 Now it's my great pleasure to bring to the stage Dr. David Healthand. He's an astronomer at Columbia University But he's also the chair of the American Institute for Physics. Please welcome David Healthand. So when I meet a mathematician, the first question I always ask is, is mathematics discovered or invented? Now as a scientist I lean very heavily toward the latter. Most mathematicians I see tend to take the former position. But when I asked tonight's speaker that question, when I first met him a little over 13 years ago, he paused, he tilted his head as he often does, as you'll see, and he smiled and he said, both. And I think you'll see that philosophy coming through in what he's going to tell you about tonight. I got to Columbia a long time ago. Monday I will finish my 85th semester there. And when I arrived at Columbia in the 1970s, I was delighted to see that unlike every other major American research university, Columbia had not decided to abandon its general education requirements in the 1960s like every other place had. And in fact they still had a curriculum in which every student read the same book the same night and discussed it the next day in a 20-student seminar. This is called the core curriculum and it was celebrated in the catalog as the intellectual coat of arms of the institution. Well, I was simultaneously appalled that this intellectual coat of arms had zero math courses and zero science courses. In fact, it consisted of seven humanities courses. Being young and naive, I thought, well, I can fix this. We have the structure. We have the philosophy. I'll just invent math and science courses, and then we'll have a full coat of arms. So 27 years later, I succeeded in adding one semester to this curriculum. So it now consists of eight semesters, seven humanities courses that haven't changed since the late 1930s. And my course, Frontiers of Science. The year I achieved that, 2005, I got a cold call from someone named David Strangway, who I didn't know. I probably should have since he was a geophysicist in charge of the moon rocks during the Apollo era when the astronauts were bringing back rocks from the moon. And as a Canadian, I've gone back to Canada, became Provost and then President of the University of Toronto and then President of the University of British Columbia. And he said, David, I've heard what you've done about integrating science into a liberal arts curriculum. We're starting a brand new university for scratch, and I want you to come out and tell us about it. So I did the obvious calculation that in 27 more years I'll be dead. So I'll never add another course to Columbia's curriculum. And therefore, I decided to go for a day. A day that turned into a decade. Because it was just irresistible to start with a truly blank piece of paper, which had none of the sclerotic 19th century ideas imposed on it as our current universities do. And try to design a university for days students in tomorrow's world. Needless to say, things were very different. We had no tenure. I've rejected tenure at Columbia 35 years ago, so that was good with me. We had no departments, so we had no silos where different areas of knowledge were walled off from one another. And we therefore had no majors. So each student had to come up with a question that embodied what they were interested in cutting across disciplinary boundaries. And we had a two-year core curriculum where there was equal weight on mathematics and science and social science and humanities and arts and language. And finally, we adopted something that was a counterpoint to the neurophysical nonsense of multitasking. We had students take four courses in a term, but they took them in series rather than in parallel. So they took one course at a time for a month and then switched to another course. The level of depth you can get to with students in a month is completely transformational from any normal university system. Well, so I kept going out. I would go out for a weekend. I'd go out for a week, but out in the summer. And I started getting nervous because all the people designing this university had not been in a classroom in 25 years. And I said, look, guys, you really have to hire faculty. And so we tried to find faculty who would come to a university that didn't exist and had never granted a degree and may never grant a degree for all we knew. And join us in this project. And one of the five people we selected was tonight's speaker, Professor Glenn Brand Brumlin, who came to us from another innovative liberal arts institution, Bennington College in Bennington, Vermont. Glenn was Canadian, so he was happy to come back to Canada, but he was even happier to have complete freedom to design courses that, again, every student would have to take because everyone has to take math, but math is seen through his eyes. Glenn is a truly remarkable pedagogue. He's also a great scholar of the history of mathematics. He has published dozens of papers. He's the past president of the Canadian Association for the History and Philosophy of Mathematics. In, he's published two books with Princeton University Press, the last one, most recent one, called Heavenly Mathematics. Yes, of course, what else would it be called? Heavenly Mathematics on the Development of Spherical Trigonometry in the 19th century, which one reviewer said, if this book doesn't recruit you to the cult of spherical trigonometry, nothing will. It's a splendid book. And if you've read it, it is a splendid book. Interestingly, spherical trigonometry, every one of you is carrying in your pocket because your GPS wouldn't work without it, but it's not even taught anymore at the Naval Academy. It's taught, however, at Quest University, Canada, where Glenn has been for the last 14 years. He's been recognized for his remarkable pedagogical skills by the Haimo Award of the Mathematical Association of America, meaning he's one of the three best university math professors in North America. In 2016 and 2017, he was named one of the 3M Teaching Fellows in Canada, which means he's one of the 10 best university teachers in all subjects in all of Canada. And you'll see that on display tonight. I'll never forget when we were discussing philosophical issues before we had students in this university about how we were going to be successful, and Glenn said, well, it's really very simple. You just have to love your subject and love your students. As an example of this, I'll give but one I could give many of Glenn teaching class, and it was the spherical trigonometry class. Now, this was sort of registered as a sort of cool class, but a really hard class. It was cool because they all got these lucite spheres and they'd carry them around like pet rocks, you know, they'd be in the cafeteria drawing triangles on these spheres. And so students would take it. And again, everyone had to take math. So none of the people in this class were going to be math majors. We