Speaker 1:	I'm thrilled to be here, I love the school. It's one of my favorite schools. My son, he's a junior in high school. This is probably his dream school to come to, so I've never been here before so I'm just really thrilled. Although I know a lot of the faculty here, they're friends of mine over the years. I had email correspondences, and actually went to the conferences. So it's good to see people here. 
	So let me tell you what the vision of today is. It starts in this era of the renaissance. Where Di Vinci was there. So here's where you'll want to think about Di Vinci's ideas. Here are a couple of things you guys know, the Mona Lisa [inaudible 00:00:43]. His study of biology and anatomy. And this engineering aspect, he wants symmetry, he wants perspective. What's really interesting about Di Vinci that I love, is the fact that he's not three different people. 
	He doesn't have a hat called the engineering hat, and then he takes it off, and he puts on the artist hat. And then he takes that off, and then he puts on the scientist hat or the biologist hat. You could see in his work on biology, his work as an artist. And you can see in his work as an engineer, his work as a mathematician. You could see all of these things blended together in all of these fields. He's really not switching gears. 
	Now what happened after the Renaissance is it's a time of enlightenment. It was a beautiful time around the 1750's where what happened? Everything got compartmentalized and broken into pieces. We wanted full understanding of why things were working. Somebody said, "Look, the sun is the center of our solar system." Why? You just can't say it, prove it, convince me. 
	So you need to go really deep into astronomy, into physics, to understand some of these things. By doing so you get a really intense sense of tunnel vision. Because you get to be good at that one thing well. So these gods, like Di Vinci who reigned over the earth at that time period, became less and less. What happens was, there's this huge chasm that's come between math. Just sort of on the one end. 
	You've seen those posters, or comic books, things like that. There's an order of who's cool, right? There's sort of a biologist, there's the chemist, there's the physi- actually these guys are getting more and more abstract. And then the truly cool people on that side are the mathematicians. And then, sort of the other spectrum, you have to draw a spectrum at the other end, one side analytic people, very intellectual, broken down, mathematicians and then the soft hearted, the warm, loving artist. Who they just want to express, that's all the want to do. 
	There's this chasm of compartmentalizing people and people like Di Vinci, these great ones, that world kind of starts to fade. So when we think of math and art coming back together again, to come back to a time of the Renaissance again, what does that look like? 
	So let me show you a couple of things that people think it looks like, and I want to give you some other options. So here's one attempt of what this could look like. Piero della Francesca, one of the most famous works if you care about art history and look at art. This is one of those great works that was the first one to really bring together design of perspective. 
	Here you see amazing perspective lines, you can actually see those lines focusing in. We take it as granted now a days, right. But at this time, it's amazing. You can see the floor, you can see the tiling pattern of the floor with the perspective. In fact you can even see the top right corner of the bell tower. All of these things drawn flawlessly. So people say, this is in some sense the intersection of math and art. 
	You can see the use of geometry in this beautiful way. One of my favorite parts is the light that's hitting on the tile of the ceiling, I think it's gorgeous. 
	Here's another one, [inaudible 00:03:53] has kind of pushed the boundaries of what math and art could be and has worked on wall drawing. Next to Williams, back up in Williamstown there's a town called North Adams where they have an acres worth of Louit's work. And it's beautiful. It's just an entire warehouse, Williams is just a small town, so you can actually have a lot of land that's cheap unlike your Philly, your Stanford, your Paulo Alta. 
	This entire warehouse has been converted to Sou Louits work. Sou Louit, here's the way he does it, his works are basically instructions on what to do. He doesn't ever paint anything, it was a beautiful job he had. So he just tells you, draw the line one centimeter from the floor connected to this line. Shade this- and then he gives it to other people and they through his approval actually draw the wall drawing for him. 
	For him the idea was, the idea itself. That piece of paper with the idea, that was the artwork. This is sort of a representation of the artwork. He sort of pushed this idea of what it means to draw on walls. So beautiful, extreme, this is this guy. So you can sort of see the nerdiness of math coming through. So yeah I could see where math and art are fitting together. Even the concept of the fact that it was these instructions feels like a theorem. There's this idea of the theorem not what's written on the piece of paper not how it's copied. 
	Here's something else, Marcel Duchamp, you see the nude descending the staircase? An extremely famous work, but here at this time at the turn of the century, this struggle with four dimensions, what that looks like when you're trying to capture an extra dimension, especially in two dimensional framework. You can see the layering, one on top of the other one of how you can keep order of time. 
	This is actually pushing a different way, by Salvador Dali. This is called the Crucifixion. And here you see, something has actually unfolded just like you could unfold a three dimensional cube and lay it flat and it looks like squares glued together like a cross. He's unfolded the four dimensional key. He's not laid it flat, he's laid it flat in space. And he's put in some sense a very religious and symbolic figure of Christ on the Cross with the Virgin Mary looking at him, that adds some meaning. 
	What that meaning is, that's not my thing, but the point is, that these guys are struggling with how to capture dimension, and capture space. Is this the intersection of math and art? 
