 However, he was being spiritually inclined. He joined the Ramakrishna Mutt as a renunciate upon being impressed by the life and work of the Vedantic philosopher Ramakrishna Paramahams. His initial name was Brahmachari Brahmachaitanya. He was renamed as Swami Vidyanathananda after receiving the saffron robe in 2008. Swami Vidyanathananda is a monk at the orders recorders in Belur Mutt. He was professor of mathematics and dean of research at the Ramakrishna Mission Vivekananda University in 2015. He is currently a professor of mathematics at the Institute of Fundamental Research, Mumbai. He has widely published and presented his research in the area of hyperbolic manifolds and ending lamination spaces. His most notable work is the proof of existence of canal thriston maps. This led to the resolution of the conjecture that connected limit sets of finitely generated flying end groups are locally connected. He is also the author of a book titled Maps on Boundaries of Hyperbolic Metric Spaces. At that, Prasamaham also received the prestigious Infosys award last year. Prasamaham. Yeah, it's a technology thing. It's tough, so I'll start as soon as the technology gets in here. No, it equals function. OK, thanks a lot. Thanks very much. OK, so I will try to begin an answer to this question, which is the topic of my talk. And I'll assume that everybody here knows calculus. I'm going to move them out, but that's about it. And there are two problems in mathematics which have a fairly longish history. One is, I mean, sort of from the beginning of mathematics there have been sort of two directions, one involving numbers, one involving shapes. So numbers have given rise to number theory, algebra, et cetera, et cetera. The other side has given rise to geometry, ecology, some sort of analysis, and so on. So both these questions occupy some of the greatest mathematical minds for any guesses, couple of thousand years, actually. So there are two very specific problems. And they were solved by two of the greats. One was Galois, the other was Gauss. And Gauss solved the geometric problem. Galois solved the algebraic problem. And being a geometer, of course, my personal bias prejudice was geometry, so I'll answer the geometric problem. And that was this. I mean, of course, nobody knew what hyperbolic geometry was, and it looked like. But the problem was posed by Euclid. So when we start off doing geometry in school at the very inception, we are introduced to Euclidean geometry, first introduction to formal reasoning. And we are given a bunch of axioms, Euclid's axioms. So there's a whole collection of that, but this is a very short summary of Euclidean axioms in two-dimensional plane Euclidean geometry. Since the previous talk was largely a life science talk, I can't resist making this statement. Apparently, the part of the human brain that is involved in asking apparently childish or meaningless mathematical questions, which very often lead to profound things, is exactly the part of our will, has a sizable overlap with the part of the brain that invites very new information as a child, that meaning, say, within the first two or three months. So asking these meaningless questions, apparently meaningless questions, is definitely something that helps us acquire a lot of knowledge. And some of our profoundest questions come from here. So I'll try to see. I mean, so Euclid was exercising presumably that part of his brain. And he posed a question. So he laid out axioms. So I'll just summarize the axioms first and then get to the question. So first of Euclid's axioms, there are four axioms, and then there's a fifth positive. So four axioms, first one says that given any two points in the plane, you can join them by a straight line. So this is what we, in today's terminology, we would call a straight line segment. The second says that if you have a straight line segment, you can extend it infinitely in both directions. So that by infinite extended object is really what we call a straight line. The first object is what we call a straight line segment. Third axiom says that given any point on the plane in the radius, you can draw a circle. Fourth axiom says that all right angles are equal to one another. What does that mean? It means that all other angles are wrong angles. But what, yeah, so what exactly does that mean? I mean, so what is the right angle? How do you define the right angle? So you take a straight line. So already you have a notion of a straight line given by the first and second axioms. How would you define it? So basically, you take that straight line, take an arbitrary point on it. So now you have a canonical angle. There's an angle coming back, coming from minus infinity to that point, and then going off to plus infinity. By set that angle, that's the right angle by definition. What the fourth axiom says is that that angle that you've got is independent of the line you chose and the point on it that you chose. So that's the formal content of this. Before I come to the fifth postulate, let's look at these four axioms a little more carefully. And what these axioms are really doing is that they are defining the players of the game. What are the objects that we are going to play with in order to build up our geometry? The first axiom defines for you a line segment. In terms of what? In terms of what are the things that are sort of going there, we have been quietly shopped under the carpet. There's a plane and there are points on it. So these are things that we don't define. But with that data, we are going to define the players of our game. Given any two points, you are able to define a line segment. That's what the first axiom does for you. So all these four axioms are in some sense defining players. Second defines a by infinite straight line. Third one defines a circle. So a circle has a center and a radius. That defines the circle completely. And the last axiom defines the notion of an angle. Now let's come to, so these are in some sense definition axioms, first four. The fifth guy is different. It's a different status. What does it say? It says that you take two straight lines, assume these are straight. And now you let them follow a particular line. And suppose that the sum of these two angles is less than 180 degrees. What that axiom says is that