didn't have any majors, so it didn't matter, but the point is they were going to take this. And in the third block, this third month of their first year, Glenn had a class of 16 or 17 students in the month of November. It's really dark and rainy in British Columbia in November. So, you know, the students are ready to do some work. Every class at Quest was either 9 to 12 or 1 to 4, 5 days a week, 3 hour blocks. Glenn, of course, has to be different. So his classes were 10 to 12 and 1 to 2, because he wanted to give students a little break and have them thinking about math with their spheres in the cafeteria all the time. And so on the nice day of class, you got a picture of this, a bunch of first year students, none of them are math majors, none of them have any particular interest in math. I just heard this class was cool. They're taking this class. Glenn puts a theorem up on the board that was first published in the first decade of the 19th century and has appeared in every spherical trigonometry textbook ever since. And he gave the students the chance to prove the theorem. Now at Quest, we do collaborative learning rather than competitive learning. So the students work together on this. I was the university president at the time. I would come home about 7 o'clock at night. This day, I went back to the classroom a Friday night, right? And all the students were still there, from 10 o'clock in the morning, trying to prove this theorem. And they go up to Glenn and they go, Glenn, there's something wrong with this proof. This doesn't work. And Glenn goes, yeah, yeah, yeah, yeah. Well, we'll work on it tonight and we'll talk about it tomorrow. A couple hours into class, Glenn is walking around the academic building. Now, the academic building is a circle at Quest. So he's walking around a circle and he's swinging his meter stick. He walks with a meter stick like this, right? The students had shown a flaw in the proof that had escaped mathematicians for 200 years. And Glenn had to redraw his proofs from Princeton University Press, change spherical trigonometry, and acknowledge the class in the frontispiece for having done this. This is the level of pedagogical excellence you're going to hear tonight. I give you Professor Glenn Ben Brummler. Good morning, everyone. Yeah, I like mornings. I think mornings are full of rich possibilities. And so I love to start always in the morning. So let's all imagine it's 9.30 in the morning. Are we feeling good? Okay, 10.30, is that better? No, still drowsy. Still drowsy, okay. You can pretend it's 1 o'clock in the afternoon. All right, this is going to be an interactive presentation. And so you are going to be probably talking more to the person beside you than looking at me. So what I would like you to do now is to look across at the person sitting beside you and thank that person for being in advance, for being so brilliant and intuitive and ready to engage. Go ahead, introduce yourself. All right, are we ready to begin? Thumbs up if you're ready to begin. I need lots of thumbs up. There you go. Okay. Pardon me? Oh, I had no idea. All right. Well, that gesture you just made if you were in certain Middle Eastern countries or in West Africa has a slightly different meaning. The gesture you made is the meaning that you would get in our culture if you were two fingers over to the right. Enough said? All right. Our symbols that we use, the concepts that we work with are all deeply informed by who we are and what culture we're within. We all see this as a positive symbol but there's nothing particularly positive about that, is there? It's just lifting your thumb in the air. All sorts of concepts we have are colored by who we are and who we interact with. So one quick example to get you started. Does anybody know what this is? What is it? Justice. This is Lady Justice. She actually has a name and I looked it up before the talk but I've forgotten. Does anybody know? Okay. I don't feel so bad now. Okay. Thank you all for not knowing the answer to that. I appreciate it. This is Lady Justice. She is, you find this statue on various courthouses all over the place and she's got these two way scales. Why does she have the way scales? What does justice have to do with waiting stuff? Pardon me? Balance, balancing, waiting the evidence. You will also notice that she's blindfolded. So I'd like you to briefly discuss with your neighbor. You've got 15 or 20 seconds to come up with an explanation for why she's blindfolded. Go. All right. So why is she blindfolded? Somebody raise your hand. Yes. Okay. Prediction in what way? Why is it important? Ah, prediction of what will happen. That's partly right. There's a little more to it though. Okay. Fair enough. What's going on here is that justice is not supposed to know who the people are who are being judged. The fact that it's a president, I should be careful given the current political climate. I did not intend that. Okay. Whether it's a prime minister or a premier or something like that, I'm from Canada. That's right earlier today. I'm not getting, I'm Canadian. I'm not going down that path either. Okay. I'm staying totally out of politics. Or am I? Nowhere to run. So she's blindfolded because that way every individual person is treated equally under the law. It doesn't matter who you are. You're going to have the same treatment. And this is a Western concept of justice where justice is meted out with respect to individual people. Justice can be defined in a lot of different ways than that. Religious communities will often define justice to mean living in harmony with God's will, however that's conceived. And today in the modern world, Western values are being challenged by various Eastern values of justice, which have to do with things like communal rights and group rights, things like karma or for instance, or restorative justice. So culture is deeply imbued in all of these various concepts that we work with. But tonight we can escape from that, right? Because what is the subject that doesn't have cultural influences in it? Count to three. One, two, three. Math. Let me try to show you that you're wrong. All right. So a lot of people believe this, that mathematics is one place you can go where two plus two is going to be equal to four, whether you're from the United States or Brazil or Asia or wherever. The question, I'm not going to question whether that two plus two is equal to four in every country. But what I am going to try to question for you is something about how we see mathematics being imbued also with who we are and what social groups we interact with. So this is the Arecibo Observatory. Anybody know where it is? Puerto Rico. Anybody know what James Bond movie this was in? You're, did you say? Golden Eye, in fact. Yeah. I had to look that up actually. I didn't know it either. There, you might have. Well, we're going to talk about that in a minute. This Arecibo Observatory in 1974 did basically