Riemann • Riemann Surfaces

Great. Okay, well, welcome back. This is part three of what's going to be a four-part series on Riemann, specifically getting into his use of abelian functions. What we're going to do today is we're going to have a bit of a recap of what the past sessions have been, and the subject of this session is going to be his treatment of singularities in particular. So we had looked at Dirichlet's principle in the past from the standpoint of metaphor, of how based on certain characteristics that had to be fulfilled, you could tell what field must have existed to create those conditions, even if you couldn't describe it directly. Dirichlet's principle did not include unusual points like singularities. So that's what we're going to talk about in particular today. I'm going to put it in a little bit of context, though. So just to, you know, the things we've talked about already have been the problems of trying to look at change quantitatively. So that came up in the first discussion around contour, where we were discussing the economic value of stages of development in an economy, and how you couldn't measure, for example, the fruits of the Apollo program in the same kinds of dollars as were spent to create it. We'd looked at some examples using the cones in economics and biology, and then we started last time getting into how complex functions work, and that's what we're going to be continuing today. I wanted to start with a quote from LaRouche on abelian functions to situate things. So he wrote in his paper, Science vs. Oligarchism, from earlier this year. He wrote, So, as in the case of the essential part of the matter I am setting before you in these two successive reports, the mind cannot be permitted to proceed on some set of its own mere impulses. The mind must discover and employ the mind's powers for the critical examination of its own powers, as from a still higher ontological standpoint than mere sense perception, such as is done according to the principle of metaphor, or Kepler's vicarious hypothesis, a standpoint of actively conscious insight into the accompanying actively functional role of metaphor, as Riemann does with the relevant examples featured in the third and closing segment of his habilitation dissertation, and his treatment of abelian functions as a mode of escape from the misleading presumed object of perception. Riemann's method provides the means of escape affected by means of a reciprocal interaction between the human mind of the scientist and the putative subject which is being examined, as Johannes Kepler had already done in the discovery of the principle of the orbital system, done through the specific role of the principle of vicarious hypothesis. So let's come over here, let's just quickly review how complex functions work. If you're not familiar with this and what I go through right now isn't enough, watch class two because that was the subject of it. Very briefly, when we looked at numbers not as existing on a line but as existing in space, we found some differences. We needed a one to do multiplying, this is one example, that if we had a and b were directions and motions in space, to multiply them we need a one, and we would say that one is to a as b is to the product. Just like if we had the numbers two and three to multiply, one is to two as three is to six. From one to two is an act of doubling, apply it to three, you get six. That's doubling and tripling put together. Here a b is a-ing and b-ing put together. From that standpoint there was a very special kind of number that appeared. So we've got the Arabic one for this direction, if we use the Roman numeral for this direction, we get this number i. If we take that i and then apply it to itself, we look at i is i, well one has the same relationship to i as i does to this leftward motion, which we can call negative one because it's the opposite of one. So i is i, or i squared equals negative one, and you can say that i is the square root of negative one. Now the physical meaning of this was the subject of Gauss's 1799 doctoral dissertation, where he took the fundamental theorem of algebra, which all sorts of algebraists were working on trying to prove it. They were just using letters and symbols and ink on paper, so when they would write square root of negative one, it didn't mean anything, and they call it i for imaginary or impossible. Gauss showed that no, they exist in a world, that what was the perceptual world of normal real numbers has a hidden domain that's lying behind it, and without understanding that hidden domain, you couldn't understand the visual world of the so-called real numbers. So that's going to be the subject as we go on here, is that there is an unseen world that you have to understand to get an idea of what ends up coming into perception. You know, we're going to be visualizing the complex domain. So one of the things that came up when we started to do that was we looked at squaring in general, and we found something that was a little unusual. This is, here we've got numbers, and here we've got them squared. Complex numbers, we're squaring each of them. We're going to walk around on a square over here. The funny thing is that the same path is basically traced out twice on the right. From an overhead view, you would just get that crescent-shaped object covered twice, and that's because for every number, there's two square roots. Like what number times itself equals four? Two and negative two, right. So we'd started to touch on this, about how when you introduce this double surface, then two and negative two, when they're squared, they both become four, but they're different kinds of four. So let's address the different kinds of singularities that come up. One of them, which we just saw, is called a branch point, and here's why. Let's take a look. Here, we're going to take the square root