also suggests new research problems in mathematics. And I'll show you at least one example of that. And they also sometimes lead to classroom activities that are fun, but also entertaining as well as educational. But again, we have the most fun just playing around with these ways that math and dance might be connected. I want to start with a story about a book that I discovered in this is the Birmingham, Alabama Public Library. I grew up in Birmingham. And about 20 years ago, I found in the dance collection that they had, which was very extensive and very unusual in having a lot of old books. And it's actually since been moved to the non-public stacks, so you can't really find things very easily. But I found in that collection a book called Rhythm and Dance Mathematics by a man named Joseph Thee and published in 1964. And I'd never seen anything written about dance and mathematics and hadn't done much with it myself, though I think Eric Stern and I had started a few years before that creating shows about dance and math. But I hadn't seen things published about it. And about 10 years later, I finally got interested in who wrote this. And I looked online and found that Joseph Thee lived in Knoxville, Tennessee. And so I found his phone number online and called him up and asked if he was the Joseph Thee who'd written this book. He said, yes. I told him who I was and that I was interested in his book. And I said, so when was the last time somebody contacted you about your book? And he said, oh, you're the first in 40 years. There's an interesting issue here, which is this question, why is it that so little attention has been paid to the connections between dance and mathematics? Now, Joseph Thee, when he wrote that book, had been studying tap dance and taking classes and maybe performing a little bit. And he also was working in the nuclear power industry using mathematical techniques to examine data streams coming out of nuclear reactors and looking for anomalies in those data streams. More recently, he's worked in the nuclear medicine imaging field and, in fact, just published a book a few years ago about the mathematics and statistics involved in that field. But he and I and a woman named Kaja Williams have been applying what I would say is Joseph's sense of how to connect dance and mathematics in a research activity where we're looking at dances and looking for what we call the center of attention, or actually what Kaja first defined as the center of attention in dance. And that would be what that is, is that you take the locations of the dancers on stage and average them and see what that center of mass or center of attention is and see how it moves during the dance. Our idea is that perhaps by looking at how the center of attention is moving and how dancers are moving around it, we might be able to identify characteristics of various dance forms or maybe characteristics of certain choreographers there. But I'm going to go ahead and play this very short clip. And what you'll see is this open circle, which is the center of attention or the center of weight, I guess you'd say, of the four dancers in this piece. It pretty much stays in one place as the dancers move around it, because this dance, which is one I made actually, was designed to be very centrally located. Anyway, here we go. But you could probably see the motion of these little circles connected by a line to the center of attention. Anyway, that's a current research project. Before I want to do some more interactive things with you all, like the rhythm tessellation. But I also want to address some of these issues I've raised here. The Bridges Math and Art Conference is a conference held once a year, usually on a different continent. And it's really the primary conference around the world on connections between math and the arts. They now have an archive for the last 20 years of papers presented. There are about 1,700 of them. And you can search them. I did this recently and found that 35% of them are on art and sculpture, another 27% on pattern and symmetry, and so on down the line, until you get to dance, about 1.7% on dance. That's about 20 papers out of these 1,700, theater even less, fewer papers. So it really is the case that even in the math and arts world, there has been little attention paid to these particular connections. We can ask, why not? There are some reasons that we might come up with. Dance is fleeting. You're watching it. And before you know it, it's gone. Dancers have moved on. Music has a very successful international notation form in theater as well as writing that can assist in analysis, say in mathematical analysis. And sexism. Well, could it be that dance is taken less seriously because like artistic work would say fabric and cloth, it's considered either a feminine or a women's art form? And it is the case, especially in this country, that many more women and girls participate in dance than men and boys. So there's questions there. What about the mind-body duality? In other words, what we might call Descartes, what some people call Descartes' mistake. Instead of, I think, therefore I am, maybe it should be, I think and move and feel, et cetera, therefore I am. And there's a way in which mathematics in the real world could be considered more messy or more complex than the kinds of abstractions that mathematical models utilize. So there are a lot of reasons why this might be the case. If you ask a dancer or a choreographer about whether they use mathematics and dance, many dancers will say, well, yeah, the elements of dance are time, space, and energy. Sometimes they'll add force, shape, and motion. Those are all terms from physics. Shape may be from mathematics and geometry. And dancers and choreographers may often say, well, we experiment in our laboratories or studios. We perfect our technical instruments, our bodies. We present our discoveries on stage. And