a publicity stunt. And they decided they wanted to communicate with aliens in the M13 globular cluster. Why they thought there were aliens there and not elsewhere. I don't know. But there they were. So they actually missed. They were aiming for the middle of the galaxy, the globular cluster, but it actually is going to be moving by the time the signal gets there and they're not quite going to hit the target. It was a very simple message they sent, but they had to think, okay, if we're going to send a message to a group of aliens who don't even possibly share the same body structure, let alone the same culture, how are we going to make sure they know that there's a message there that's there to be understood? So what they came up with was this image over here. And this is what they sent to that globular cluster. Anybody recognize anything in it? There is, in fact, a human being. So they're telling the aliens what shape we are. Any other things that you see here? Yeah, at the back. Oh, very good. Nobody got that in the earlier sessions, so you guys are smarter. Right here, this is, in fact, the double helix. But where did the message start? Pardon me? McDonald's! No, that's not McDonald's. Yeah, I sure wouldn't that be something. We go to the M13 globular cluster and the first thing we see is a drive-through. Oh my goodness. Okay, now this is actually a representation of the observatory itself. Okay. But where do you start? What's the beginning thing that's most independent of culture? Well, the people who put this message together started with this collection of symbols up here, right at the top. The pink dots are not important. They're just indicating places. But I wonder if anybody can tell what's going on in the red dots. Are those the binary numbers? All right, we definitely need a quick round of applause right there. Indeed, these are the binary numbers. This is one. That is one zero. The number two, one one, two plus one is three, and so on all the way through. It was felt that the one thing that could attract an alien's attention, aliens have to have the same numbers as we do, so they'll recognize this and then they begin talking across cultures with math. Well, it turns out, I think David mentioned that a quest university, the university I teach at, all students and even all faculty will motivate their own studies also with a question. And so I have a question. Would you like to know what it is? All right, my question is, okay, now you can't see this very well, but the question is if a space alien were to land on the lawn outside, these are space aliens. Of course, all space aliens look like humans in gold jumpsuits, but if they were to land outside and walk outside of their spaceship, would we understand their mathematics? And that's the question that motivates my studies as a historian of math. So it's not, again, not trying to prove that two plus two equals five, but what I am trying to ask is, when we look at mathematics, do we all in fact see the same thing? And so I'm going to try to push that a little with three different examples for you today. So are we ready for our historical tour? All right, first one from ancient China. China is probably the best culture in order to test this question, because almost every mathematical culture that developed to a certain extent ended up interacting with the West to some extent, but China didn't so much. There was a little bit of interaction here and there, but for the most part, Chinese math developed entirely separately and on its own and for its own purposes. So it's a great way to take a look at this question, see if there's a different understanding of mathematics in this culture, and if the answer had been no, I wouldn't have been here, would I? So let's take a look at this a little more. This person is, now let's see, do we have any native speakers here who could help me pronounce this name? The name of the person is given right here. Do we have any volunteers? Okay, that means you're going to have to live with my pronunciation. My limited Western vocabulary has restricted me. I'm going to do my best. Are we ready? Are we ready? Liu Hui. Yeah, people are laughing at me. I probably muffed that completely, but I don't care. This is the best I can do. All together now, Liu Hui. There we are. Liu Hui was the major figure in ancient Chinese mathematics. He was basically the mathematical Einstein of ancient China. He actually did not write a book. You'll notice I've mentioned the name of the book here, Zhejiang Xuanzhu. This means the nine chapters of the mathematical art or the nine chapters of computational prescriptions. In fact, people love this book so much that many math books after this one was written in China were intentionally written to have nine chapters, maybe so people could confuse it with their big one and buy that one instead. So this book more or less formed the basis of Chinese math. And guess who wrote it? Liu Hui? No, he did not. That was a trick question. We don't know who wrote it, but there wasn't such a concept of authorship in Chinese math or in Chinese culture in general. So this book was out there, and it would be totally fine for you to take a book that you had, write down all sorts of extra comments, explaining what the book was about and how to understand it, and then release it as your own book. That was fine. It was copyright did not exist. And that's essentially what Liu Hui did. He wrote a commentary on the nine chapters, and that commentary became the most important text in Chinese mathematics. So we're going to start with a warm-up question from this. So again, look at your neighbor and express joy with your excitement for doing some math with that person. Here we go, the warm-up problem. This is not, by the way, from this book, but you'll see more later. Here we are. Suppose you have a building and you want to find out how tall it is. Well, it's a sunny day. The building's casting a shadow. And as you can see, the shadow is 54 feet long. And supposing you're five feet high, one thing you could do is just keep walking further and further away from the building until the tip of your shadow touches the tip of the building's shadow. So you're five feet tall. The shadow is 10 feet long. The building's shadow is 54 feet long. Look imploringly at your neighbor and say, do you know how to do this? You've got 30 seconds. Find for me the height of the building. So, the triangle can be five feet high. Yeah, from the foot of the person to the end of the shadow. Person standing vertically. So that's good. On the 10 inch, then from the left. Ding, ding, ding. If you have an answer, please raise your hand. We've got some other answers. I'll get back to you a little later. Yeah, right there. 