of numbers. We're taking the square root of two, of one, we're about to take the square root of zero, and we get numbers over here. Square root of two, one, zero. But look at what happens if you continue in a straight line on the right. Right, I was taking the square root of positive numbers, but after I passed zero, once I started looking at the square root of negative numbers, what happens? You get two possible outcomes. Yeah, and you might say, well, there were two possible square roots of one in the first place. It could have been either one or negative one, but once I chose one of them, I said, I'm going to take the positive one, then I'm just going to start walking towards zero. I'm just going to get the neighboring square root each time, nothing unusual. But on reaching zero, well, the square root of negative numbers is in the complex direction, the imaginary direction. So the path, you'd say there's a fork in the road, you'd say it branches. So for the square root, zero is a branch point, because if you walk straight towards it, what seemed to be a simple path, an unambiguous one, all of a sudden just split into two. Here's another view of this. So we're heading towards zero. Once we get there, you can see how the path branches out. So that's an unusual place. You can't just continue walking through it, because Dirichlet's principle says that once you've got the boundary conditions, you've got an idea of what must occur inside one of these fields, and you can extend it. Everything should be known about the way you'll proceed, except this is an unusual place where that doesn't happen, where a straight path splits. Here's just another view of how there are two different numbers that when squared both have the same value. So here, one and negative one both come to one when you square them. So without the two different sheets, it would really look as though there was only one point on the right, because two numbers that are, you know, one's negative of the other one, their squares are the same. Now we're taking a look at cube roots. So again, we're going to walk from one to zero, and then at zero, we're going to go to the negative numbers and look at their cube roots. That might seem a little strange. Maybe let's go through that. For example, this number right here, does anybody have an idea of why that might be the cube root of negative one? Why this number times itself, so to speak, three times would give you negative one? 60 degrees. Yeah, it's 60 degrees. So from one to that number, and do the same thing again, and then do the same thing relative again, and repointing to the left. Here's just good old negative one. Negative one times negative one is positive one. Reverse again, negative one. So this is also a cube root of negative one. How about this number down here? Same thing. From one to the number, to its version of itself, and a third time, and we're pointing to the left. Here's a view of the branching. Are there any, this stuff might be, there might be some questions I'd like to know right now. Is this okay, this being the cube root of negative one? Would anybody like to go over that again? Matt said it was 60 degrees. I'm not sure why that is, that the cube root would be a 60 degree angle and not some number. Well, let's take a look at it over here. I got room right here. So here I've got one. If I take that same distance and I go up here, and I call that number just for fun, well, maybe not very fun. Okay, I'll call it A. Okay, that's not so fun. Now, from one to A, if I do that same relationship again, like if I wanted to double twice, I would say one to two, two to four. So multiplying by four, that's doubling twice. If I A twice, I'd go from one to A, and then it's as though A was my new number one, and from its point of view, I would draw its A. Then from that A's point of view, if I did it a third time, I'd say, let's consider that to be one. Well, boom, here's A. So that relationship of one to A, one to A, one to A, done three times, gets us pointing at negative one. Oh, right, right, right. I see, because when you multiply negative one by negative one, you get one, and if you multiply by negative one again, you get negative one. So you have to start on the opposite side of where you, you have to end on the opposite side of where you started. So your motions have to be divided in such a way that it takes you to the opposite of where you started. Yeah, after doing it three times. Right. Because negative one also works just fine. Let's take a look at that. Let's say I had one, and let's say I had this as my A. Well, or maybe to be special, we'll call it B. So again, if that was one, this would be its B. If that B were one, this would be its B. So this change from one to B, done three times, has us pointing to the left. So it's the cube root of negative one, or it's the number, it's the change that done three times would give us negative one. Okay. Jason, the sheets on the right, that's the cube, that's the complex plane cube? Yes, but it represents the cube root. It's as though this, like, if you look at this from the top down, as we had in the earlier slide, you're just moving from one to negative one. Revealing it this way shows that you have to think of the normal plane as having three sheets if you want to look at the cube root of a point. So that's why of a particular point is seen from the top, there's actually three, you're actually really seeing three points, you just didn't realize it, because the cube root of any number has three answers. The reason I picked one to start with was saying, let's say I picked a specific one, because there was three cube roots of one as well. But I picked the one that's just a simple number one. And then the funny thing is that when I get to zero, I can't continue anymore. Like, imagine if I had neighbors, if I said I want the cube root of one, well, that's