	Well here's a very different take on it very recently, this is Julie Mehretu's work, I love her work, she is a MacArthur Genius Grand winner. She won it a few years ago. This thing, does not do it justice. It's almost the size of a wall, it's huge in it's immensity. But you could actually see, I think it's called Stabia or actually, Stadium is the translation if you want. 
	But can't you just see this enormous stadium maybe where gladiators are fighting, you can see the energy all around, you can maybe see the flags and the banners all around. It's a very abstract in it's sense but if you can see the movement of space, it's sort of Marcel Duchamp in a very massive setting. So maybe this is math and art. Maybe this is what it means to talk about math and art. 
	It's a very recent work of art. What does it mean to kind of bring back the Renaissance again, to bring these guys back together. Here's a work from an architecture perspective. From a work of sculpture. This is Anish Kapoor, this is in Chicago. I actually moved to Chicago around the time Michael Jordan moved there. The big question is, did I go there because of him, or did he go there because of me? 
	This is amazing, you could actually walk under this little bubble and the perspective and distortion of these skyscrapers are gorgeous to just stand- and you can actually see the deformation of space and deformation of that tiling on the floor on this beautiful sculpture. 
	This is another very recent work, Rem Koolhaas's work. This is the CCTV Tower, this is in Beijing. This is the central tower which broadcasts all of the national TV and information from China. So what China's trying to say is, look if you go to the top it really doesn't mean you're any far better than the person on the bottom, because it's just a loop anyway. 
	Those in bureaucracy and in power are basically the same as the working mortals who are down on the first floor, right? Whether that's true or not is a different thing, but this is again, beautiful architectural work. If you want, you can kind of turn up the nerdy scale a bit. [inaudible 00:08:20] is that math and art? [inaudible 00:08:20] right, does that mean something? Are we now, bringing Di Vinci again in terms of building types of things? 
	These are all clearly beautiful things. All wonderful works, but are we really at the edge of math and art? So this is my perspective on it, my perspective is the following thing. See this arrow is very rich to me. Math has influenced amazing works of art. But what is this era? So what new math have we come up with because of works of art?
	It feels like they're just sucking us, you know what I'm saying, mooching. They do things, nothing wrong with that, sweet. Take it my friend. But if you think about Di Vinci again, he blurred the lines between those worlds again. So what does it really mean to bring math and art together? Can you create a work of art that actually needs, not math, not uses math from 20 years ago or hundreds of years ago. But uses math that needs to be built in order for the work of art to exist. What would that look like?
	That's what Di Vinci had, that's what I'm excited about. So to me, I just want to give you some steps towards that answer. That's what I'd like to do. 
	So it could start with bees. And I think about punchlines to some of this is nature holds the key to actually making the step between those two worlds. It's done in some beautiful things, I just want to talk a little bit about it and hopefully have a little bit of time for you guys to ask me some questions. So here's the problem, that's a beautiful problem to think about. 2D patrician problem, that's what it says. What is the most efficient way to cut up the plain into equal areas?
	So when I say the plain I don't actually mean a sheet of paper, I mean the plain that just goes on forever. So think of it that way, no boundary. So you have a pencil and you could actually mark up this plain, the chalkboard. You want to cut it up so that each piece that you cut up has the same area as every other piece. Cool? The only constraint is this word efficient. What that means is you want to use the least amount of pencil lead in order to accomplish it. You want to use the least amount of chalk in order to do so. 
	That's the question, okay let's take a look at it, this is my first shot. If I had to draw this, right. If somebody said, cut up the plain into equal areas, this is what I would do. Just a perfect grid. If you think about this for a second, you know that they're all equal areas. But the one issue is this corner, these corners. Look at the amount of ink that's used for what it's really capturing. 
	You see here, the sides, the ink that I'm using to capture these guys makes sense. You have to track those regions. But that decent intense concentration of ink right there, but what am I really getting out of it? I'm just getting a little corner. It feels like I'm wasting so much ink to get that spot. To actually make that partition. A great way to do it is sort of, instead of doing it this way, the dream would to actually just draw a circle. 
	The circle would avoid the corner completely. But the problem is, you can't patrician space using circles or you're going to have all those leftover holes. So you want to think of these circles filling them with air. And what you would get aren't perfect squares but you might get something like hexagons. 
	Now if you look at that corner again, notice that it's a lot less in than the one before. That had like four lines coming together. Now you only have three lines coming at a corner, do you see the hexagon looks more circular than a square does. So the circle is the dream, we can't get there but at least the hexagon has it. Is this the answer? 
	Is the tiling of the plain by regular hexagons, notice everything is the exact same area, the best way of cutting up the plain, partitioning the plain into equal areas using regular hexagons is best. Is this true? Well this is called the honeycomb conjecture. Because bees make something like this with their honeycombs. That's the conjecture. 
