B. Riemann and the complex sphere | Sociology and Pure Mathematics | N J Wildberger

Hello everyone, I'm Norman Weilberger. We're looking at the sociology of pure mathematics, investigating just how the social aspects of the subject direct its movement. Today we're going to continue with our discussion of the 19th century, introduce a very important figure, Bernard Riemann, who proposed an ambitious new theory of metrical structure in higher dimensions. It became very important for Albert Einstein's general theory of relativity in the early 20th century. And it also introduced all kinds of new ideas about the nature of space, and as we'll see today, also a little bit about the nature of arithmetic, which ends up getting involved. So he's famous for introducing things called Riemann surfaces, which are two-dimensional surfaces, which arise in his work in complex analysis, that is, function theory in the complex plane. And we're going to be talking a little bit about that and give you a very important example, the Riemann sphere, which is really interesting. So he introduced this more abstract idea of the Riemann tensor or metric, and ultimately that led, after many decades, to people slowly coalescing towards the modern notion of manifold, which is these days sort of the preferred way of thinking about the possibilities for the shape of higher dimensional geometry. In my view, the theory of manifold really needs to be looked at very carefully. To me it's a suspect notion, although most of my fellow pure mathematicians will not agree. So ultimately we want to address this, so Riemann is a very important figure in this discussion. Underlying this development is a deep philosophical question as to what is the real nature of the space in which we live in. This is a naughty problem to get one's hands around, because, you know, what is space? How do we even start to study that? Typically physicists are looking at the physical space, but they are motivated by looking at things inside the space, like matter, objects, forces, trajectories, and so on. There's also a corresponding notion of mathematical space, which is built up from these various possible geometries that we've been talking about. So each of these geometries provides us with at least a potential framework for for modeling the physical space in which we live in. Now on the physical side, I mean, we have these objects, we have rulers and protractors, we can talk about motion, we can look at the swinging pendulum, or the force between two objects, or we can talk about the motion of a planet around the Sun. So these are all aspects of space, because space is the environment in which all these things are happening. But if we strip space from the particles and the possibilities for measurements, then it's much more amorphous as to, you know, what actually we're talking about. And in the 20th century, of course, with modern physics, this becomes a crucial, really naughty question, as people start to realize that the essential, sort of small-scale nature of space is really complicated, and probably has some kind of quantum limits that prevent us from drilling down beyond a certain point. And also that, you know, on the large scale things become really strange, and then again there, we run up with limits against, you know, what we can actually do. But okay, so aside from these things, there's sort of mathematical things, which sort of, or mathematical objects, which somehow bridge the gap. They're mathematical things in some sense, but they're occupying physical space, like a platonic solid, like a cube, or a sphere, or maybe a little bit more complicated, an ellipsoid, or hyperboloid, these quadrics that we've talked about. There's curves, there's the idea of the plane, and you know, a grid, or maybe in some higher dimensional version of that. And then there's other kinds of shapes that one can start studying, and that in the 19th century people became more interested. For example, this thing here is a donut, or a torus. Here is a more sophisticated version of a torus, or a donut. It's like a donut with two holes in it, okay. This is the kind of object that became quite interesting in Riemann's theory of Riemann surfaces. So there's all kinds of possible, you know, approaches to this nature of space. But one of the things that Riemann, I think his work sort of opened up, is the idea that there's a kind of a local question, you know, like we just look at, you know, some small part of our space, you know, what is the nature of that? But there's also a global question, which is quite different from that, which asks about the large-scale structure or geometry of space. You know, like, is it shaped like this in some sense, perhaps in some higher dimensional fashion? Is it shaped like this? Is it shaped like this? Shaped like this? So when we try to get a picture of the entire thing, ideas of Riemann start becoming involved in that. They give us a language for talking about the shape of the space in which everything is happening. All right, so I do want to give you a more precise feeling for what's going on here. So I want to introduce this Riemann sphere, but we're going to do it sort of one dimension at a