one. If I want the cube root of, you know, 0.9, you know, I could say it's some number that's over here, but instead I'm just going to go with the neighboring one. And then once we reach zero, because see the orange and the blue plane aren't actually connected. So like this path is no more straight ahead than the other ones. Really the orange, the orange colored, it ends right at zero, you can't continue on, it's a dead end. It's like you've got a, you know, stairs on a spiral staircase, and you're walking along the stair towards the center of the staircase, you just hit a wall, right, you can't continue that way. So all three of these are branching. So zero for a square root, for a cube root, zero is called branch point. And Riemann would say it's a branch point of order, here it's of order three, because there's three branches. Like a branch, it turns into three smaller branches. That's why it's called that. Now we're going to take a look at when we've got, oh, this is just another view of, there's always three points that have the same cube. Anyway, all right, now we're going to look at logarithms. So this probably needs a review too, because this is not a commonly used concept. So sometimes you say, like, what if I asked you two to what power equals eight? How many times do I have to double to get to eight? Okay, I heard three. That process you just did in your head is a logarithmic process. And you would say that three is the logarithm of eight. Here it is kind of written out. So, yeah, two cubed equals eight. If I said the log base two of eight, so if my base multiplication is two, how many times would I have to do it to get eight? Well, the answer is three. There's a very special number, now known as E, that's considered to be, it's called the natural base of the logarithms. And I was not planning to go through why it's such a special number, but we can do it in the discussion period at the end if you want, because it's a pretty important point. One very unusual thing about this E, this natural base that I wanted to look at, and I'm also not going to get into why this is, but although it might seem hard to believe, if you take this E number, which is between two and three, it seems like an unassuming character. If you take it to an imaginary power, you end up getting the trigonometric functions out of it. Yeah. So, again, why that is, I'd like to do at the end, but I can look at what happens when you use it. In other words, if I've got, in my complex domain, where I've got one in this direction, and my Roman numeral one or my I in this direction, this spot is E to the zero. Over here, I've got E to the I times pi halves, because that's going to be... Okay, why pi halves? Using 360 degrees to measure a circle is a little bit arbitrary. It's a good number. It's close to the number of days in a year. It has a lot of factors, so it's easy to divide it up, but there's nothing special about 360. A more natural way of measuring how far you've walked along the circle would be to measure how far you walked. So, if the radius of a circle is one, what's the... Does anybody know what the total circumference is? Yeah, it's two pi, actually. Yeah, I'm sorry. Well, from the diameter to the whole thing would be pi. If we're using the radius as one, then this total circumference is two pi. So, a fourth of the total circumference is two pi divided by four, or pi halves. So, this point is e to the i times pi over two, because it's... Okay, what's... We've got our cosine of pi over two, and then in the i direction, the sine of it. So here, you know, we're at 90 degrees. Cosine zero, the sine is one. So, we're going to move the sine in the imaginary direction. So, if you're taking an imaginary exponential, you're moving in a circle in the complex domain. Why that is, let's talk about at the end. Sorry, I know that's a... That's just a big thing. But I want to look at something funny about it, which is that this is another question, like the square root or the cube root, that has multiple answers. If I asked you, if I said e to the what equals one, there's a few answers. I could say that my mystery number was zero, meaning if I don't do any e-ing at all, I'm still at one. So, either the zero is one. Also, it could have been two pi i, meaning if I just went around the circle, the whole circumference, I'd be back at one. If I went around at four pi i, I'd be back at one. So, this question has not two answers like the square root, not three answers like the cube root, but it actually has an infinite number of answers. So, the logarithm has an infinite number of branches, you could say, because the logarithm of any number, I've always got as many answers as I want, because I can keep just going around the circle again and again. Let's take a look at what that looks like on a Riemann surface. So, on the left, we're just moving in the i direction. This is the exponential over here. So, you can see we're just going in a circle again and again. So, zero, now this is two pi. This is about six up in the i direction. Once we reach two pi in the i direction, taking the exponential of it, we got back to one again. That's the top down view. This is the side view of it. So, remember with the square root, it was like a spiral staircase that after two stories, you were back at the bottom again, and you just kept repeating yourself. Now, it's an infinite spiral staircase. So, zero, again, is a very strange point here, because if you actually walk to zero, you could pop out on any of these branches. So, it's like an infinite branch point, or branch point of infinite order. So, you can think about maybe there's other ways of representing the fact that that's happening. And we talked before that maybe putting something on a cylinder, if it repeats again and again, might be a better way of thinking of it than spirals that go forever. Okay, now this