	2000 years old. That's pretty stunning that nobody knows the answer to this thing. 2000 years. And it actually came up because a roman soldier was writing to his wife in a letter and he wrote, by the way honey, take care of the family, and arranged things this way, 'cause you know, the bees know what they're doing. I'm paraphrasing. But its roughly what it is, translate Latin. So it actually became the standing conjecture that people thought was true and eventually people looked at it and said, you haven't really proven it. Nobody had a clue what to do. 
	It's pretty stunning that such a simple problem, the way of partitioning just a piece of paper, using chalk, you don't know. Thomas Hales proves it. Just revolutionized the world. He's a professor at the University of Pittsburgh. The proof is 20 pages long and it uses extremely elegant computer calculations. Just simple ones like, you would believe and understand. Absolutely stunning. Quite recent. Alright, it's great, it's wonderful. 
	So then the questions the following thing, what about 3D? Take space, what is the most efficient way to partition space into equal volumes? If you actually think about it, the honeycomb conjecture [inaudible 00:13:53] because bees are 3D. Their honeycombs are actually 3D. But they need an entrance to it. So we actually look at the surface of their entrance. What is the most efficient way to partition space into equal volume. 
	It turns out, that the reason this was motivated was because of issues with ether and foam. Have you guys ever heard of that stuff before? Well ether and foam is what people used to think, like Lord Kelvin of the Kelvin temperature scale, people used to think the entire universe was filled by, we sometimes call it dark matter. We're trying to figure out what's out there, this stuff. People say that air- well [inaudible 00:14:34] goes through it. It actually goes through stuff that you can't see. 
	It's called ether and foam, fills it up. So we wanted a way to tile the entire universe would stop so you could put planets in the stuff and atoms in the stuff. And so what would be tiling the universe, this would be the natural question. 
	So it turned out there was no ether. It was a beautiful [meta-samorli phonetic [00:15:00] experiment that says there is no ether. But still the question came up, what is the most efficient way to partition space into equal volumes?
	So there's one attempt, let's do it again. What did you do, you chop it up, take a knife, cut up the cheese into little cubes. Most natural thing you'd do and again if you think about those corners, notice the amount of surface area that it's taken up. Now be careful, now my knife cuts aren't just a thin grid lines that you're seeing, but its actually the entire boundary. The surfaces of the cube. Just like the squares are the surfaces of the lines. 
	So that's all of that stuff, if you look there and see all of these surfaces meeting at that one point, and you feel like you can just make this spherical somehow. So how would you make this spherical? How could you make this more spherical? The problem is when you do that it needs to tile space, remember we need to have equal volume in every piece. 
	So how'd you do it? It's not obvious but there must be some hexagon-y things in there. Because hexagons were really sexy in 2D and you have like 2D things, right. If you actually look at a grid, doesn't that look like one of those 2D grids? So you must sort of see honeycombs showing up in 3D, though what would it look like? It turned out it looked something like this. 
	And Lord Kelvin had a conjecture. Now I need to tell you about Lord Kelvin. Lord Kelvin dominated, he was bigger than- he dominated his era. So when Lord Kelvin spit, people were like, that is the greatest thing I've ever seen. So Lord Kelvin said, this is the thing that tiled space. It turns out that there is a tile to this, you could actually take these guys and tile space. Beautifully tiled space, all of the same, perfectly regular, gorgeous. 
	And so Lord Kelvin said, this is the one that works. Everybody said, the Lord has spoken, this is right so let it be said. And it was done until people started questioning it far later. But does it really- is it really the best one? Are we missing something? Here's the punchline. This never shows up in nature. There is no Kelvin tiling that shows up in nature. Ah, well, I take that back. It does, you sort of push certain things. But it's not the most natural place to find out, to artificially rig it. 
	What's interesting though, this hexagon, this 2D hexagon that we saw and this 3D Kelvin cell has a natural generalization in math, it's called [inaudible 00:17:32] always you can realize permutation. So if you take the permutation 123, 321, 213, all that stuff, you can actually take it as coordinates and 3 Dimensions. Like 1,2,3 is a coordinate in our 3. 2,3,1 is a coordinate in our 3. If you look at all those coordinates and you look at the convex cell you'd actually get a hexagon. If you look at 1234 and all those coordinates that are 4 you actually get the Kelvin cell. 
	Permutations, realizations of permutations in this beautiful way and all of a sudden we're getting all the things that feels like solving the puzzle? It's gorgeous. So there's natural [inaudible 00:18:09] it's called [inaudible 00:18:11] all that stuff. So is this the right answer to the Kelvin cell? Here's what it would look like if you're in it. So this is what the universe would look like, what foam would look like if you're standing inside it. Of a tiling of this Kelvin cell. 
	125 year old problem, Kelvin's conjecture partitioning space into equal volumes using the Kelvin cells is the most efficient. So now comes ware and Thelin. Ware and Thelin, PHD students and the professor working together, they come up with this thing. What is this thing? So first of all, it's a bunch of dodecahedron. Dodecahedron are 12 sided dice that you might have seen before. But these dodecahedron aren't perfect dodecahedron. They're not regular, they're a little irregular. The angles are off a little bit.