time. And I'll start with the planar version, which is more familiar and it's easier to understand. But already some really interesting arithmetical consequences arise, and the role of infinity becomes very interesting. Okay, so let's look at the unit circle in the XY plane, and we'll think about parameterizing it in a slightly different way from usual. Usually we sort of start at this point here and draw lines through it and parameterize things in terms of a parameter on this vertical axis. Now we're going to look at the North Pole up here as being the point that we're looking at things from. So we're going to draw lines from the North Pole to various points on the horizontal x-axis, say that one there or that one there. And where these lines then meet the circle is going to be what we consider the transform of the point on the x-axis. Okay, so we're defining a mapping S standing for stereographic, and S of X, so it takes this point here on the x-axis and sends it to the corresponding point on the circle. And here is the formula for it. The x-coordinate is 2x over 1 plus x squared, and the y-coordinate is minus 1 plus x squared over 1 plus x squared. So for example here x is just maybe a little bit less than a half, and then if you plug in x equals a half here then you're going to get a point which is the x-coordinate is positive, the y-coordinate will be negative, and it'll be at that point there. Now an interesting feature of this transformation is how it relates the image of a point x and the point 1 over x, the reciprocal, which is on the other side of the point 1. So here if x is a little bit less than a half, then 1 over x will be a bit bigger than 2. And I'm claiming that if you look at the image of x and the image of 1 over x under this transformation, that they end up being aligned. The line joining them is parallel to the y-axis, or perpendicular to the x-axis. This is an interesting sort of feature of this stereographic projection. Okay so what can we do with this? We can use this to rethink some things about the number line. So here is our usual number line, equally spaced points 0, 1, 2, 3, that are marching up and negatively minus 1, minus 2, etc. going in that direction. So now what we can do is we can use this S mapping, okay, this mapping S that we're just talking about here, we're thinking of the number line as being just a copy of this thing here. And this S mapping is allowing us to map the points on this number line to points on the circle. Okay, with the point 0 getting mapped down here, the point 1 getting mapped to where it is, the point a half getting mapped somewhere over there, point minus 1 being mapped over here, the point 8 over there for example will be, you know, some point up here that will be there, point minus 11 way over here will be some point over there. Okay, so this in in mathematical terms this is sometimes called compactification. Compactification. We're compactifying the number line by sort of wrapping it around in a highly nonlinear fashion, but that gives us room to remarkably add one more point, infinity. That's really a special feature here, right? We're getting this new point on our extended number line or number circle we could say. So this is a pretty interesting because of course traditionally we think of, you know, infinity is playing sort of a strange kind of role. Does infinity really exist? But anyway here we're seeing it in this compactified number circle. So this is a basic two-dimensional version of which the Riemann sphere story is then the three-dimensional version of that. Now I can't help but share with you an interesting aside, some more information about this planar stereographic projection which connects it with the inversive geometry that we've already talked about. Okay, where you're inverting in a circle. So this planar stereographic projection, the mapping I called S, can actually be thought of as inversion in a suitable circle. So here's that unit circle that we've just been talking about. Here is the x-axis and here again is the same picture that I've shown you already. There's a point on the x-axis there is S of X, there's the image of it on the circle. Here's another point, here is the image of it on the circle. What I'm claiming now is that this mapping from points on the horizontal axis to points on the circle actually can be thought of in terms of inversion in this green circle whose center is the point here, the North Pole, and which passes through this point here and this point here, the points 1 and minus 1 on the x-axis. So that's saying for example that if you look at the image of this point here, do you remember how inversion works, right? So the inversion of this point in this circle. One way of thinking about that is you create tangents to the circle through that point and then you join up those tangent points and then where that meets the line from the center of the circle to the given point, that is then the inverse of that point there. So that's pretty interesting and in fact so then it turns out that the property that I told you that you could investigate, that the relationship between the images of this point and this point, if they're reciprocals, ends up being reflection