is fun. So, here I've got, here I did two logarithms, and they're right next to each other. I'm going to walk in this figure eight path along it, and maybe we got back where we started, maybe we didn't. This is a side view. It's kind of like a parking garage, with the ramps, you know. No, I want to go to the exit now. But for each one, I could have just kept going up and up and up, and then I can go back down the other ramp. I could have gone back down the other one. So, there's just, I want to show what it's like to have a couple of them. Okay, so that's, those are our branch points. I want to take a look at poles. So, yeah, in this one it did come back. That, that, that route would have. I didn't make another route that didn't, but I guess if it just stopped right here, it would have looked like it came back from the top view, but it really didn't in the surface. In other words, by walking around, this is an important thing. Let me go back to this, actually. Like I was saying before, once you choose which sheet you're on, it's like you're saying, I picked a specific answer. Like if I'm standing at point one and I say, what's the logarithm of it? Well, it could have been zero, could have been two pi i. Let's say that I said it's zero, okay? Right here we're at zero. Now, if I just walk around, if I just walk around zero and come back to the exact same point one, I necessarily would have been at two pi i now, in terms of what the logarithm of the number was. So coming back to the same point, the value changed. So both, if you just ask the question out of the blue, what's the logarithm at this point? You could have had a bunch of answers. Also, if you picked one of them, and then you just walked around to neighboring points, the way that the function works, you're going to be a level up. It's like the square root which went through last time. If you start looking at the square root of one, then you walk to negative one, and then back to one again. If your first square root was one, now it's negative one, just by always looking at neighboring square roots. It's the way of always picking one of the two. So this one, okay. Now, poles are another strange thing. I'm kind of saying poles here. A pole is a place where a function becomes infinite. So if I said one over x, what's its value at zero? What's one divided by zero? It's pretty big. How many times does zero go into one? Well, it's a lot. Oh, I didn't mean to go ahead. Okay, yeah, so here's a pole. Spin around a little bit. So it's a little bit of a funny character when you approach it from one direction. I kind of wish it wasn't spinning. All right, stop. Anyway, over... Okay, here's from underneath. So here's zero as we approach it from the positive direction. So I'm heading towards zero from, let's say, one. I'm doing one divided by one, one divided by half, one divided by a fourth, one divided by a tiny number. It's way up here at infinity. One divided by a tiny negative number is down here at negative infinity. So that's what a pole looks like on a plane. It doesn't look as weird on a sphere. This is two different poles, and I'm going to pull them together and see what happens. Again, it's spinning a little bit much. Here's five of them. You know, it's when they come together, they make a nice flower. You know, it just keeps going to infinity. It just chops off at a certain point to be able to show it. The reason I was showing that is I thought it was interesting that nothing that unusual happens when they all come together. That is that if you said, if I said, how infinite is one over z versus one over z squared? You saw when I pulled the two different infinite poles together, you know, they just kind of came together. Nothing exploded when they came together. So you can think of this as actually being two that are very close to each other, instead of one that's doubly Polish. Okay, and then here, this one's a very strange one. This is an infinite pole. So here, this is e to the one over z. So at zero, one over z becomes infinite, but we're going to do e to it just to make things more complicated. Now what's happening is we're zooming in. We're actually getting, we're zooming in more and more on the origin. The whole size of the whole circle here is shrinking. You can see how it's just, as we zoom in, it gets more and more and more complicated. Isn't that weird looking? Yes. Yeah, that's actually like an infinite number of poles all together. So there's really, if you wanted to say, what's, what shape might that actually be? It's not one that, uh, that's one that you can't really make a Riemann surface for. I want to look at that one again. That was a, so here, you know, you get your positive, you got your negative and we're zooming in more. So this lobe will get cut off as we're closing in on the center. You just keep going. If you keep zooming in, it just looks like nothing. You just can't even see. It's just like a bunch of things that are all pointing basically vertically. Jason, when you said we were zooming in, what was the, was the surface changing as we were moving closer to it? Or were we just looking at us? We're just, it's not changing. Like basically what we're going to do is we're going to, we're going to look at this region more towards the middle of the, of the thing. Unfortunately, the grid isn't changing. I've been, I should have done it. So that occurred like at the beginning, the radius of this circle was two. By the end, the radius of this circle is one millionth. So we really zoomed in. It looks like the surface is changing. I don't know if it's just an illusion. It is changing because, um, it's rescaling it as we're moving in. I've got an interactive one of these. We can, uh, play with at the end so we can adjust it more and move the camera around and whatnot, which I think will be helpful. Yeah. It's, it's a peculiar one to look at. This