	Remember, I need to really worry about angels in this world, I need to tile things, it needs to match up perfectly. So they're not, dodecahedron, regular, leave it right there. The other ones are two hexagons, hexagons are good. We've seen hexagons. Surrounded by pentagons that kind of wrap around it. So there are two of these guys and a bunch of dodecahedrons that are irregular fitting together. So notice this is a weird object. 
	There's a software you can just download right now, that Ken Brakke came up with called the Surface Evolver. What they did was, where Thalin said, why does it have to be that the tiles have to be perfectly the same? What if the tiles were different a little bit?
	All we had to had was same volume. It didn't say same thing. It said, cutting the same volume. What if this was the thing? And it turned out it's tiled space. In a funny way, but Ware Thalin tiled space. At the moment Ken Brakke Ware Thalin sent their work to Ken Brakke, to say, could you run this for us. This was before computers were in the palm of your hand. 
	The moment Ken Brakke saw this he knew they beat Kelvin. And they did. 0.03% better than Kelvin. It's awesome. So there it is. There are couple of catches, my friends. Catch number one, is the fact that although it beat Kelvin, we don't know if this is the best. It's better but we don't know if it's the best. 
	Catch number two, Kelvin still is winning the game if you want everything to look the same. If you want sort of the hexagon version of everything to look identical, Kelvin is still that unique cell that still does it. Let me push ahead and tell you how it shows up in nature. Here's what it is, this is actually a [inaudible 00:20:54] molecule. It's a crystal that shows up in Arctic Circle 2, so in gas pipes that run in the Arctic Circle, is these kind of crystalline structures naturally show up and that is the Ware Thalin structure. It shows up in nature. 
	Let me show you where this is actually. This is from this book, the Nature of Chemical Bond by Linus Pauling 1960. Who's Linus Pauling, Linus Pauling was one of the rare people who has won two Nobel Prizes. One in chemistry and one in Peace. And my middle name is Linus. My mom named it after Linus Pauling. She loved his work. But what's more interesting is that this book, The Nature of the Chemical Bond, is sitting in Thomas Hales' office shelf. 
	Remember Thomas Hales that crushed the 2000 year old honeycomb problem? The one who's mastered the stuff so well? It's his dad's book that he had in his office shelf that he never flipped through. He said, if he had flipped through it, especially if he had devoted your life to studying something, the moment he saw that picture, he would have had it. But he never had a chance to flip through it, and it was right there the whole time behind him. Amazing. 
	So let me show you where this has shown up now, remember going back to math and art. Here's one thing that's really pushed us. I'm not sure if you know what this is, it's the water cube. But it's the Beijing aquatic center. This is where Michael Phelps won all those awards and crushed those records in Beijing in the Olympics. This is what it looks like inside the water cube. That's where they were swimming. This is what it looks like if you look at the ceiling. This water cube is Ware Thalin. 
	What they did was, they built the entire water cube, not through the tiling of hexagons or cubes but they built it out of Ware Thalin, because it has this intense most rigid structure available for the least amount of metal needed. So pretty. 
	Alright let me move on to trees. So here's what I mean by trees. To me trees are, I really believe this, I tell this to my students, as ubiquitous as numbers are to us today. If you go to the grocery store and give them five bucks for something that costs 10 bucks, they give you five back. You would expect that to happen, you would expect people to just have a simple understanding of possible subtraction, multiplication. Numbers are just natural. Maybe by the time you get to working with quarters, decimals, at least whole numbers it should be doable. 
	I think we will be as proficient with the language of trees in 100 years. I think data is so prevalent today that numbers aren't enough to capture the piece of information on this, that we will start understanding how to speak the language of trees. It's a good thing to know. So what I mean by trees? Here's what I mean by mathematical tree, there's seven things, you can think of the seven species, seven languages, seven objects that are somehow related to one another. This could be your family tree if you want to be. 
	This is called a root, you could actually draw a little leaf down here and actually make it the root of the tree, and those are the leaves of the tree, where the information is that you fully understand. The top level is your understanding of what's going on. These are the internal branches. And that's the structure of the tree. Now to me trees are in space not in paper. So let me just explain to you what I mean by this. 
	These two trees to me are identical. It just says, 1,2,3 are all friends with one another. These two trees are also identical. There's one which is kind of branched off, and the two and the three are over there. Whereas these two trees are actually fundamentally different. One says the two and the three are brother and sister and this is one and the two are brother and sister. I just want to visually make sure we're on the same language. 
	Okay here is a tree, the tree of life, probably one of the most famous pictures of the tree of life that we know. Here we are animals, next to fungi, slime, kinda cool but if somebody says, where do you stand? This is where we are. Sometimes depending on who's drawing it and how it's drawn, the internal edge links are really important. In fact that's the punch line of usually some of these trees because what they measure is evolutionary time. 
	You might not care about the internal instance at all, in fact this particular picture was probably drawn so you can draw all of these on the boundary really well. This might not have much of a meaning in terms of these particular edge links. But if you actually talk to an evolutionary biologist those internal edge links capture something that's important to me. It took a long time before those two species diverged. And it took a long time before something else happened. 