these are all terms from the sciences. In one sense, it may indicate just that sciences are more powerful, better funded, et cetera, than the arts in this society. But also it does show that dancers and other artists do pay attention to and are aware of the role of science in their art form, or our art form. But dance is more than those physics terms indicate. We could say it's artistic expression, which comes out in a concert. Exercise, political protests. For example, the Native American ghost dance, athletic competition. You see in, say, Olympic figure skating. Psychological work as in dance therapy could be religious expression, like in liturgical dance. Ceremony, social interaction, like in ballroom, expression of sexuality in striptease, theater, community event, like in square dance. There's a musical form, tap dance, or percussive dance forms that people stamp their feet, for example, like flamenco, tap dance, spartanacium, and so on. History also, the recounting of tales and legends. So dance is a lot of things, but I've liked to focus on the math connections, because I'm also a mathematician. I'll have that interest. I once said to a friend of mine who's here, Emily Ratliff, a friend from long ago, from high school, that I tend to think of how I think about mathematics is very similar to how I think about dance. I tend to feel and see shapes shifting in space, in both cases. And I don't know if you remember, but you said, well, that's how you think about it. But that doesn't mean everybody else thinks about it that way. So there are some warnings here. We can delve into these connections if we're interested, but maybe it's not for everybody. Science and the arts, in a larger sense, are both ways of understanding the world. Math and dance, in some sense, both explore and play with metaphors about the world. And all of these ways of thinking about things are now encapsulated in this term embodied cognition, which has become popular partly because there's my different pictures of the mind and the bodies dancing, because there have been many new discoveries in neuroscience that connect, that explain connections between the mind and the body. Oh, here's a long list of ordinary human activities on the left, related mathematical ideas or subjects in mathematics, just to focus on a couple of them to show you some examples. Human activity, making connections in mathematics. That's a subject called topology. And there are some dancers making connections. And altering shapes, or in mathematics, differential geometry. Here's a dancer named Maria Basel in one of our pieces, Shifting Shapes, and actually a very political and emotional piece about the loss of a child in warfare. And we're looking at objects, or maybe really examining objects, relates to symmetry. And here is a picture of two dancers. One of them, Jenner Purcell on the left, is now performs in New York. Anyway, I wanted to move into some interesting aspects of dancing mathematics related to symmetry. Here's Felix Klein, who in the 1870s reformulated geometry in terms of the symmetries at play in those geometries. And the Common Core Math Curriculum now also has refocused attention on geometry using symmetry, even though the approach is 150 years old. And we can also say that the symmetry approach connects to kinesthetics as well pretty easily. So symmetry. Here is a credit card. There are a number of ways to insert a credit card into a gas pump. Now, this is, if it's not a chip card, how many ways are there to insert a credit card into a gas pump? Only one of which is the correct way. Four ways. And you may have to try it three different ways before you hit the fourth. Well, I made myself a little square credit card. Maybe the only one in the world doesn't work. But why are there no square credit cards? Well, how many ways would there be to insert the square credit card in a gas pump? Eight ways. Yeah, seven of which would be wrong, and only one of which would be right. Who knows? Maybe someone will come up with a triangular credit card. Or I guess more likely they'll be doing away with credit cards entirely. So there are the four ways for inserting the credit card in. They show you what are called the four symmetries of a non-square rectangle. Well, there are also four symmetries for bodies moving in a straight line or, say, in the plane. There are more symmetries in three-dimensional space than two-dimensional space. But first a word about a very famous dancer, Doris Humphrey, who was one of the founders of modern dance, now more or less called contemporary dance. And she very famously said, symmetry is lifeless. Well, what she meant was bilateral symmetry in one body, or in which stage left and stage right are mirror images of each other. But if you examine her works, you'll see in her choreography much more than just this one reflection symmetry, all kinds of symmetries. So symmetries in a line. I'm going to put a little bit of music on. Let's see if I can get this going. So I'm going to stand here. And if I raise my right arm, I want you to raise your right arm. So you have to imagine you're in back of me. And if I raise my left arm, raise your left. And if I lean to the right, you lean to your right. So follow along. That's pretty easy to follow, right? And sometimes called translation symmetry is if the P was a person seen from above with its right arm out. And it used to be called slide symmetry in education. What we're going to do is make distinctions between facing. We were facing you and I the same way. And you all have to imagine you were. And orientation, meaning right arm or right side or left side, same thing. We were facing the same way and had the same orientation in our body. Well, what about if we are mirror reflections of each other? Give you a little musical background, Bobby McFerrin. Be my mirror image. Oh, you don't have to have one of these things. So my right arm