one's a little bit weird though, because from one side, it's just completely flat. And on the other side, it's this sheer wall that goes both up and down. So, okay. So that was, those were, those were some of these strange singularities that take place. So you may have heard the phrase that you can understand everything from the boundary points and the singularities. Those were some of the singularities that Riemann actually dealt with, uh, branch points, poles. And then that really strange one at the end was sometimes it's called the essential singularity. Anyway, um, now we're going to take a look at, uh, some other types of topology, some other ways of looking at these surfaces instead of with the corkscrews, how we can look at it in another way that might bring out its, its characters in a simpler way. So let's do that. So, okay, here we've got a simple surface. So it's a pizza. It should be filled in here with some pepperonis and things. Now, some things about a simple piece of, uh, you know, a pizza is one, if you, if you make any cut that goes all the way across it, you'll now have two pieces of pizza. So if you have a friend over and you have a pizza, if you cut it, you know that you'll have two pieces and you can share one of them. Another thing is that if you, let's say you put a rubber band on your pizza, you could shrink it as if you, if you just let it, uh, if you let it pull itself inwards, it could shrink down to a point. Okay. That there's no reason why that'd be interesting yet, but we'll see. Okay. All right. Now we've got an unusual pizza. So here you got, you got your, uh, you had a pizza and you want to share it with your friend. Um, and your friend's in a very good mood because, and they said, you know what? Why don't you cut the pizza and you take whichever piece you want. Okay. And you say, okay, I'm going to cut the pizza and I'm going to take the piece I want. Maybe you do it like this. Maybe you give them just that little sliver, but with a pizza like this, you can be really greedy because you can cut it right here and there's still only one piece of pizza. So that's a difference. This is a different kind of surface than the original one. The hole in the middle means that it's actually called doubly connected because it's, it's even if you, you can cut all the way through it and it's still one piece. Now, if I had to make one more cut across it, I would have to make two pieces of pizza, but the first one, it's possible to make a cut that doesn't. So it's doubly connected. Also, as you can imagine, if I had a rubber band, you know, around here, it couldn't shrink down to a point because there was a big hole in the middle. Okay. A third surface. Now you've got three friends, you get, anyway. So you can cut it so that you split into two pieces, but it's actually possible to cut it, not just once, but twice without actually making two different pieces of pizza here. Could have cut, I could have made those two cuts. The importance about the cuts is that they go from one boundary to the other. They go. Yeah. In other words, I didn't just sort of go, you know, it's called a cross cut or a transverse cut. It goes, yeah. From the, from the boundary from one end all the way to the other. Um, ooh, I should have shown another. Well, okay. Here's another way I could have done it. I made two cuts and it's still one piece. If it's connected, it means that from one, from any one point, you can get to any other point, which here we can from any place right now. It's more of a pain to get over here if you're here, but you could go around. Now here's something funny. Um, yeah, it's not that funny. Okay. Um, now that was a triply connected surface. It's triply connected because even after I cut it twice, it's still simply connected. So working backwards, take away the two cuts and it's triply connected. You know, another way of looking at it might be to say that there's, um, there's sort of three independent ways of getting from place to place. We have to get from here to here. I could just go straight there. I could go around this guy. I could go around that guy. Almost like there's two different kinds of independent periods in it or types of connectedness in it. Now this is a, now this is a, uh, let's take a look at a closed triply connected surface. One that doesn't have a boundary. A Taurus. I mean, it doesn't have a boundary. In other words, like there's no, you don't hit a wall if you're walking on it like this. If you were an ant walking on this pizza, you know, there's definitely an end to the pizza where you just can't keep walking. A Taurus doesn't, you know, there's no walls in it. The whole thing is sort of, the whole thing is the surface. And again, look, I can make one cut and it's still one connected piece of Taurus. I can make another cut. And even after this, the whole thing is still connected. So if I was going to share, you know, if I had a bagel and I wanted to, you know, or no, or if you had just the crust of a bagel, let's say they sell that for people on diets or something. I don't know. It's just a crust of bagel or maybe just a frosting on a, on a donut. And you said, I want to share this with you, friend. I'm going to make two cuts. I'm going to, I'm going to cut it and we'll split it up. And so first, if you, anyway. Okay. So you make this cut and, you know, it didn't, you didn't actually cut the Taurus into two pieces. All right. There's a, you know, there's a, there's a slit across the top, but it won't fall apart. Even if I cut it here, it doesn't fall apart. You could unfold it. Like imagine, so I've made this cut so I can unfold it like a tube then. And then this slit along the top is like I had a tube and I just slit it. Then it would unfold into a big rectangle. So it's still, so this again is triply. But it has, they have to be perpendicular to each other. If you made two