	Whether language has evolved, cultures changed, editions of textbooks, whatever you want to capture. There's something going on interns of that term edge links. 
	Here's an amazing article that brings these things together. It's one of my favorite articles. It shows up in nature, let me just read to you the authors. Adrian Barber and Christopher How, Department of Biochemistry in Cambridge. Norman Blake, Humanities Research Institute. Ah that's interesting. But Peter Robinson Division of Learning Development. What? Who are these guys getting together? 
	So here is the question that was asked. There were about 60 copies of the Canterbury Tales. Do you know Chaucer's Canterbury Tales? Famous work that was written a while ago, and the issue was, nobody knew which one was closest to Chaucer's original book. There are all of these different copies that existed. Each one was off a little bit. There's some errors. But the question is, which was the error and which was the original? Was one corrected by a scribe? Was the error propagated by a scribe?
	No tools that we know was good enough to understand this thing. In terms of carbon dating, that's a crude techniques that places things in big timeframes. Not the sensitivity of when Chaucer might have written something like, you could use works of linguistics itself. Sort of how the construction was formed. But some of those issues are so close to one another in that same temperate, it wasn't that the works evolved over time. 
	So here's what they did. They did DNA analysis. They did phylogenetic analysis. So phylogenetic, philo, trees. So genetics, trees constructed from genetic information. Here's what they did, they took these manuscripts and had to have DNA data, they had to take the data from the pieces of paper or from the ink. The data was the text. Like the THE BOY. That was the data. They fed these 60 manuscripts, they actually took the white [inaudible 00:27:43] prologue, one of the big chunks of text that we're constant in all of them. Fed them into a supercomputer that understood DNA sequencing and biogenetic analysis.
	Here's the outcome, so the text or the leaves of the tree, and here is the structure itself. Notice that at the center there's some error, those propagate it. And the moment that error was propagated, it kind of went to all of these manuscripts. They're all related. Like you could actually if you want resolve the [inaudible 00:28:13] and blow it up here, but it doesn't matter. You basically see, they're all close to one another but all the errors come from there. 
	And then there's another error or something that got propagated which spread to these guys and similarly the black, and the gray, and the green. What's interesting is, and this is what most scholars of Chaucer usually dismiss, is that the red ones are the ones that are closest to the original central node. 
	So people think that the red texts are as authentic as possible. This is tree data. So what I want to show you is how to build such a space of trees. We're all good at understanding the spaces of numbers. Take the number line, here's zero, here's 50, here's -2 and you can construct a world of numbers. 
	How do you construct a world of trees? Every point in your space is not a different number but it's a different genetic tree. And why would you want to do this? If somebody gave you a bunch of numbers, you could tell the distance between two numbers. It's this minus this. Well if somebody gave you two distinct trees, you'd like to find the distance between those two trees. Then you could actually average trees.
	So here's an example. Imagine there are 10 kinds of potatoes that I grew in my backyard. And I know how I crossbred them and grew them. I have their phylogenetic information in my pocket. A and B are brother and sister. C's over here and this is the mom and how they're all related. All they do is they give you the 10 potatoes and they go for it. Tell me. But here's what I can give you, I can give you genetic information. You say, okay if we're talking about tomatoes, there's a gene that can measure the thickness of the tomatoes skin. How are they related? You can ask me that question. I'll say, great, I'll take the gene that measures the thickness of the tomato skin for each one, do a genetic analysis and I'll give you a tree. 
	Here's a tree, this one, tomato one and two are best friends. They're brother and sister. And then you say, well what about the gene that measures the color? Well in that one you get a whole different tree. One in seven are brother and sister. For each gene that I'm picking out you get different trees. How do you get the original truth? Well the answer is, maybe you'll want to try to average it. 
	Here's one way to go about it. You want to create a space of trees. So here you want to find the distances between trees, you want to find averages of many trees. So if these are my trees that we talked about, that's what's going to matter in my space. Those internal edges. The leaves are good, they're important. But this edge link doesn't really matter. If you take one and stretch it even more, the relationship between one and two are still going to be the same. 
	These edges are going to matter. These internal edges, is what I'm going to parametrize my space by. That's what's going to give definition to my space. So here's what I'm going to do, check this out. One, two, three, let's pretend I have three species. They're all together. I can't tell the difference between any one of them. One, two, and three are all brothers and sisters all together. 
	Here's one thing I can do. I can start pulling one and two by the hairs and lifting it out. As I do that, notice you can now say, one and two is actually distinct from three. How far can I pull one and two away from three? As far as I want to. I can keep pulling it. Now for every time I pull it, I get a different point in my space of trees. 
	So this point right here is when the distance between one and two is say, five. And this point right here, the difference between one and two is seven. Does that make sense, guys? So actually increase that length from one and two away from three, this point keeps track of where I am. Okay I can do this. What else can I do? Is it also true that I can pull one and three out? But at the same time I can pull two and three out. So I can start here and I can pull one-two, one-three, or two-three, away from one, two, and three. As far as I want to. So the space of all ways three species can exist is this trivalent to that. It just goes on for ever. 