is up. Your left arm should be up. If you're sitting in the front row, you don't have anybody else to follow. If I twist to my right, you twist to your left. If I twist left, you twist to your right. OK. So I was facing that way. You're facing this way. The facing is opposite, but also orientation in the body was opposite. I had my right arm up. You had your left arm. Reflection or mirror symmetry. And let's try one more. I'm going to raise my right hand. And I want you to raise your right hand and repeat after me. If I lean to my right, you lean to your right. Like in the game Simon says. So I'll just move back and forth in this plane. So I'm leaning to my right. So you should be leaning to your right. Yeah, it's a little hard. I'll try and step back. So you're rotated 90 degrees for me. Yeah, now if I move out of the plane, it might get a little harder. If I twist to my right, you twist to your right. I'm twisting to the left. That might be a little harder to follow. What would the letter P become in this case? So we were facing same or opposite ways. Same or opposite orientation. Actually, it rotates 180 degrees to become the letter D. And so we've got 180 degree rotational symmetry. Like I said, with you all and I, that was more like a 90 degree rotational symmetry. David's going to come help me, I think, demonstrate this. It's the symmetry of shaking hands. We rotate halfway around. The overall shape of the two bodies is the same. I raised my right and he raises his right. We could rotate around this point right here, take each other's place. Whereas if we're doing mirror symmetry, notice that the arms are away from you all if we rotate around. And the overall shape is such that the arms are near you. So they're really different. And there really is a difference between those two kinds of symmetries. Now that we've got same, same, opposite, opposite, opposite, same, you can see that something's missing. What is it? Same opposite. So let's see. I guess I should face the same way as you. And if I raise my left hand, you want to raise your right hand. So if I lean to the right, you lean to your left. So see if you can follow this. And I'll just stay in this plane. If I twist to my right, you twist to your left. If I move out of the plane with my right arm forward, you should have your left arm forward. So that one, people usually have a little more trouble following. And oh, the P becomes the letter B. I don't know if you can see that if you're in the back. But it's called glide symmetry or footsteps symmetry. I'll try and demonstrate why it's called footsteps, or we like to call it footsteps. When I walk forward, right now my left arm is up, and my right foot is forward. And as I take a step, I move in the same direction, but I've switched sides. So it's like move forward and reflected to the other side. So this is an example of glide symmetry. It's as ordinary and everyday as walking, as footsteps. Let's see. What I'd actually like to do now is show you something about how these symmetries combine now that we've had our little bit of an introduction. And usually what we do is we get everybody up and playing around with these. But there are too many people, and we don't have enough time. So David is going to come help me. Chandler, would you help as well? And the three of us are going to stand in a line, like right here. OK. And so let's say Chandler is going to make a shape. Go ahead and make some shape with your body. There you go. It could be simple and nice that his hand is up. Everybody can see in the back. And David is going to what? Oh, yeah. OK, you want to do a spot. OK. David is going to do the result of applying a glide symmetry to Chandler. So they're facing the same way, but he raises his left arm. And I'm going to take that shape of David's and rotate it 180 degrees. So right here we've got a glide symmetry followed by a rotation. And David is going to step out of the way and relax. And what symmetry do you see between Chandler and I? Does everybody see that it's mirror? We're facing opposite ways. He's got his right arm up. I've got my left arm up. So we've got a mirror symmetry going. So no, don't go away yet. I'm going to make this work. I'm going to try to build this chart. So we did glide, which is right here. Let me put it a little higher so everybody can see. Glide followed by rotation. And the result was M for mirror. Everybody see that? So the glide was between Chandler and David. The rotation was between David and I. And the result was the mirror between Chandler and I. Let's do another one. So now I'm going to ask somebody here from the audience to choose a symmetry for Chandler and David to exhibit. Yeah. Mirror. So they're going to mirror. Mirror and then what? Oh, OK. So they're going to be mirror symmetry. So I guess they have to face opposite ways from each other. So they're facing opposite ways. And one has his right arm up, the other left arm. And you want to me to be the mirror of David. So here I am with the right elbow up. And the result is translation. So mirror followed by mirror is translation. Mirror followed by mirror is a T. Let's do one or two more. What? You will get to that. Yeah. You're way ahead of me. OK, another one for David and Chandler. Yeah, in the back, yeah. Glide. So go ahead. Ah, there's glide. Now, another one for David and I, that's not rotation because we did that already. Mirror. So I want to be mirror symmetric with David. Do I face the same way or do I turn around and face that way? And do I raise my left arm or my right arm? Let's vote. How many say left? How many say right? The left's win. OK, so we've got a glide followed by a mirror. And David steps out of the way. And the result is we're facing opposite ways. Do we both have the same arm raised? So rotation. And David, come back. And let's look at this again. So we've got glide followed by mirror is rotation. But David was just