27 feet. How do you know? The similar triangles. Oh, you've said the famous phrase. Say it again. A similar triangle. How many of us know similar triangles? Everybody knows similar triangles. All right, so can you fill it out a little more? How does that help us? You can see that the two better sides, the building and the person, they are proportional to the 54 and the 10, but easier said, five proportional to 10 as the height to 54. So obviously it's half of 54. It would be half of 54. Exactly right. This is half of that. And since this big triangle is the same shape as the smaller one, then this has to be half of that. Do we agree? Okay. Do you think we could do some Chinese math now? Not at all. It is so obvious to us looking at this. This is a similar triangle. You see those two triangles that are the same shape. There is not one text in all of ancient Chinese mathematics that even registers the existence of similar triangles. Gas before. Now, what are you going to do? Well, it's not that they couldn't do this. It's that when they looked at this diagram, they didn't see two triangles in it. They saw something else altogether. And here is where it all begins. This diagram has a rather cumbersome name. It doesn't roll off the tongue very well, but I'm going to show it to you anyways. In Chinese, I'm sure it's much smoother, but let's see if we're going to be using this term frequently. So here we go. You ready? You ready? The out in complementarity principle. Say it with me. Yeah, that really sounds gross, doesn't it? But it's not that bad, and it's an amazingly powerful technique. Here we are. So we have a rectangle, and that rectangle has a diagonal line cut down from the top left to the bottom right. Now, what I'm going to do is I'm going to move the laser pointer from that top left corner downwards on the diagonal. And when you're ready, shout out a moment and I will stop. Okay, so you're going to choose a random point on that diagonal. Ready? Ready? Here we go. Pardon me? Pardon me? Oh, there we go. Okay, we'll stop right there. Okay, very good. Okay, so at this randomly chosen point, draw a vertical line, draw a horizontal line, and that's going to break up our rectangle into two smaller shaded rectangles. And how many triangles? Four of them. Okay, now I'm going to claim that this rectangle here is exactly the same in area as this rectangle here. Now, you get why, but I need everybody to get why. So again, look to your neighbor with some puzzlement possibly. See if you can think about why those two rectangles have to be exactly the same in area. I'll give you 30 seconds again. All right. Do we have a solution? Sort of. Okay, I think and the young will lead us today. So I don't know your name. What's your name? Mark. Okay, Mark's going to tell us why these two are equal. Well, let's take the bottom triangle. We know that the, like we can find out the area of both triangles really easily. I mean the rectangles. Sorry, sorry, sorry, sorry. So we take the base of A which we found out is 54. Oh, we're not using numbers though. This isn't the same rectangle as before. Okay, let's just use that as an example. Okay, okay. 54. Let's just use an example. Okay. So you see those two triangles A that make up a rectangle? So you take the base of the lower one, which is let's say 54 as from the previous problem. And the height of B is five from the, again from the previous problem. Okay. And so that means the bottom rectangle is 54 times five. That's this area. So, and now let's calculate the top. The height of triangle A is 27 as from the previous problem. And the, and the base of B is length 10. 27 times 10 equals 54 times five. And there you go. Okay. A round of applause for Mark right here. Would you like an easier explanation? Okay. Would you agree with me that this diagonal cuts the entire rectangle in exactly in half? Okay. So the triangle up here is exactly the same as the triangle down here. Okay. This triangle is the same as that one. That triangle is the same as that one. So take the top half triangle, take away A and B, and you're left with this guy. Take the bottom half triangle, take away A and B, and you're left with this guy. Therefore, that has to be the same as that. That's another way to put it. All right. So, when the Chinese mathematicians are calculating or looking for the height of the building, they are not seeing similar triangles, they are seeing that. Okay. Now, do we remember how tall the person was? Five. Five. And how long was the shadow? And how long is it from here to here? It was 54 all the way. We've got to take away 10, so 44. So, find me the height of the building using the out in principle. Talk with your neighbor. Go. All right. Does anybody know how to do it? Who is more than five feet tall? Okay. Sorry. I got to share the wealth right over here. Yes. Okay. The triangles have to be the, the rectangles have to be the same area. Yes. The one on the bottom is five by 44, which is 220. Indeed. The one on the top right would have to be 10 by x. Whatever to get 220 would be 22 high. So, it's 22 vertical and it plus the five. Exactly. This rectangle here is 220, so the one at the top must be 220. This is 10 feet, so this is 22 feet, right? But I thought the building was 27 feet high. We've got to add the person's height, don't we? Okay. So, this out in principle works just as well as similar triangles, but when you look at the diagram, you're seeing different things. So, let's look at a more sophisticated problem. And for this, you have a clipboard with a diagram on it and some little rectangles attached to it. This problem is from a sort of an appendix to the nine chapters written by Liu Hui, entitled the Sea Island Mathematical Manual. And it makes me homesick to look at this because back in Squamish, my hometown, there's this giant cliff face that overlooks the downtown area. Last I heard, we did not have a little bit of water passing through the cliff. If we do, I'm definitely in trouble when I get home. And we are on this other piece of land over here. And the problem that Liu Hui asks is, how can we find the height of the cliff without being able to go to the base of that cliff? Now, this is a problem that you find in various different cultures. And in fact, it totally shocked me just yesterday. I was translating some work from the person I'm studying right now on my sabbatical. And this problem turned up. This exact same problem just turned up at the same time. And this was in Italy in the 15th century. So what's the problem? Well, here's what we've got. Liu Hui says to solve it. I was about to write on here hoping that would work. That's not going to work, is it? Okay, here we go. You have a pole. It is five-boo high. A-boo is a unit of measurement. It's a little more than a meter, about a pace. All right? Now, if you put that there, and it's five meters or five-boo high, you back up until your sight line up to the top of the cliff goes exactly through the top of your pole so that this is a straight line. Let's suppose that takes 123-boo. Now, if you look at this triangle, that's five, that's 123. What's your conclusion