in here, is then a special case of a much more general property that inversions have, which is that they respect themselves inversion in circles. So anyway that's a little bit of an aside for those of you with a little bit more experience, showing again that mathematics is a really all-embracing subject. All aspects of it sometimes sort of interact in surprising ways. It's not just that there's these separate disciplines, but they actually do interact and it's actually a very rich story. So this view of the number line as a number circle with an additional point at infinity is quite disconcerting perhaps because it does suggest that there might be a valid arithmetic that does actually include a new number called infinity. And so a natural question is to what extent can this be done? To what extent is there really a consistent arithmetic that does involve a new point called infinity? And does one have to, you know, go against the the laws of logic in order to do this? So many of you will know that I'm sort of an opponent of a lot of sort of infinite thinking in mathematics, but some people sometimes call me a finitist, which is not really the case I think. I'm quite happy to entertain notions about infinity as long as we can be completely precise about what we're talking about. I don't object to using infinity in some context. For example here this is a quite a reasonable context. There's nothing magical about infinity because we're able to pin it down very precisely in terms of the geometry of this circle. What I do object to is the claim, the specious claim, that we are able to do an infinite number of things and get outputs which we can then use for other calculations. That is a different use of sort of infinite things and that is I think hugely problematic logically. So what about this question is there actually a consistent arithmetic which includes infinity, sort of extends the usual arithmetic with ordinary numbers? And the answer to that is partly. And so there are a couple of operations that we can introduce in a consistent way but there's others which we can't. And a way of thinking about this in order to try to make sense of some equation like 2 plus infinity equals infinity, what does that mean? So one way of thinking about that and to try to convince yourself that this is sort of consistent with the usual arithmetic is to think about maybe not infinity immediately but rather some point on the number line which is pretty close to infinity. So a big number like 8 but much bigger which is big positive or a big negative number. And ask yourself okay if we're moving towards this point infinity on this number circle does this law still work? In other words suppose you took 2, the number 2 and you added some really big number like a million. A million is really close to infinity up there somewhere. The result of that is 2 plus a million or a million and 2. Where is that going to be on the picture? Well that's also going to be close to infinity. And so the closer you are to infinity, in fact it doesn't matter which side you are, if you add 2 you're also going to be close to infinity. And that is then the starting point for thinking that you can probably make this a valid kind of arithmetic. Similarly it's not too hard to show that 3 minus infinity equals infinity. That's a reasonable kind of thing. 5 times infinity is infinity. If you take some big number, like big meaning close to infinity, so either a big positive number or a big negative number and you multiply it by 5, well you're gonna get another big positive or negative number. If you take the product of two such big numbers you're gonna get another big number. And here's a really interesting one. You've all been taught since early days that it's completely forbidden to divide by zero. It doesn't make any sense. But I'm claiming that in fact this does have a consistent meaning in this larger arithmetic where you're willing to include this infinity. And what's the meaning? The meaning is that if you take 1 divided by some number which is close to 0, either close on this side or close on this side, well then the reciprocal is going to be a big positive number or big negative number. And as you get closer and closer to 0, well the reciprocal is getting closer and closer to infinity. So it becomes consistent to say that 1 over 0 is infinity and similarly 1 over infinity is 0. However, this is not a complete consistent arithmetic because infinity times 0 cannot be so easily pegged down. If I take a really big number close to here and a really small number close to here and I take the product, I have no idea with just that information where that product is. That product could be very big, it could be really small, it could be, you know, somewhere in between here. So this one here is not a valid rule in at least, you know, in the obvious way in this geometry. And similarly infinity plus infinity doesn't work consistently because this possibility has to include not just the sum of two big positive numbers which admittedly is a big positive number or the sum of two big negative numbers which is of course another big negative number, but also the sum of a big positive number with a big negative