saying, what about the other direction? There's also rotation. I mean, mirror, which is David and I, followed by glide, is also, and David steps out of the way, is also rotation. So it doesn't matter whether we go glide, mirror, or mirror, or glide, we get that rotation, thanks. So therefore, that was glide, mirror was rotation, but also mirror or glide was rotation. So anything that we have, so to speak, above this diagonal is going to be below it. Let's see. Let's do one more. Somebody else give us a suggestion for David and Chandler. Yes. Translation, OK. Is he doing it right? Is that right? OK. So what about between me and David? Glide? OK, so do I face this way? Which arm do I raise? Left. So translation followed by glide is? Glide. But what about glide followed by translation? Glide. So we get two more. Translation followed by glide was glide, glide followed by glide. Anything along the diagonal is going to be translation? Ah, let's try it, and then you can explain what you're thinking. So we did mirror, mirror on the wall. And which one do you want to do? Rotation, rotation, or glide, glide, or rotation, rotation. All right, they're going to be rotational opposites, or rotate each other. So you want to, there you go, OK. And then I want to take Chandler's shape and rotate it. And the result, when Chandler steps out of the way, is translation. So why do you think, then here's a nice question, why do you think that anything followed by itself gives you translation? If they both reverse direction, then reversing direction twice gets you going in the same direction. And it's the same with orientation, is what he said. So if you reverse orientation with one of the moves, then doing it again, that double reverse gets you back to the same. So there are t's here. And I'm going to just fill in this because we are short on time tonight with what goes in these places. The t is like adding 0 to a number. It leaves the number the same. We're multiplying a number by 1. Leaves it the same. And m followed by rotation. Anyone guess what that's going to be? It actually is a g. And that one we knew was an m. And so now that we've got this table, we can say, what patterns do you see in this table of symmetries? And let me, I'll ask for some responses here. So what are some patterns that you all see? Raise your hand if you've got one. Yeah, right here. The first column is like the same thing on the last column. So this column is the same thing but in the opposite direction. What? The first row is the opposite of the last row. And what were you saying? It's what? She knows the name of it. It's the Klein force symmetry group. Yeah, it has a name. And what about this row and this row? In this column and that column? Yeah? You say it's mirrored across either diagonal axis. Across the other diagonal axis as well. In fact, what you all were just pointing out is that you've got 180 degree rotation. So you've got all these symmetries in the table of symmetries, which is itself pretty interesting. Any other patterns you all see? Yeah? This was on the left quadrant and the opposite quadrant. What were you saying is the bottom right quadrant? So let me divide it up into its quadrants. And so you see that aspect of the quadrants being the same as what's diagonally opposite. And let me go back then to, yeah. So that pattern with the quadrants, upper left, lower right, is the same or the same upper right, lower left. Similar to when we add even and odd numbers, if you add an even to an even, you get an even. And if you add an even and odd number, you get an odd number. What do you get if you add two odd numbers, like 3 plus 7? You get an even number. Well, what about this? When you multiply positive and negative numbers? Positive times positive is positive. Positive times negative is negative. But what's negative times negative? Positive. So this particular structure is very common, is ubiquitous in mathematics. And we've kind of gotten to it by looking at bodies moving in space, which is kind of interesting. Not only that, but some of the patterns like this reflection along this diagonal may be puzzling if you just look at the symbols here. But when we looked at bodies in space, you saw that it was obvious that this followed by that has to give the overall result that's the same as this one followed by that one. So the bodies moving in space actually make clearer why it is that we get this symmetry of the symbols. And sometimes we tend to think, or we tend to be told sometimes, that mathematics can only be done via symbols and that the symbols tell us everything. But in fact, we can see that the kinesthetic approach, the approach of how we move our bodies in space, can be more revealing and teach us more about what's really going on here. Does that make sense? I think it's an important lesson. Another lesson is that when we move from one form of representation of a concept or idea to another one, in this case, from movement of bodies in space to symbols, we notice new things. So one form of representation, symbols or bodies in space, is not better than the others. But the act of going back and forth can be quite revealing and give us new perceptions. And to me, that's one of the real powers of mathematics. It's not necessarily in solving equations, but in learning new things by looking at things differently. So let's see what else we can look at. The fact that mirror followed by rotation was glide. Rotation followed by glide was mirror. Glide followed by mirror is rotation. Or you can go in the other direction. You get a really interesting triad of operations here. And we can also look at connections to the math curriculum. So as I said, we've found ways of making connections between how we move our bodies in space or on the stage to the math curriculum. If we reflect across the x-axis, the y-coordinate changes to its negative, if we reflect along the y-axis, then the