about my drawing skills? Nuanced is, yeah, that's a nice way to put it. Okay, nuanced. That's actually accurate because if I drew this diagram to scale, you'd never be able to make heads or tails of it. So bear with me on the scaling. All right, now it turns out that's not quite enough information. So Liu Hui backs up a thousand-boo and does it again. Five and then does the same procedure and this time gets 127. So if that's 123, that's 127 because clearly that's just a little bit longer than that, right? Okay, now, from this, we have enough information to solve the problem. Now, if you were to look at this problem before you'd heard any of this Chinese mathematics, what are the two words that would come out of your mouth? Similar triangles. There are two pairs of them and that's how you would solve it today. With this triangle and this one and this triangle and this big one here. Oh, and that was somewhat painful. You don't know why it's painful. A few months ago, I hurt my arm and I can't lift this arm above my shoulder right now. Three days ago, trying to compensate for it, I hurt my other arm. You have no idea how long it took me to put on this sweater this morning. So if I grimace, it's not because of you, it's because of my shoulders. All right, so this isn't, in fact, what we see. What we see then in the text, Liu Hui explains how to solve the problem. He says, multiply the distance between the poles by the height of the pole as the dividend. Take the difference in distance from the points of observation as divisor and divider adding the quotient blah, blah, blah, blah, blah, blah, blah. He gives us an algorithm for how to calculate the solution. He does not explain how it came about. So we just have this pattern of calculation. It generates the correct answer. But of course the obvious question is why? Well, this is where historians of mathematics jump in with great excitement because we can try to explain why. And you can explain it perfectly by assuming that Liu Hui is using similar triangles. And that is, in fact, how the entire book, not just this problem, but the entire book was explained using similar triangles in publications until about 30 years ago. So we had a complete understanding of Chinese mathematics, and it was completely and totally wrong. That was finally started to be corrected in the late 1980s, and it was using what principle? The out in complementarity principle. The algorithm will give you the right answer, but if there's more than one way to demonstrate the answer, the algorithm isn't going to tell you which of those methods is the one the person actually had in mind. My personal hero of this story is a colleague by the name of Frank Sweatz. S-W-E-T-Z. You're Americans. Z. Okay. You're welcome to move to Canada. Okay. It's beautiful. It's a little cold. But it's colder here actually. I'm not so thrilled with the weather here right now. Anyway, Frank Sweatz published a couple of books in which he used the similar triangles theories to explain several different Chinese texts. And of course, he got it completely wrong. So did everybody else. The thing that made Frank Sweatz my hero was that once it was found, once this other interpretation came along that indigenous Chinese math, historians of math came up with, he not only admitted that he was wrong, but he published several papers explaining how he had fallen into this trap. And that must have taken an enormous amount of courage. So Frank, wherever you are, you are my hero for this. Okay. So how did they do it then? Well, you have this diagram on your paper in front of you, correct? You're going to need the clipboard. You're going to need those little rectangles. What I would like you to do to begin is to put the two blue rectangles into the diagram as I have shown you here. Now, they're different dimensions. So one of them is going to fit in one place and the other is going to fit in the other place. Once you've got them into the diagram, raise your fingers in the air, and then of course your papers are going to fall to the floor. Okay. Yeah, we've got thumbs up. Yeah, thank you. Okay. All right. Now, I'll count to three. I want you to explain to me, shout out why we know these two rectangles have the same area. One, two, three. The in and out complementarity. Yeah. Just say out in. I'll know what you mean. One, two, three. Out in. There we go. You can see this rectangle right here, cut it in half on the diagonal and those two blue rectangles are equal. Okay? That's the first step. Now, take the vertical blue rectangle and slide it over to the right. Now, first question. How far is this? Why is it going to be four? Exactly. When this triangle was over here, this length was 123. This length is 127, so if this is 123, obviously this is. Four. And it clearly looks like that, right? Okay. Now, you also have two yellow rectangles. Now, I'd like you to add the two yellow rectangles to the diagram as indicated there. Now, I need you to be very careful and don't jump to conclusions. In other words, don't just randomly shout out your two newly favorite words, which are? Out in. Explain to your neighbor why the two yellow rectangles are equal to each other. All right. If you have an explanation for why the two yellow rectangles are equal, then raise your hand. Hi. Because if you consider the entirety of the rectangle, you've got a diagonal going from top left to bottom right. Uh-huh. And so the rectangles formed by the blue and green together on the bottom has to be the same area as the rectangle formed by the blue and green up on the top right. And if the two rectangles combined are both equal to each other, then? Since we know that the blue ones are already equal, we can conclude that the green ones are already the same. Round of applause right there. Beautiful. Okay. That's exactly right. So now we know that the yellow rectangles are equal to each other. Again, one more time, turn to the person beside you and ask that person, how high is the cliff? Go. Yeah. All right, does anybody have the height of the cliff? Yes. 1,255. Can you explain how you got that? I did it with the yellow rectangles. The bottom one is 5 by 1,000, so the square foot is 5,000. Since we already realized the other one is only 4, the only number that it could be equal to 5,000 square oomphies is 1250. And we already, I'm sorry, 1,200. Indeed. So we already know that this is 5, so you add 5 to it. A masterpiece. Round of applause. There we go. Beautiful. Okay, so we would have solved this in a manner that was completely different. We would have seen different geometrical facts in the diagram. Thankfully, we and the Chinese mathematician still come out with the same height of the mountain. Okay, second example. My second example has a tendency to provoke some rather fierce reactions, shall we say. I actually once had a pair of students arguing so much that I had to restrain one of them. So, that was before I hurt