number. And if you take that combination like 8 plus minus 10, you're going to get minus 2 which is nowhere necessarily close to infinity even if the two ingredients were close. And similarly infinity minus infinity is not defined. So it's intriguing and in fact with Riemann's three-dimensional version of this, this sort of partial arithmetic with this concept infinity ends up playing actually a crucial role in the function theory that he is studying. So it becomes not just an interesting kind of academic question, it becomes a central feature of the complex function world. So let me now explain to you Riemann's idea of the Riemann sphere where he does what I've just showed you but basically starting with the complex number line to begin with rather than the usual number line. So here is the usual complex number line which is to us a plane. It's got an x-axis, a y-axis, and a typical complex number x plus iy which we may call z, it's the usual format for it. z equals x plus iy has a component in this direction and a component in this direction and crucially in this direction here. There's an i, that magical new number that has a property that i squared is equal to minus 1. So Riemann is going to do the same thing that we kind of did but it's going to take this complex line, it's really a one-dimensional line, and make it horizontal and then he's going to, over the origin 0, he's going to make another axis. Let's call this the w-axis. So there's the w-axis here and horizontally there's this plane, we'll call that the z-plane, and there's x and y coordinates in this horizontal plane. So this plane is the one that you're seeing here, so there's the point 1, there's the point minus 1, there's the point i, there's the point minus i. So this circle here that you're seeing here is the usual unit circle in the complex plane. And now Riemann says okay now let's take this sphere, the sphere which is centered at the origin here and which has an equator which is created from this complex plane and here's the North Pole of that sphere. And now we're going to do the same kind of thing that we did. Now this is like the stereographic projection of the sphere but that then allows us to basically coordinate the points on the sphere by complex numbers because the numbers in this horizontal plane they are basically complex numbers. So we start with some complex number, say let's say this point here, it's a little bit hard to see, this point here, we'll suppose that that this is a point on the on the horizontal plane. Maybe we could think about it as being out here. Actually in this diagram it would be slightly inside. And then you connect that with the North Pole, you get a line and where it meets the sphere, it comes out the sphere, that's S of the point Z that you're starting with. So what that ends up doing is allows us to to map the sphere via the plane with the addition of this additional point at infinity. So we're moving from the complex number line to the complex sphere, it's called the Riemann sphere, which is now a sphere whose points are indexed by complex numbers. So for example here that might be the point 1, that would be the point minus 1. Along this circle here there would be I on this side and minus I on this side. And you know points over here might be like 3 plus 4 I or you know 18 plus 27 I, that kind of thing. And in addition to the regular complex points there's a single North Pole up here, this new point which has now the the name infinity. So this is really interesting because Riemann is interested in the idea of functions on the complex plane and this gives him a way of not just visualizing but also sort of connecting these functions with geometrical objects. So let's have a little discussion about functions. The notion of a function is a hugely subtle one and is full of pitfalls and logical difficulties having to do with what the actual explicit concrete definition of a function is. But loosely we think of a function in the usual way, it's something that takes some input of a certain kind and outputs something else of possibly a different kind or maybe possibly the same kind. Now in high school you're used to thinking about arithmetical functions given say by polynomial equations like y equals x squared. Okay so let's just remind ourselves about the different ways that we think about such a simple function back in high school. Perhaps the simplest way is to think about creating a table of values where we record different x's and different values of the function applied to those x's, in this case x squared. So 0 squared is 0, 1 squared is 1, minus 1 squared is 1, 2 squared is 4, minus 2 squared is also 4, 3 squared is 9, etc. So we could create some points like that. And then we could take a table like that and try to create a more visual representation of the function by essentially plotting those points in a two-dimensional plot. So the point 0, 0 is right there, point 1, 1 is right there. So what we're doing is we're replacing the x squared with y or interpreting the x squared with y. So the y component here is the x squared value while the x component is just the original value. So minus 1, 1 that's that point there, 2, 4 is that point there, minus 