x-coordinate changes to its negative. If we rotate around the origin, both of them change to their negatives. So we get a new set of three symmetries as well, or really a total of four, if you include the identity. And these are the same symmetries as the symmetries of the credit card or the symmetries of dancers in space. We can also look at symmetries across these diagonals. And what that does is switches the coordinates. So these things you may have studied in math and pre-calculus or whatever in algebra can be examined and played with by just moving around on a stage or in space. And interestingly, some of these same symmetries are at play in the dance form of square dance. When you have eight dancers, four couples, you can see some of these reflections or rotations operating. We can also look at how these symmetries play out in the visual arts. Here's a triangle, any triangle rotated 180 degrees and then moved around will be seen to tile the plane that has covered the plane without gaps or overlaps in a pattern that could continue indefinitely in all directions. And it's the same with quadrilaterals, even if they're concave. If you take another one and rotate it 180 degrees, attach it to itself and put in multiple copies, you get what's called a tiling of the plane. What about pentagons? Well, there's some wonderful history here. And Cindy Lawrence was pointing out to me that some of these pentagonal tiling discovered by this woman, Marjorie Rice, are on exhibit here in the museum. In the 1970s, there were about nine kinds of pentagons known that could actually tile the plane. And Marjorie Rice was a homemaker, had not studied geometry, had studied geometry in high school, but had been 20 years since she had, didn't have a math degree, but she got interested in this question. And she actually discovered four new kinds of pentagonal tiling that mathematicians had missed. But in my mind, that wasn't the only important thing she did. She also created beautiful, Escher-like designs based on her tiling patterns. So she did both the mathematical and the artistic work as someone not officially trained in mathematics. Here's another one of her pentagonal tilings and Escher-like butterfly design based on it, a design of bees based on another one of her tilings and so forth. Well, we created a show a few years ago called The Daughters of Hypocha about women mathematicians, their mathematical work, their life stories that in many cases had been overlooked by historians. And we wanted to highlight Marjorie Rice's work. And so a software designer named Kevin Lee, who I think has done some work with MoMath, also allowed us to use some of his software which hasn't been released yet actually. And what this is, I'm gonna show you this excerpt from this piece. What you'll see are dancers doing a kind of, what's called a structured improvisation in dance where they're improvising to music and they're watching their images on a screen in back of them and then there's a smaller TV screen downstage of them that they can watch. And they're manipulating their image in a tessellated pattern. So at the beginning of this excerpt you'll see a very simple tessellation design because we discovered that audiences thought we had created these beautiful video designs prior to the performance, but in fact the dancers are creating them as they go. So at the beginning you'll see something very simple and then it morphs into a more complicated tessellation design. So it turns out that it takes actually a good bit of practice in the studio for them to do this in a comfortable, a way that's comfortable to watch. They're really moving very slowly. And these are Laurel Shastri and Maria Basel, two dancers who do a lot of work with us. Okay, so I wanna go on to something that's a little more interactive with you all and show you a way of using symmetry and bodies that may be a little bit surprising. This is something that was brought to us by Scott Kim also, did I mention that already? He has, yeah, who's worked with you all and who brought us something that he was calling finger geometry. This is a finger cube created by some students in Puerto Rico a few years ago when we were doing workshops there. And do this, push your first and second fingers together like this and let the index fingers go up and the second fingers go down and you get a kind of diamond shape. So if you do it sideways, it's kind of hard but if you face your hands down and push those fingers up, it works pretty well. And if you rotate it, you'll see it's actually a very nice square and David's gonna join me, I think. If we open our hands, you get a shape that's kind of like gold posts and if two people then side to side move their hands together, you get a really nice square that in an animated way sort of opens and closes and flower in or flower out like so. Okay, so there's, did you all get that? I think so. So there's a fairly simple one. Now, Scott, tetrahedron. And so there is a way to make a tetrahedron with hands that I'm gonna show you and it goes like this, you put your thumbs together, put your first and second fingers out like a pair of scissors and now watch very closely because this is a little tricky. We face each other, one of us rotates his hands 90 degrees so my hands are up and down, his are right and left and we join the eight fingers together and you get this really nice tetrahedron. So try that with somebody near you. Thumbs together, first and second fingers pointed at your partner, face your partner. One of you rotates only 90 degrees. Did you get it? Okay, face me. No, don't rotate, only one of us gets to rotate. So try that. So there are a lot more things we can do with hands and so forth but now they're playing with these polyhedra, this tetrahedron. I need somebody to help me out with this. Let's see, who can I choose? Would you come on up? So David, could you hang on to this? And what's