my shoulder. Okay, so everybody raise your right hand. I hereby solemnly swear that I will treat my neighbor with the respect that she or he deserves. Even if that person is completely wrong. Okay, here we go. So our second example today is from the person who is considered by some to be the father of the math that we do today. This is Euclid of Alexandria. Yay? Sure, give him a cheer. Why not? You did. Good. Well, he's named after this guy. Lived in the third century B.C. He is to ancient Greek mathematics what Liu Hui was to Chinese mathematics. He wrote the most important book in Greek mathematical history called The Elements. And you can see a page of it here. It's hard to read the text, but at least you can see the diagrams. This, by the way, is Euclid's proof of the most famous theorem in mathematics, which is... The Pythagorean theorem. They actually, Liu Hui had a proof as well. And it turns out I did not plan this in advance. But there's an exhibit just in the corner down over there that behind... We also go back when the museum's open that actually has Liu Hui's proof of the Pythagorean theorem. It doesn't label it as such, but it is Liu Hui's proof. This proof is called the Bride's Chair. It's the most famous diagram in mathematics. Why is it called the Bride's Chair? It actually does. And there she is. All right? I kid you not, that's why it's called that. So when I got married, oh, hello these many years ago. How many years is it? Don't tell my wife I didn't know that off the top of my head. 25, 26 years, something like that. Yes, I celebrated the 25th anniversary. When we were getting married, I asked her, what would you think if I got you a chair that looked like this to sit in during the wedding ceremony? Not everybody here is old enough to hear the reply that my wife made to me. So that did not work. Oh, she gave me a thumbs up. I love that. No, I'm not telling you. All right. Now, Euclid's Elements is talked about a lot. You'll find it referenced everywhere. But when you read about Euclid's Elements, you will hear them talk. Out of the 13 books, you will hear them. He'll people talk about books one, two, and three. Maybe book six, and that's about it. Out of the 13. So the question is, why don't we talk about the others? Well, there's a very good reason. Even though Euclid was at the heart of what made mathematics what it is today, he was still very much a foreign person compared to us. He did not think like we do for a lot of what he did. And what I'm going to show you straight out of Euclid's Elements is going to cause you to harbor negative feelings to the person sitting beside you. So, here we go. It's going to start in a very simple way. First of all, notice that there is absolutely no practical implication that you're seeing in any of this. It's pure geometry. Euclid is writing his geometry intentionally not to connect with the physical world in any way. He was trying to show what the power of logic can be to explore mathematics, independently of the physical world. Leo Wei didn't care about that. He wanted to actually solve construction problems. So, Euclid is very, very tough on his logic. He's very precise. And he starts with a series of definitions of basic terms. He tells you what a straight line is. He tells you what an angle is. And this is where he's telling you what an angle is. A plane angle is the inclination of lines to one another when two lines in a plane meet each other and are not lying on top of each other. Makes sense? That's what an angle is, right? The inclination of two lines. Agreed? Then he goes on to give a second definition. And when the lines containing the angle are straight, then the angle is called rectilinear. And even Paul Zeitz is confused about that. And if Paul Zeitz is confused, you can be sure the rest of the world is too. What is going on here? What does that second definition do anyway? Well, the key phrase is right here. When the lines containing the angle are straight, Euclid's use of the word line is different from what we mean by a line. If I drew this object, is that a line? We've got some yeses and some noes. According to Euclid, this is a line. This is actually the original meaning of the word line. What we now call a curve used to be called a line. And what we now call a line was actually specified as a straight line. So we use the words curve and line. There's still some languages in which the word line, lien in French, for instance, actually means a curve. And if you want to specify a straight line, you have to say lien doit, straight line. So this first definition, a plain angle, is talking about a construction like this, where you have two curves that cross. The second definition is talking about what we think of as an angle, where two straight lines cross. Agreed? Okay. Let the games begin. All right. This diagram is from Euclid 3, verse 16. Now, I love watching American football games, because these people are holding up these signs that say John 3, verse 16 all the time, right? Whenever I see those signs, I always think of this theorem. I don't think that's what they intend. Okay. So this is actually Book 3, Proposition 16 of Euclid's Elements. Start with this red object here. Is this red object a line? Yeah. It's a line. Remember, a line can be curved. So that's a line. Is that straight line that the circle's lying on? Is that a line? Okay. So here's my question. Is that the inclination between the circle and the line on which it sits? Is that an angle? Ooh. Take a look at the person sitting beside you. Convince them why they're wrong. Go. All right. Do we all have an opinion? New Yorkers. You're in New Yorkers. You always have opinions. All right. If you do not have an opinion about this question. Okay. That means everybody except you has an opinion and is going to be forced to make a commitment to this. I just noticed that one of those lines is red and the other is blue. I promised not to get involved in politics. That was an accident. Okay. How many people think it is an angle? Raise your hand. Almost everybody on that side. Hardly anybody on that side. That's really weird. How many people think it is not an angle? Again, a lot of people over there. Are you picking both? Okay. You're picking both. Okay. Someone who said that it was an angle. Explain why. Only because the second definition talked about a rectilinear angle with the straight lines. So that left for if they weren't straight lines. So they're not both straight lines, right? Right. Yeah. That would be the other kind of angle. Exactly right. A round of applause right here. That was half-hearted. I think those of you who had a different opinion weren't clapping one more time. There we go. Okay. She's exactly right. It's not a rectilinear angle because they're not both straight lines, but it perfectly follows the definition of a plane angle. And so in fact, this is a plane angle. Now, by the way, it's called a horn