of your first kind of cut. Right. Yes. Yeah. If I did this and then I made another one close to it. Yeah. Then I would have had that thin piece that would, that would come off. Also, you know, there are cuts you can do right away. If I just went like that, you know, I could cut a little circle out of it if I wanted to. But it's just that it is, you could do two cuts if you wanted to. That wouldn't break it apart. So it's triply connected. Okay. Oh, great. So now we're going to take a different view of all of these things using the Riemann sphere, as it gets called today. So this is where, because one of the things with the plane of numbers is that the plane goes on forever, right? Using this trick here, not a trick, using this technique. So this is a projective sphere. This should, you know, continue out forever. We've got a light bulb at the top of the sphere. It's shining through it like a stained glass window, casting a projection on the ground. As this red bug is walking along the window, it has a shadow. It's appearing on the ground from a light right at the North Pole. See how that works? So as we're getting up towards the top, I should have made, okay, these meshes don't correspond. I should have done that. In other words, these circles of latitude here, you know, the equator, it comes out to this. How about this circle right here? If you project a light from the North Pole through, you know, 80 degrees north latitude, it's going to go really far out before it finally reaches the plane, right? What if I just, you know, I stood on the North Pole and I took one step over to the side and there's that light right on the North Pole and it's shining at me. It's basically going to go just forever. It's going to go out forever. It's never going to reach the plane. So this point right next to the North Pole, it just projects, it's basically parallel to the plane. It will go out literally forever in that direction. From right next to the North Pole infinitesimally to the left, it will go out this way forever. The North Pole itself becomes a really interesting location because it's infinitely far out on the plane, but it's not in any particular direction. It sort of corresponds to just every possible direction literally infinitely far away. It also means that those aren't different places anymore. So that if you went beyond infinity out that way and came back on the other side, you can do that on the sphere here. You just walk over the North Pole. So that comes up with, you know, Kepler used that. We didn't use the sphere, but with the different conics of, you know, you take two foci and put them together and you got a circle, pull them apart, you have an ellipse. You take one focus and put it infinity, you've got a parabola. Once you bring it back from the other side of infinity, you end up actually having a hyperbola. And you bring those two foci closer together, you still have a hyperbola. You bring them so that they touch each other after having passed through infinity, and the hyperbola flattens out so much you just get a straight line. So, you know, that's Kepler using this, actually going through infinity, and then you come back from the other side as a real thing. So when we use this sphere, it's going to change the way that we can look at things a little bit. For one thing, infinity is an actual point. That's really interesting. But let's take, I want to take a look back at the squaring with those two surfaces that crossed into each other. And let's see if we can change our way of looking at it. First off, I'm going to, I'm going to go ahead and cut the sphere in half. So I'm drawing a meridian line here. Then I'm going to split it open. You see I'm stretching it. The color, this is still the red color. I'm just sort of, I'm just, I'm basically compressing the whole sphere. I've still got what was the whole sphere before. All right. So now this is, this is no longer walking from 1 around to negative 1. This is walking from 1 to negative 1 back to 1 again. Remember with the square root, when we walk from 1 to negative 1 and then back to 1 again, we had the negative square roots, the opposite square roots. We had to walk around a second time to really get back to where we started. So let's, if we take this, if we add in then another half of a sphere that acts the same way, we can put the square root as just living on this sphere. We don't need the two surfaces that somehow go through each other without crossing. Yeah. Okay. Let's, I'll say it one more time, then I'll draw something up. Well, this is 1. This is also 1. The negative 1s are over in this, these directions. This is i and negative i. So when I walk, I have to walk from 1 to i to negative 1 to negative i to 1 to i to negative 1 to negative i to 1. I have to walk around from 1 to 1 to 1 twice. It's like making two normal circuits to really get back to where I started from. When you're taking square roots of numbers, you really do have to walk around twice to get back where you started from. Because there's the, there's the two, the two different, the two different sheets. Let's take a look at it over here. So here I've got my, I got my numbers. I've got 1, 1 plus i, i, negative 1 plus i, negative i. I'll just label 0. Negative 1 minus i, negative i, 1 plus i, right? I'm going to take all those numbers and I'm going to go ahead and square them now. No, oops. They're 0. If I square 1, I get 1. If I square 1 plus i, I get 2i. I'm going to color code these in a second. If I square i, I get negative 1. If I square negative 1 plus i, I get negative 2i. And then when I square negative 1, I get 1. So in other words, when I walk from along this path here, I walked along, I've already done the whole circuit. So here, here I've got the normal numbers. Here I've got their squares. By the time I went from 1 to negative 1, I'm already back at the same number 1 again. 