your name? Sophie? Sophie, we've got this tetrahedron here and these PVC pipes, I'll hold it up so people in the back can see, can you fold this to make that tetrahedral shape? You wanna play around with it and see if you can. Ah, tattoo right there, okay. Yeah? Very quick, very nice. Take a bow. Thank you, Sophie. Yeah, when we use this in our stage shows sometimes with someone from the audience and if you hold it like this, whoops, it's hard to see in fact what happened naturally there is what people often try and do to hook those two things together. And Sophie just saw it in this form when very young children are there will sometimes hold it like this because then it's easier to see, right? How it goes together. But in this form, it's kind of paradoxical how tricky it is. Kind of play with these fingers and PVC pipe led us to try and make dances with PVC pipe as props or hands and we actually have created a whole bunch of dances based on this and so let's see where I am. I'm actually, I'm gonna show you an excerpt. This is one of our dances for hands in this case hands coming through a screen created about almost 25 years ago but it's a piece we still perform that people enjoy and like I say, it's one of a number of dances with hands making shapes. So let's try and get this excerpt going. Okay, so that's a short excerpt and sometimes we make geometric shapes, sometimes have little hands walking along, just play around with it in order to see what kinds of things are fun to watch. This led to, like I was saying before, a mathematical exploration which was this, are there ways to use fingers or PVC pipe to construct say all the platonic solids, all of these polyhedra and the first thing we did was we put big sections of PVC pipe together in what we call twosies or threesies and invented some dances with them. Let's move on. Oh, we also made a cube, here's made this cube for the Children's Dance Foundation in Birmingham, Alabama but also at Bridges Bant we played around with it and I found a way to supposedly make all the platonic solids with six dancers but we have never performed this because I think you'd have to perform it underwater or in outer space where there's no gravity. Anyway, the mathematical exploration that it led to was looking at what could be called the skeleton of the polyhedra, that's the set of edges that make up each polyhedron and if you look at all of the platonic solids they each have a number of edges that's a multiple of six and here is a set of photographs and I'm only showing photographs rather than a video excerpt because this piece was done with these PVC pipes painted with glow paint and using black lights and it really would show up even less well than some of these other clips. So here are some photographs and each dancer is holding three sections of PVC pipe. There's an octahedron and the same sections made a cube. Cube flattens out into this wheel or into this interesting shape but we also make shapes like this rowing set of shapes or wings because we are not just trying to demonstrate say the math curriculum but as I said several times trying to play with the imagery in a way that's fun for us and entertaining for our audiences. Let's move on and here's some other ways in which we've played around with these polyhedra. Doing okay? Doing good, okay. So string polyhedra with loops of string and we're actually gonna show you some more of the polyhedra. We showed you a little bit at the beginning when David pulled the string out so David and Chandler and I had a chance to rehearse this a little bit. It's not really a dance but just a kinesthetic construction of polyhedra. We all put our right thumbs in and grab two strings and come together and each grab one string at the top and open that up. And that's actually an octahedron. It's a little hard to see. Usually the octahedron is held so that one point is above but we're showing it in this form where it's called it's threefold symmetry where you see it sitting on a face and I actually need one more person to help me. Come on up. What's your name? Come back here where I am. Come back here where I am. What's your name? Jonathan. Make a loop with your fingers with your right hand. Use your right hand up there and make a loop with your fingers there because I'm gonna be pulling on these. Now this octahedron has six vertices. It is a triangular antiprism. A prism would have vertical sides but he's noticed that you've got these triangles sort of oddly at, I don't even know what you would call it but each one is slightly off kilter from the other one but it has six vertices which is why three people can hold it with their six hands. The cube, how many anyone know how many vertices the cube has? Eight, one, two, three, four, or two, four, six, eight. If I could add two vertices, we might get the cube. What I'm gonna do is I'm gonna squeeze this and squeeze this and now we actually have a cube sitting on a vertex. Usually the cube sits on a face but we've got it sitting on a vertex and it does collapse down into a wheel which we can show but if I let go and they pull it taut you actually can see a star of David or a six pointed star if you separate your hands a little bit Chandler if you go, move it down, everybody can see, thanks. Great. So there's a nice six pointed star and again, to, what? Put head inside. Yes, again, to make it into the cube all we had to do was squeeze a triangular face like that. Anyway, thank you Jonathan, great. So again, is that dance? Not really, is our possible. Here are some other dance pieces that involve this. You saw a video of that, that's from a whole show we did involving loops of string. I'm just gonna show you a little bit more. Scott Kim and I heard that there was somebody in our area who was teaching giant string figures to dancers and we invited him to a rehearsal and Greg Keith was his name and he came to the rehearsal and started