	Now let's think about this thing. Draw any tree that you want to with a new species. Just any tree you want. With an internal edge link. I guarantee it has to be somewhere on here. Either it has to be two and three have to be pulled away, one and three, or one and two have to be pulled away. Or if nothing's pulled, if it's just right in the center. So this is the space of all ways you can have three species. Makes sense?
	Alright it's interesting, we can actually do averages, we can take two points. Say you have a tree that's over here and another tree that's over here, you can actually find the average between those two trees, what that average tree would be like. So you have three different trees. You can find the average of three of them. Or four of them, or five of them using geometric ideas. Really beautiful. 
	Alright let's see what the space would look like if you have four of them. I have four, one, two, three, four. Now what ways can I pull it? Let's think about this thing. I can do this, I can pull one, two, and three out. Isn't that true? I can grab those three things by the hair and now how far can I pull one, two and three? I can keep pulling that forever. But just for the sake of drawing it, I've pulled it up to a certain point. 
	Imagine this thing goes on forever. But I can also, at the same time pull out just the one and the two. So I can do both of these things. So what happens if I do both at the same time? What happens if I do one, two, and three and then I pull one and two. Then you end up somewhere in here. Not only can I do this, and this, but I can do a combination of both. It's like moving into an X cornet and the Y cornet in this two dimensional world. 
	So this point right here is some combination of me pulling just the one and the two. And then me pulling the one, two and three. Every point on there, on this quadrant, remember this quadrant goes on forever. It's a quarter of your plain, keeps track of how to pull out these trees in this particular way. But there are other trees that are related to this next to here. Remember the Trivalent graph that we saw earlier? That's going to show up here right? 
	'Cause if I have one, two and three, can't I pull out one, two, three in different ways, the two-three, the one-three, and the one-two? It's like a mini previous example showing up in this bigger example. So you're going to have these three quadrants. These quadrants are going on forever fitting together like this. 
	But not only that, it gets more. If I take two of those guys there's a way to walk around and I get a whole pentagon's worth of those guys fitting together. If your brain is starting to hurt, excellent. 
	That's the point, right? This space gets really complicated in terms of what you're going to have. So particular, remember how we had three quadrants that I glued together? Here you're going to have 15 quadrants you're going to be gluing together. I just showed you a sample of what looks like a round one quadridentate. Around one of these quadridentate passages you're going to have this [inaudible 00:34:36] and actually a pentagon nest. But if you look at the whole world you have to deal with 15 things. And each one of these aren't squares, but they're quadrants that I have to glue together. 
	And here's the catch, they all glue at one point. Do you believe that? And that point is where one, two, three, four have no internal edges. As you start pulling one, two, three, four in different ways, you're going to land in one of these guys. As you shrink and expand and pull and expand you're going to walk into this world. 
	So what does this world look like? Well it looks something like this. So those are my 15 things. This for me is a quadrant. It's drawn incorrectly and it's squished but I'm a [inaudible 00:35:16] so I don't care. It's four sides, there's my quadrant, these two guys kind of go on forever over there. These two are my squished quadrant. But I need to draw this. You can sort of see pentagons, you guys see little pentagons around there? The red one is sort of a pentagon flower, all these things. You sort of see Tribalentness right? There's like a tribalentness there, another there. In fact I haven't finished this yet, I need to glue that A over there. I need to glue the B over here and glue the C over there. 
	When I do that I finally have it. So of course you can't draw this in 3D without self intersection. You need immersion, so higher dimensions to actually visualize this thing. What's important is, if you think about this thing, there's some pattern that this is just stretching the pattern in space. So what you want to do is sort of slice right here and look. What is it that it's taking a cone point from? What is that object?
	That object turns out to be the Peterson graph. You might have seen it in graph theory, you might have seen it in discreet math. But this graph is a really important, beautiful graph. But someone's Peterson graph is keeping track of my entire space of trees. In fact here's a different way of thinking about the Peterson graph. It's the space of all trees where the sum of all internal edges has to add up to one. 
	You could sort of think of it as intersection with a unit sphere if you want to think of it that way. So there's my Peterson graph. And just because this is the math talking, I'm throwing out this theorem here, beautiful results, this space that splices the spaces of trees is homotopic to a bouquet of spheres. The way to think of this guy, this [inaudible 00:36:59] I talked about over here actually shows up in the space of trees. That hexagon you saw, is really this blue hexagon from the bee's part. It shows up here, and those red ones combined with those blue ones forms those pentagons. 
	Remember these pentagons, that's what you were seeing before, these bouquets. They're associate hedra if you care about that word, you should. It's a beautiful word. It turns out you can collapse all the associate hedra down and keep the boundary of the hentagon and you end up with these bouquets here. 
	That's my little [inaudible 00:37:29] so there'd be spaces of trees that are really important. So how does this fit in to this world of math and art? Well 
Speaker 2:	[crosstalk 00:37:43] you just showed us T two, T three, T four, right?