1 squared is the same as negative 1 squared. It's just 1. But I've still got all these other numbers down here. If I square each of them, if I square negative 1 minus i, I get 2i. Maybe I should be going through why that is. Well, let's look at this one. If I square negative i, I got 1. I got negative i. Negative i is negative i is negative 1. Let's take a look at the negative 1 minus i. Zero. All right, I'm at negative 1 minus i. And here's 1. So 1 is to negative 1 minus i as, sorry, let me draw that again with the square. Better grid. Okay, zero, 1, negative 1 minus i. Now I'm going to let negative 1 minus i be the new 1. I'm going to make its grid. If this was 1 and this was negative 1, then this is i and this is negative i. You just have to kind of tilt your head the whole way around. So from this guy's standpoint, negative 1 minus i right here. So 1 is to its negative 1 minus i as negative 1 minus i, if it was 1, would go to its negative 1 minus i, which in the blue original grid is 2i. Let's say at this point right here, it comes over here. Negative i squared is negative 1. 1 plus i is 2i and 1 squared is 1. So when we go around the path for the second time, when we take a look at all these numbers, their squares just do the same circuit again. So it's indistinguishable. It's like, oh, let's play a riddle or let's have a guessing game here. This isn't a very fun game. But anyway, let's say for whatever reason you're saying, oh, I'm thinking of, hey, guess what? I'm thinking of a number. And your friend who's having a lot of fun playing this game with you says, oh, you're playing 20 questions. And they say, hey, what's its square? And you say, I guess you can only ask yes or no. Anyway, they say, what's its square? And you say, oh, my number squared is 1. And then they wouldn't know if it was 1 or negative 1. So if you've only got the squared numbers, you don't know which, was it the red path or the green path? You have no idea. But here's something interesting. If you walk along it smoothly, if I'm walking along, I'm sorry, let me start with this. Let's say, OK, so I'm starting in the squared world. I'm starting at 1. And then I say, let me say that I'm at 1. I'm at the square of 1. I'm that kind of 1. There's two kinds of 1. There's that square of negative 1, 1. But that's not me. I'm the square of 1. OK, well, just so we know what we're talking about here. So then let's say you go for a walk. You walk up to visit your buddy 2i. And hey, how are you? Well, that 2i is going to be 1 plus i squared. It's not going to be this one. You didn't, out of the blue, wind up way down here. Because you're just taking a nice little walk. You're visiting all your other number friends along the way. And you get to this guy. Then you say, oh, good to see you. I'm going to go visit my friend negative 1. So you go to your friend negative 1. Well, that's i squared. It's not the negative i squared. It's not that kind of negative 1. Because you're just going, you're visiting, going house to house along the way. Keep going. You visit your buddy over here, negative 2i. You say, how's it going, negative 2i? Say, oh, pretty good. I'm this square of negative 1 minus i. Say, oh, that's negative 1 plus i. Oh, that's great. See you later. So then you walk back home to 1. But guess who's there? You found someone in your house. Say, wait a minute. I live here. And then the other one, the imposter, the other one says, uh-oh, no, no. I'm the square of negative 1. I'm that kind of 1. And then our protagonist says, well, I'm the square of 1 kind of 1. So the point being that just by walking around and always looking neighborly, going from house to house, you all of a sudden you're in a different neighborhood. You realize, wait, this isn't my neighborhood. This is one that looks just like it, except it's got different people living in it. The green people. So then if you say, oh, you say, well, let me, instead of retracing my steps, why don't I take my same walk again and see what happens? So you go to visit your friend 2i. And it's actually negative 1 minus i living there. It's not your normal 1 plus i you were expecting. Then you take a walk to visit negative 1. And negative 1 says, oh, yeah, I'm negative i squared. You know, you're getting freaked out. You want to get home. You come down here. Sure enough, you know, you're basically, you know, it's going to be someone unusual. And sure enough, it's negative 1, 1 minus i. You're like, yeah, that's what I expected. Then you get back home and you're actually back at home again. All right. So if you're looking at what's the space of square roots, you get scared. I say, well, there's something weird going on here where it seems like every number is actually 2. But if I go around, if I go around 1, I'm just in another world for that circuit. And then I'm back to the one I expected. You say, well, how is it? How can that be? So the one way is to have the two, the double corkscrew thing where you got your ramp that goes up the parking garage. And then somehow this ramp, as it goes around, it comes back down to the bottom floor, even though it's going up. Now, that's the Riemann surface I showed earlier. Hard to imagine because how does it go? How does it cross back down without ever going? You can't really do that. You could imagine a sort of a world like a Pac-Man world where you go up one story, you go up the next story, and then you just go off the screen and you're back down here again. You could imagine the square root world as being like that. I'm saying this is another way of looking at it. It's a world where the, so here the sphere is this world. The plane is all of the square roots, right? The plane, if it weren't colored unusually, well, actually it doesn't really matter. Okay, anyway, so the plane is all of these numbers that are going to be squared, and the sphere is when they get squared. You