2, 4 is that point there, 3.9 is up there somewhere. There is however another way, a third way of visualizing what a function is, which is to think of it as a mapping or a transformation of the underlying space. So here's our number line and we can think about this function y equals x squared as taking a point on the number line and moving it physically to another point. So for example 2 gets sent to 4, 3 gets sent to 9, minus 2 gets sent to 4, minus 3 gets sent to 9, while 0 and 1 are sent to themselves. So visually you can kind of think of taking this number line and sort of folding it over, like this negative part gets folded over onto the right side, and then there's also a corresponding stretching going on. All right so these are three different views of the usual function y equals f of x which is x squared. So now let's look at that same function but now thinking of it as a complex valued function. So something that takes complex numbers and sends them to complex numbers. So we might write w equals f of z where f of z is z squared. Now we're thinking about z as a complex number and w as a complex number. So we can visualize that, at least partially, by creating a table like we did last time. But now with values of z and w ranging not just over rational values but also complex values. So we can include something like i, i squared is minus 1, or 1 plus i which is 2i. Or if we square 3 plus i, what is that? The square of 3 is 9, the square of i is minus 1, so that's a total of 8, and twice the product is 6i. So this is another way of trying to get some sense of what this function now looks like. How about if we want to graph it like we did for y equals x squared? Well that's a bit more problematic because we now need a z-plane, which is like the horizontal axis, and we need a w-plane, which is now the vertical axis. But each of those are two-dimensional from our point of view, so they don't fit so easily onto representation like this. But nevertheless we can sort of still think that, say, a point like 3 plus i, 8 plus 6i, is represented by a pair of complex numbers. So that point there, which is this point here, has a z coordinate, which is 3 plus i, and a w component, which is 8 plus 6i. And then when we think about creating a lot of such positions and then sort of joining them up, joining the dots, we end up getting what's to us a surface. It's no longer a curve like it was in the rational case. We get a surface, okay, which roughly we can visualize something like this. And this is the origin of this geometrical, almost topological, aspect of function theory that Riemann realized was going on in the complex world. Now if we go back to the third form of thinking about a function as a transformation or mapping, that becomes actually more useful now because, first of all, we can visualize the complex plane as our Riemann sphere. So there's that Riemann sphere again. And now this function, z goes to z squared, can be thought of as something that moves points from the sphere to the sphere. And what's that going to look like? Well a good starting point is what happens on the equator, which is just the unit circle. If you take a complex number on the unit circle, say that one z, okay, then its square will also be on the unit circle and will be a copy of that sector, so you basically double that sector. So as you go around the circle, z squared goes around twice. So if there's one, then there's a kind of a wrapping that goes around, the function sort of wraps the circle around itself twice. And correspondingly what happens to the entire sphere? Well the northern hemisphere also gets rotated around, but also what happens is that the points on here also start moving up to the North Pole. So if we square something like that will it move around here, but also get closer to the North Pole, while points in the southern hemisphere, if you square them, they also rotate around, but they get closer to the South Pole. So this is an interesting thing because now we can also augment our understanding by asking, well what happens to the infinity point? So we can augment our understanding by adding basically the point infinity and the square of infinity will be infinity in this case. This is one of these situations where this arithmetic actually is defined, this z squared function actually is well defined on the Riemann sphere at infinity. So that's really attractive because now the entire sphere is being transformed. And Riemann realized this was especially useful for more complicated functions like the function 1 over z. Usually in high school you have a problem with 1 over z. At 0 it's not defined so you have this sort of gap in the graph. But when you move to the Riemann sphere like this, something like 1 over z can also be represented as a transformation and it has a continuous aspect because the reciprocal of 0 is infinity and the reciprocal of infinity is 0. So it becomes actually a nice transformation, actually essentially a Mobius transformation actually, of the sphere. So this is a very attractive kind of thing because it's giving us a larger view of geometry but also the arithmetic is intimately connected with it. I'm Norman Wildberger, thanks for listening.