angle. Can you tell why it's called a horn angle? Because it kind of looks like a horn, right? Okay. Who among you then decided that it wasn't an angle? Okay. Could you explain at the back why it's not an angle? Okay. So I'm thinking, if this is an angle, what's the angle? And it's going to change the closer you get to it. And so I wanted to think of it at the point of tangency that the angle would actually be zero because that's exactly where it's going to touch the curve. And I noticed in the definition it says and are not lying in a straight line. And so I thought maybe at the actual point where the angle would be formed, it would be a straight line because the angle would be zero. And that's why I voted no. Lovely. Another round of applause right there. Okay. Now, one way of saying what the gentleman asked is, well, if it is an angle, how many degrees is it? Well, how many degrees is it going to have to be? Let's take a look. This is what Euclid actually says in Euclid 3, verse 16. Take a line, the line on the floor. Draw any rectilinear angle coming out of point A upwards from it, forming a rectilinear angle here. What can you tell me about that second green line? What's it going to do? It's going to cross the circle. Does it matter how many degrees that is? No, it's always going to cut inside the circle. And that's exactly what Euclid proves, that if there is a rectilinear angle there, it must cut inside the circle. So, if this is five degrees, it's going to cut inside the circle, right? Which means the horn angle is less than five degrees. If it's one degree, it's going to cut inside the circle, which means the horn angle is less than one degree. If it's .000001 degrees, the horn angle is going to have to be less than .0001 degrees. What number is it that's less than any positive number, and yet greater than zero? Oh. You thought you had me until I said that last phrase, didn't you? Okay. So, Euclid concludes that the horn angle then is less than any rectilinear angle. But you are all making an assumption that it is in fact the right thing to do to measure angles using numbers. Euclid's conclusion here isn't that the angle is zero. His conclusion is that the angle exists, but you can't measure it with a number. Okay. So, did he know zero? Did he know zero? He had ways of expressing concepts related to it. He didn't have zero itself. That's true. Okay. So, there were fierce debates about this. Whether or not it was legitimate to call a horn angle an angle. They happened in Euclid's time. Euclid probably didn't like this either, because as soon as he proved this theorem, he basically did the equivalent of putting his fingers in his ears and yelling out, but lots of people did. And all the way through the 18th century, people were arguing about it. It's totally feasible to consider this to be an angle. It means you end up doing different sorts of things and seeing different sorts of things. You end up dealing with concepts in math that are called infinitesimals. And I'll leave it at that, because we have one last example to go through. So, that was our second example, and nobody's gotten hurt, so are we ready for the last one? Okay. Back to Liu Hui. Liu Hui, what was his most important book? The Nine Chapters. The Nine Chapters. Have we seen an example from the Nine Chapters yet? No. No, but now we will. This is in a section called Construction Consultations. And it has to do with three-dimensional geometry, at least in our language. So, name me some three-dimensional shapes. Cube. Ellipsoid. Oh, Lordy. I don't know how to write that. Sphere. Pyramid. Okay. There are some others. Do we know how to find the volume of a cube? If that's A, that's B, and that's C, what's the volume? A times B times C, right? These are the primitive shapes we work with. These ones you see on the right. Only one of those shapes is one of the primitives in Chinese three-dimensional geometry. They have a different set of shapes that they work with. One of them, thank goodness to you, is the cuboid. Okay? The same shape here. So I'm going to call this a cuboid rather than a cube, because a cube means all the sides are the same, and I'm not going to require that. All right. Here are some of the other shapes. This is a keandu. All right. Now, a keandu in the little plastic kits that some of you ended up with on your seats, you will find some of these shapes. Now, it took me 25 hours. I kid you not to make these. So please be gentle with them. All by myself. I actually learned that you can use permanent injury from the little serration thing at the end of a roll of tape. Really, I'm not kidding. I actually did. Okay. So how many keandu are in your kit? Oh, great. The kits have gotten mixed up. There should be four. Share amongst your friends. Okay. You may have to share amongst your friends. You should find four. Now, anybody want to take a guess what keandu means? Triangular prism. You would think it means a triangular prism. In fact, it means an embankment dike. We're still connected with the physical world here. This is what the shape of an embankment dike was. So, what's the shape of an embankment dike? The shape of an embankment dike. All right. Our third shape, you will also find in your kit, it's called a yangma. And you should find several yangma. I can no longer vouch for how many are in your kit. Three. Three? Okay. Now, I'm going to need four volunteers to have the full blown kit and work with it up at the front at this point. So, I need four people. Right there. One. Okay. Two. I need two more. Right there near the back. Right here. Well, I'm not taking Paul. He's too much trouble. Right there. Okay. So, come on up. You guys get to sit over here. And you're going to be live on the front. Pardon me? No, don't bring it. We've got a complete kit for you over there. Okay. Warm up problem. First of all, remember that the only shape we can find the volume of is a cuboid. Length times width times depth. Agreed? Agreed? Okay. What I would like you to do first is to do some practice in combining some of these shapes to make cuboids. Cuboids are what we want. We know how to find their volume. How many can do does it take to make a cuboid? Okay. You guys show us up on the screen. You see it right here? Okay. Round of applause for my friend Mark. Yay. Okay. Now for a slightly harder problem. How many yangma does it take to make a perfect cube? Did you do it? All right. How many people have made a cuboid? Okay. You guys show us up on the screen. You see it right here? Round of applause for my friend Mark. Yay. Okay. All right. How many people have made a cube out of their yangmas? We've got one right here. Have you guys managed to do it yet? They're working hard. We've got somebody else back there. Okay. And over here. All right. I'm going to give you a hint. Can we switch to the other screen? Okay. Watch this and be amazed. Are you ready? I need a drum roll for this. Do I get an ooh for