teaching us string figures and Scott and I wondered whether there were more obviously mathematical string figures that we could create and he had taught us traditional string figures so let me try this again. Traditional string figures from around the world tend to be representations of animals or people, sometimes abstract designs. The string is, it's a little stiff, let me see if I can get a gun, there we go. So here's one that I invented and first one's hard to see maybe it's a nine pointed star and this is a seven pointed star and this is a five pointed star and this is a three pointed star and this is a really pointless star. So Scott and I wondered about what else was possible and we invented the string polyhedra I just showed you but there are a lot of really wonderful string figures for example this one's called the Philippine Noose Escape, oh I'll do it again. The traditional string figures that move are a lot of fun, it looks like I'm gonna strangle myself but not so fast or this one is called mosquito when you try and catch a mosquito it slips away. Anyway, less about that, more about some of these dances. There's a dance Eric Stern and I did with bungee cords and another one with PVC pipes and here are some puzzles like puzzles that we created and I've been using straws and pipe cleaners so one of the puzzles is this and you've got in your handout you've got pictures of these as well which of these can be folded into a tetrahedron there's some more on that post or you can just look on that handout and if you find that too easy then you might try and figure out which pairs of those could be folded into an octahedron or a cube because the tetrahedron has six edges or six straws the octahedron and cube each have 12 so it would take two of them well here's the math problem that this generated the math research problem and it was this say when you look at the platonic solids each has a multiple of six number of edges and so I wondered what is the largest structure that could be multiple copies of which could be folded into each of the platonic solids I call it the greatest common decomposer or edge GCD and here are six possible what are called trees each of which has six edges or straws and multiple copies might make things actually each one of them two copies of each one will fold into the octahedron or the skeleton of the octahedron and each of five of these will make the cube or the icosahedron four of them will make the tetrahedron there are two that won't again it's a puzzle for you to figure out which ones but there's only one that makes the dodecahedron and there it is down in the bottom and it's also what's called the Dinkin diagram for E7 of all of them have symmetry except that one and so it could be that this one which I called T123 because it has these pendant edges of lengths one two and three it could be that the lack of symmetry actually makes it more versatile symmetry may be a limiting factor in this case so like I said this was a kind of research problem here's how it works there it is making the tetrahedron two of them fitting together to make the octahedron two of them fitting together to make the cube five of them floating together to make the icosahedron where those X vertices you have to imagine are joined together in back and here is five of them fitting together to make the dodecahedron so a kind of fun example and it turns out that this particular T123 is what I tend to call it decomposes many different kinds of polyhedra and here are a bunch of the platonic solids and Archimedean solids that it decomposes the edges of here are these other tessellations but that wasn't all what happens with these kinds of problems is once you've solved something you ask yourself other questions and like in the arts once you've made one picture or one dance you can ask well what else can we do on stage and so I wondered about something derived from something similar to the game of asteroids when the rocket goes off on the right it appears on the left in that old game of asteroids and that's like taking this cylinder this rectangle and joining the edges to make a cylinder now when it goes off on at the top it reappears at the bottom and so that would be like joining this circle in this circle to make what's called a torus and if we look at what's called a torus graph that could be wrapped around this donut shape torus could that be decomposed by that very special graph T123 that special tree and it turns out that it can be there are multiple copies but that's not all we could also tessellate this cubic lattice in space with that same T123 which I found really intriguing and again it seems to work in a lot of these cases but I don't know why and so that's the ultimate question is why is it so versatile and not to be stopped there I thought well what else could we do let's make an alphabet each one of those letters is composed of one copy of that T123 tree inspired by Eric Domain and Scott Kim's uh... play with fonts actually I'm gonna show you one more excerpt because I think it demonstrates something that I find really important and I hope you'll be able to see this this is from our show the Daughters of Aipache again it's about women mathematicians throughout history and uh... we have a lot of storytelling uh... about uh... different mathematicians and also in some way shows something about their work like Marjorie Rice's work with tessellations and at the end of this show the four dancers in the show talk about they wrote some script and they created some dance uh... that we melded together uh... about their sense having been in this show about how mathematics uh... exists in the world of dance the music is actually by by heart uh... who many of you may know for for much of her interesting mathematical work but you may not know she's a very accomplished uh... composer so I'm gonna show you um... a little bit uh... just a teeny bit