have to imagine that the plane only went around the colors once now. Sorry. But then if you went around the plane halfway, you would have already gone through all the colors on the sphere, right? So you go around this halfway, and you've already gone through all the possible colors of the squared numbers. So here you go through all the colors twice to get back where you started. So that's what the square root world is like. It's like this, you know, it's like this, we took the sphere, you know, cut it in half to make some room, and then brought in another sphere and glued it on top of it. So hopefully that makes some more sense after this. I'd like to remake this picture with, maybe I'll put it on the web, in the video version, with the plane being the right colors, so you can see how it goes from one to the other. Okay, now let's take a look at what's coming up next here. Oh, it's the square root, perfect. Okay, so here's the guy taking his walk. So yeah, visiting his friends. Oh, what? See, they cross here, and it looks like they're crossing, but they really don't. That's the thing, like, these are not, there's not one place, there's actually two totally independent places, it just has to somehow get back down somewhere, so you have to draw it like that. But you know, he started in blue, and when he got back here, it's orange. He's really not on the blue plane, he's on the orange one, he's cutting right through it. So he's in, he's at his, that other guy's house. Okay, so, yes, yes, let me see, okay. Now, okay, I'm realizing that was a lot of, a lot of particular details, and the conspectus putting it all together is going to have to come in the next class. I can say a few things about it, which is that you've got a, with the sphere thing, for example, you've got, you've got the way things seem to behave. You're just sort of playing with things, you see what happens. You sort of have a perceptual feel of what occurs with the process of squaring, you know, or cubing, or complex exponentials and logarithms that we saw, or the poles, or any of these, these strange kinds of things. And you say, is my idea of the space that these things exist in, is that a true one? I've got the idea there's just a plane of numbers, is that really where they live? Well, when you're talking about square roots, they don't really live on a plane. They live on a double sheet, or this sort of double sphere thing. If you're saying, where does the exponential live? You know, if you're, if you're walking around and you're doing logarithms, you would keep, every time you went around and walked through the neighborhood, it would be different every time. You would never get back. That was the one that had the infinite, right? Well, you can just, you can just walk backwards when you get back home. So, yeah. But always, there will always be something new. So wait, so what would that look like on the sphere? You'd have to make room for a third band of color, fourth band of color. Eventually, we'd have to have an infinite number of bands of color. So that's not really going to work that way anymore. You get, with the elliptical functions, you get two kinds of periodicity. Here, we only saw one kind of period with the logarithm. In the imaginary direction, it keeps going around and around and around. In the real direction, it just goes out and in. It doesn't repeat. If you've, if you've got something that's repeating in two different ways, then, you know, I've shown before, you can use the torus as an idea of imagining the world that that lives in. The torus has two totally independent types of independent periods on it. When you get more, and I guess this will have to come up next time, you end up with, you know, toruses with like multiple holes in them. It's having multiple independent paths of connectivity. The importance of all this, just I guess say something about that right now, is that these singularities are an essential part to go back and look at Riemann's approach from the habilitation dissertation and his view of what he goes through in the abelian functions paper of how do you understand the space that something exists in? You've got the Dirichlet's principle, which I'm going to come back to next time to look back at that, which gives you that metaphor sense, but you also have the unusual aspects inside of it. And they're best treated topologically as what kind of unusual, what kind of connectedness of space must correspond to this phenomenon that we're seeing. So Dirichlet's principle gives, you know, lets you know that given a boundary, you can fill in what's in the middle, but it doesn't give you much of an idea about what shape, what kind of space you're in. Like the fact that you might be in a double-holed torus, for example. So they're both needed. This is more the analysis situs that lies behind things. It gives you an idea of what the shape is that they're in. And then I do think that, you know, not to overdo it with connectivity and economics, but I'm sure if you think about it some, you can think of examples about how there's many kinds of economic developments that change the connectivity of functioning in an economy. Simple examples like bringing in rail, where, you know, if you take a look at the Midwest, you're able to have a different kind of production. With efficient rail and efficient transportation between factories and production sites, you no longer are only making finished goods. You're able to make intermediate goods and actually ship them around and have a, have that kind of, you know, part-based mass manufacturing. But anyway, I think that's probably, I don't want to say too much more about that. I want to see if, because I bet there's a lot of, there might be some questions about all of this, including a couple of things I promised we could come back to. So I guess that's, we should do that now. Thank you.