that? All right. So three yangmas put together form a little cube. Okay. Yeah. That one's a bit of a surprise. I should mention one other shape that exists. And I need your help on this because my imagination just isn't good enough to understand this. The next shape they worked with, which you don't have in your kit, is called a bianow. And it's basically a yangma cut in half according to one of those two slices. Can anybody guess? I will actually fall over in shock if anybody gets this. What the word bianow means. Yeah, you didn't get it. I'm not going to fall over. You ready? Now, I don't know if I can ascribe this to cultural differences or not, but I'm not seeing the foreleg of a turtle there. Okay. Can anybody tell me what this shape is? Frustrum. Oh, I heard the magic word. What is it? Frustrum. It's called a frustum of a pyramid. Basically imagine somebody building a pyramid and running out of money. Okay. Now, first of all, you don't quite have enough pieces to make this frustum in your kits, but some of you might, now that the pieces have moved around a bit, see how many cuboids, kyandu, and yangma it takes to make one of these frusta. See if you guys can build it over here. You guys are live on camera. Okay. Don't touch it. It's beautiful. All right. We have formed a frustum. How many cuboids did it take? One. One. How many kyandu? Four. How many yangma? Four. Four. So, can we switch back to the screen? So, you can see that here. You can see one of the kyandu and one of the yangma is highlighted. I'm going to call the length of the top of the frustum a. I'm going to call the length of the entire base b. And I'm going to call the height. Well, you tell me. What letters should I use? I don't want to use c. It's a height. Okay. I'm going to use h. I would have used c, except that I've already made the next slide assuming that it's h. So, can I use h, please? Okay. Thank you. All right. You see the little cuboid in the middle? Ask your neighbor. What's the volume of that cuboid? I'll give you a hint. The top of that cuboid is a square. Okay. So, what's the volume of that cuboid? A times a times h, right? Ah. Okay. I didn't actually plan that. That was a coincidence. All right. Now, what I'd like you to do next is to consider the following shape. Now, in the interest of time, I'm going to build it with your kit over here. And then I'm going to ask everyone to find its volume. So, can I have your frustum? Can you guys see the frustum on the screen? Okay. Move it back over there so they can see it. It fell apart. Gas before. Gas before. All right. So, what I'm going to do is I'm going to break up the cuboid. And I'm going to put 2keandu over there. Now, what have I formed? It's another cuboid, isn't it? Look at your neighbor. Now, look at your other neighbor. Aren't you surrounded by wonderful people? Find me the volume of this cuboid. That's right. All right. Let's reconvene here. We have an answer for this volume. And I have no idea what your name is. Pardon me? Biyush. I didn't hear that. Biyush. Biyush. Okay. I'm really glad I got that right. Okay. Tell us. What's the volume of this cuboid? It is BAH. B-A-H. Explain why it's BAH. It's because, thanks to this suggestion, the base is the problem of being an example. This is the entire base of the frustum. That's B. Right. And we know the height is H. The height is H. And that the base is the height. And A is the width of this. The width is H. And there you have it. So the volume of this cuboid is B times A times H, which is BAH. So we've got AH and we've got BAH. Okay. What does this have to do with a frustum anyway? Well, we're almost there. Almost done. A frustum, remember, is a cuboid for Kyandu and for Yangma. Agreed? Okay. Well, we just found out that the cuboid is AH, A squared H. Agreed? And the other cuboid we just did is, okay, A-B-H, B-A-H, same thing. It's a cuboid and for Kyandu. Agreed? Okay. One last cuboid I want you to form. Okay. Such a cheerful sound, isn't it? Okay. That's B, right? Yep. That's B. Agreed? Yes. And this is H. What I'd like you to do, I want these people actually to build it. The rest of you might need to do this more mentally. I'd like you to build a cuboid that has the base of B and B and the height H. What shapes do you need and how many to fill in the frustum to make it a cuboid? Go. Yeah, he always did it though, so I'm going to try to make sure he doesn't. Okay. They're almost there on the video screen. Can you feel the tension? We're reaching a climax here. Okay, folks, keep a close eye on the front of the screen. They've almost got it. Almost there. Oh, we're approaching the finish line. Are they going to do it? Are they going to do it? I feel like it's over time in Game 7. Do they have it? I'm not cutting 16,000 pieces of Velcro. And there it is, everybody. All right, they did it. They formed a cuboid. What's the volume of this cuboid? B times B times H. Pronounce that for me. Okay. How many cuboids in it? One cuboid. How many kyandu? Eight. Eight. How many yangma? Twelve. There we go. And now comes the favorite moment of my life. Please do not tell this to my wife. But this is one of those moments that gives me the great happiness. What does any of this have to do with a frustum anyway? Well, watch. Suppose we add these things together. If we add these three, we obviously get that. How many cuboids? How many kyandu? How many yangma? So what? Say it again, Mark. This. There we are. This. Thank you. I have PowerPoint skills. Yeah. But this is the real moment. This is exactly three of those, isn't it? Wait. This one doesn't make sense. This that we just built is three frustums. That plus the other two cuboids is three frusta. Frusta is the plural. So, divide by three. And Yohui has found for the first time in history the volume of the frustum of a pyramid. Let's give him a cheer. All right. This is the first time it was proved. This problem goes back many, many centuries. In fact, this problem was 2,000 years old by the time Yohui got his hands on it. In one of the oldest Egyptian mathematical documents in history, this is from about the year 1850 BC. Almost the oldest mathematical document there is. You can clearly see that that is the frustum of a pyramid, can't you? He actually gives in this document an algorithm to give you the volume. He doesn't tell you how it's done. Yohui is the first person ever to explain why this thing is true. And this is a totally different way, a gain, of thinking about mathematics that is, again, quite different than how we do it today. So, there may be only one mathematical reality, but there are infinitely many mental spaces in which we can play in the mathematical playground. Math is just too great, too vast to be contained by a single person's thoughts or even by a single culture's thoughts. So, when we're at home or in school or here or in the world, let's open mathematics to the full range of human ingenuity. Thank you very much.