Speaker 1:	Exactly. Yeah. So and this is what I mean by T. Not the entire space but that one dimensional slice of the of the level. 
Speaker 2:	Where's the root?
Speaker 1:	The root is not here. You can take any point as a root and take the cone of this over here to get the entire space. All of that stuff is trivial [inaudible 00:38:04] this is the non trivial part. It's better to talk about this, it's a beautiful simplicial complex. It's a pure simplicial complex, you can view beautiful geometry on this. If you want the whole space let's take a cone over a point and you get the entire thing. 
	And here, if you think about it, all that infiniteness that I had to keep drawing is gone. This is the underlying structure of what's really going on. 
	Let me show you why this is kind of exciting, there's this beautiful software that you can use. I started a little company. This is a fun little thing that I did based on the structures and spaces of trees. Here's what it does, and this is a particular thing that it did for Williams. This is my college so I was able to have this data. 
	On the left side of the circle, every alum that we have records for at Williams, about 16000 alum who gave us full complete records. On the left side of the circle are all the majors at Williams college broken into some big categories. For example, there's a major called culture studies. Which would be anthro, socio, asian studies, epicanthic studies. On the right side are all the careers at Williams that you could go to as an alum. Again broken into 15 big categories. The law, government, banking, finance, K-12 education, health, medicine. So this tells you, using this underlying phlogogenic software, what's happening? It tells you what the alums have done. To me there are two nice things it tells you. 
	One, it tells you first of all that a huge number of people from the health/medicine people who are doctors basically are all coming from bio and chem. 50%, right? Which makes sense because of the pre-med requirement. That requirement basically forces you into one of those fields. It's not true for law. You could do anything you want, take the test to go into the law school and be happy there. Unlike pre-med. The second thing it tells you is, almost every piece in substance coming equally from every other piece. 
	I find this really fascinating. So here's english and lit, spread the joy to the world. There all these guys, going into different things. Now of course, they might be a little bit more pushed into law or writing, but overall, it's not a huge weight, like the bio and the chem was. Here's history, broken equally, you see there's a little more into law and history, that's naturally selective, that makes sense. 
	Econ, again, more into banking than the rest, that makes sense but there are a bunch of economists who are doing art and entertainment and writing and health and medicine and others. [crosstalk 00:40:39]
Speaker 2:	So is mathematics. 
Speaker 1:	Yeah, mathematics actually turns out, the Williams thing is really small but does exactly the same thing. The reason, that should have been combined with other things but we don't have the heart to combine it. Leave that baby alone. 
	Here's from the other side. If you want to do sales, what do you have to do to go to sales? Do you want to do banking, or do a college education? It's really pretty but this underlying visual display tells you something and it's actually coming from this tree structure. Let me keep going and close out with this last piece here. 
	Origami. So this is a work by Robert Sabuda, my favorite origami artist. Popup books which themselves are works of art, absolutely gorgeous. Unfolding ferns, if you think about how plants grow, it turns out when plants, if you make them grow in a linear way, they'll absorb too much photosynthesis when they're young and it'll kill them. They don't actually need photosynthesis 'til later on. But if you wait 'til they're older, for them to actually start absorbing the photosynthesis, they can't grow the leaves fast enough. So what they do is they actually curl up their leaves in origami folds. And then they unfurrow at the right time. 
	Have you ever walked around and all of a sudden it's spring. It's not green and it just turns green. That's what's happening. They already have their leaves folded up but at the right time it just unfurrows together. It's really stunning. That's origami folding. Protein folding inside your body, it's really origami folding going on. Folding and unfolding. If the protein comes out of your body as it's made, it's actually made into little rods that turn and form three dimensional keys and interact with enzymes. If it actually unfolds the wrong way, you get Alzheimer's or Mad Cow Disease. 
	So we would love to know how folding and unfolding works from just a mathematical viewpoint. This is the core [inaudible 00:42:47] stent. It was a stent design based on a student who was walking through. She was a student at Oxford, but she was walking through her friend's town in Tokyo. Went to the origami museum nearby and was inspired to create a stent. A stent is something you put inside your arteries to unclog it, to help the passages go. But this thing becomes this thing you just pull it. All based on these origami folds. Beautiful techniques. 
	The James Webb Space Telescope should be in space in 2016. It's about the size of a tennis court. It's going to crush Hubble in terms of how much data it can capture. But how do you fit a tennis court into a satellite to launch? Well you have to fold it. Those crease lines are really important as to how much data you can lose along those folds. You want to be really careful about how that works. 
	Eric and Martain Demain, Eric Demain was a professor at MIT. He got to be a professor at the age of 20. MIT gave him a job, he was I think 19 when he got his PHD, and MIT said, come, we would love for you to be a tenure track professor here at MIT in the computer science department. One of the best computer science departments in the world. And Eric said, it's alright, thanks. MIT said, what do you need? He says, well my dad's looking for a job. Okay, well, what does your dad do? Well he's an artist. He does glass blowing. So MIT built his dad a glassblowing studio two floors below his office. 