of this before we call it quits I've always liked math I thought of it a lot like three puzzles but I always thought of it as black and white right or wrong and it requires a lot of memorization now I see it's much bigger it involves imagination creativity thinking in new ways it's in how we specialize in rhythm and grace to make a tap dance how we arrange the answers in patterns on the stage how we flow through shapes in space and time as a child it was just as soon that boys were better at math the stereotypes and the teaching of the subject itself were very limiting I've never thought of myself as a mathematician but as an artist yet as a dancer choreographer human being we are immersed in a universe of mathematical concepts I now have the vocabulary to understand this Cindy wanted me to bring it to a close but show you a little bit more about one of the other mathematical elements that we've played with I'm gonna steal a little bit more of your time just to give you all something to do circle your hands so they're both going in the same direction now reverse them so there are two ways to circle but we can also circle in the same way so they're half a circle apart or we can reverse that so now we've got four ways they're going in the same direction but they're out of phase well we can also circle our hands so they're going in opposite directions and reverse that now try and do that out of phase so they're going in opposite directions and you can reverse that there are eight ways to circle the hands or the arms if we have more space and that got us got me thinking about ways that we can employ circles here's something really interesting uh... about a circle if i circle my arm all the way around keeping the arm going in the direction of the palm notice it's turned over in order to get it turned right i have to go in back of myself after complete what you'd think of as two complete turns because in reality in three dimensional space it takes not three hundred sixty but seven hundred and twenty degrees to get back to the beginning so we've i've made a bunch of dances in the last ten years that deal with circling patterns and uh... i found these light balls online i don't know if you can see them let me uh... black this out see it a little better uh... these are designed for people to use in swimming pools throw in the swimming pool they float and they're real pretty and all the reviews that i found online said that after five seconds my light led lightball stopped working uh... because it got waterlogged but on stage we've had no problems and so we turn the lights down which we can't do here and uh... create all kinds of interesting patterns with the balls in space including that one that i was just showing you and you can actually use uh... both of the arms uh... in that seven hundred twenty degrees circling pattern and all kinds of other interesting patterns uh... that are fun to watch in the dark anyway i i think that brings me to the close i think we may have time for a few questions we have time for two questions so two people raise your hands i'll bring your microphone this microphone to anyone who has a couple of one question or comments or comments or reviews i think everybody's being very polite away we got one back here hi i'm actually a dancer uh... and like laban and the kinesphere i'm also thinking about alwin nicolai's dance i can't remember the name of it what he uses all the ropes and strings yes and the way i can't remember the name of it i can't either so just the because laban has notation right so there is notation for dance yes laban's notation and the ways in which um... applying what laban does with the kinesphere to what you're talking about with the platonic cells and the polyhedra yes this so it because you can also use weight and force to these things too so not only are we making these shapes but you're also using the sense of dynamics in movement is more of just a commentary on on connecting it a little bit more to dance than the mathematics and thank you for inspiring me so let me say a little bit about that so rudolf laban is one of the most prominent examples of uh... choreographer movement artist who did explore mathematical elements of dance and movement and he located the human body within various polyhedra there's a wonderful photograph of him sitting in his office surrounded by models of polyhedra which looks a lot like a very similar picture of mc escher sitting in his office surrounded by models of polyhedra you can almost not tell which is which and uh... for example the right and left points the up and down points the front and back points are the six points of an octahedron and he said that uh... is a very stable position if you reach to those positions whereas the points of the cube if you're surrounded by them are on diagonals and he said when you move to those points brings about motion and so uh... leban and his followers develop some ideas about this there uh... exploration of symmetry tends to be about one's one body within say a polyhedron and our emphasis has been on how symmetry has is used among multiple bodies in choreography so we have had a different approach and you're also right uh... leban and his followers didn't just delve into uh... what are the directions or the locations in space but what what's now called effort shape which is the effort uh... heaviness or lightness the shaping qualities and so forth so right and he did develop the most widely used notation form lab of notation but there are literally hundreds of notation forms in use in a few and you probably know i'm sure ask any dancer or choreographer what kind of notation they use for their dancers most people say i just made something up and so there really are hundreds of notation forms and uh... none of them have taken off like musical notation say or writing all right let's give a hand to our speaker i'll say her okay thank you