	Eric and Martin are good friends of mine, they wrote about 300 papers together. Eric won the MacArthur Genius Grant about a few years ago when he was 24. I don't think MIT was foolish at all to do what they did. It's amazing. To me these guys are basically the new Di Vinci in terms of pushing the boundaries of math and art. 
	This is a work of theirs using cutting edge origami folding, but it's in the MOMA. Museum of Modern Art. Really without that math, you can't do this. But it's not the math alone, somehow, it's the minds of mathematicians and artists working together. There's a theorem called the one cut theorem and it was Eric's PHD thesis. What it says is, if you take a sheet of paper and draw any boundary, namely this boundary with this darkened edge, is it possible to fold a sheet of paper and in one cut, cut it so that the turtle falls on the floor and nothing else?
	He proved it to be true. So you can write your entire name, as long as it's straight pulling the lines, you can write your entire name, and it doesn't have to be connected, complete disconnected things. There is a way to fold that piece of paper, make one cut and your entire name falls beautifully on the floor. And the rest of it is that piece of paper. Gorgeous results. And that's the intersection between math and art. 
	Let me close with a couple little results. [crosstalk 00:45:45] theorem, it's one of my favorite results. Kochi says, two closed convex polyhedron [inaudible 00:45:53] corresponding convert must be identical, what does that mean? That means that if you have pieces of a puzzle, have you ever done those snap kits where you can glue pentagons together or any kind of Lego folding. What it says is, you have pieces of a puzzle, and those puzzle, at the end when you've glued it together formed closed, convex polyhedron. All you need to know are the pieces and how they fit together. Which touch which? You don't need to know the angles which they fit. 
	In fact if you just know the pieces and exactly know which is going to touch which in which order. The entire object is going to be ridged. There's only one way you can possibly do it with such a simple instruction. If you have two of them, that are component and the piece and instructions are the same, and they both have the same pieces of the puzzle, then you only have one of them. They must be identical. 
Speaker 2:	You can still build a [inaudible 00:46:49] but only one way. 
Speaker 1:	Exactly. If you have such polyhedron, you might think of something like this, the instruction kits, these all glue together exactly the same way, they both have the same gluing instructions as to what glues next to what. They both have the same piece of the puzzle, don't have two different ones. That's why you need the word convex. 
	You need the convexity to make it work and you need the word close. If this didn't have a top and a bottom, imagine you only had four squares, you can kind of imagine, squishing and opening the squares and deforming it. 
	So I need close also. So that's [inaudible 00:47:21] rigidity results. It's a very beautiful, famous theorem and it's not that hard to prove. But then the question is, what happens if you don't have the word convex? If you allow it to be non-convex? Can you have polyhedral that aren't ridged? Can you have polyhedral that flex? 
Speaker 2:	[crosstalk 00:47:43] you can't get from one to the other. 
Speaker 1:	Exactly right. Bob Connolly professor at Cornell asked this question. He spent 20 years of his life thinking about an answer to flexible polyhedral, he believed it was true, he poured his heart into it. He went to a conference and he heard a rumor that somebody had proven this result. He tracked the rumor down, and the rumor was about him. He was relieved but he pushed ahead and was actually able to find one. He found something based on the cards four dimensional [inaudible 00:48:18]. Take this four dimensional model, took pieces of it that made it work and made it into 3D. People were able to shrink it down even more, this is called a [inaudible 00:48:27]. 
	My students actually built this for me. This is out of metal, this is out of plexiglass, and it actually flexes. If you move the joint it flexes beautifully. There's the second question you get, from that beautiful result. It's something called the Bellows conjecture. While it's flexing, does the volume change?
	Nobody would have thought of that question if it had not been for the theorem to come up with it in the first place of this result. Let's look at it one dimension less. This is what it is per square but these are my hinges. Let me flex the square and notice that the area completely changes. You can make it as close to zero as you want to. Is it true for 3D when I flex this guy, goes through this flexible motion, is it a Bellows going in and out or is the volume fixed? 
	Beautiful work by [inaudible 00:49:14]. Bellows theorem, the volume of a flexible [inaudible 00:49:16] does not change. Stunning. Amazing work. It's closed with the slide, David Huffman, passed away from the Huffman codes, this is some of his origami pieces he had way back when. This is the cutting edge of things we have no idea about. This is from a flat sheet of paper. 
Speaker 2:	He did this in 1950's?
Speaker 1:	50's, 60's. At that time. But we don't understand how to get curvature in some of these things. How do you pull it off. If you remember Eric and Martin Demains work, again that was from a flat sheet of paper. They're getting these beautifully curved objects. They're beginning to explore what David Huffman was coming up with just as an artist. 
	So I just want to close with my favorite quote here from Princess Bride. Life is pain, anyone that says different is selling something. When we try to fit math and art together, I think there are these easy ways of saying, oh this is math and art. Now you've used it in this pretty way. But to really reclaim the time of the Renaissance, to go back to how Di Vinci was, that's a very difficult thing to do. But that doesn't mean we shouldn't stop trying. 
	Thank you for your time.