then these two lines, if they are extended in this direction, they're going to intersect. This was the original formulation of the fifth postulate. There's a crucial thing here. If these two is less than 180 degrees, it's going to meet on this side. By the same axiom, if it's less than 180 degrees on the other side, it's going to meet on that side. What if it's exactly equal to 180 degrees? Because then we have no idea. So the two lines don't know which side to meet, so they don't meet. That's not a proof. But essentially, you can crank it up to a proof. And that reformulation that the lines do not meet is called, I mean, that's really called the parallel postulate. So the fifth postulate is equivalent to, say, that through a point, so do not meet gets a special name that's called parallel. And that's what it says. You have your straight line. You have a point outside it. There's exactly one line through this point on the plane, which does not meet this line. Basically, from here, you draw any line. They just make an angle such that the sum is 180 degrees. And then that line will not meet. So the fifth postulate says that through a point, not on a straight line, there is one and only one straight line to the point parallel to the given line. This formulation of the fifth postulate is called the parallel postulate, maybe later. Now, what was the problem that Euclid posed around 200 dc or earlier through the fifth postulate from the remaining postulate? This was his problem. If he was, yeah, I mean, I'm not sure. I mean, I have not read the original. But I guess, in hindsight, the problem should have been posed as, can one prove the fifth postulate from the remaining postulate? The sort of inside this question, there's a kernel of perhaps the deepest theorem of mathematical logic in the 20th century lurking. So yeah, 200 dc, 20th centuries is a kernel. All right, so now that's before we get on to the maths, let's see the answer a little bit of story type. So what's the history of the problem? Zivians of people who are current. Finally, it was cracked by Gauss. Started thinking about parallel lines in 1792. In 1824, he wrote to a friend. Tarnus was a friend. He writes, the assumption that the sum of three angles of a triangle is smaller than two right angles leads to a geometry, which is quite different from our geometry. At that point of time, our geometry meant Euclidian geometry. But which is itself completely consistent. So the sum of the angles of a triangle, three angles of a triangle is equal to 2 pi. That's also equivalent to the parallel possible. So basically, you look at a triangle, draw a line through here, this angle is equal to this angle, this is equal to this. So sum of three angles, this is 180 degrees. And from that, we can go back to the parallel possible. All right, so there's a little bit of mathematical gossip. Perhaps an apocryphal Gauss did not publish this one. And there's sort of questions as to why he did not publish this one. And there's a sort of a slightly naughty explanation given. By 1824, Gauss was an established mathematician, and he did not want to appear wrong in public. There's some reason that is given, perhaps, perhaps not. However, Gauss is not, so basically, the discovery of this geometry where the sum of angles is less than 180 degrees is not a popular literature attributed to Gauss, which is, so who are they attributed to? People who published it first. But before, they published some other history. So you know of these hyperbolic functions, hyperbolic trigonometric functions, cosine hyperbolic e power x plus e power minus x by 2, sine hyperbolic e power x minus e power minus x by 2, et cetera, et cetera. So these functions came in trying to compute areas and other menstruation problems of the foregames. So in the 18th century, some of these things were actually computed by Lambert. And again, then after that, the discovery was made simultaneously by two people, a Hungarian mathematician by the name of Wally I, and a Russian mathematician by the name of Lavazhevsky. Lavazhevsky's book is completely unreadable, because it's got all the sort of bits. Apparently, somebody, I think, Gauss really went through that basically in order to understand Lavazhevsky's work, it's extremely computation-intensive, very technical. And in order to understand what he's trying to say in 500 pages, you have to read all 500. So there's no way you can sort of get the crucial idea. I mean, if you, I mean, people who do math research sort of, well, OK, let's get the main idea, then we'll try to work out the medium. You can't do that. You have to go through every tree individually in order to understand the thoughts. That's sort of a style that the publicists get adopted. As good or as bad as any other, I guess. Anyway, so he published his paper on the principles of geometry in 1829, 13, and a couple of years later, Absolute Science of Space was published by Bollywood, which is much more intuitive. Anyway, at this point of time, what were they trying to do? They were trying to formulate a geometry in which the fifth postulate would be violated, which means what? Which means that, OK, reformulation of the parallel postulate sum of the angle of a triangle is 180 degrees. So they were replacing it by the axiom that the sum of the angle of a triangle is strictly less than 180 degrees. That's what they were doing. And then they were trying to bend up a geometry. So less than 180 degrees has nothing to do with hyperbola. So the term hyperbolic geometry came much later when hyperbolic geometry was really put into a certain context. And again, in hindsight, it's a wonder that it was not discovered earlier because almost all geometries in a certain fairly precise sense are actually hyperbolic. This Euclidean geometry is some exceptional thing. And it is only this sort of perverse habit that we have grown into that tells us to think that this Euclidean geometry is the natural thing and that polygeometry is not. So I'll try to justify that so that it's not just a sales experiment, mathematically accurate. OK, so it was formulated by, I mean, the word was coined by Felix Klein. But the person who really brought it into mainstream mathematics and tied it up with zillions of other areas is Pankai. Just after Klein, Klein had an ongoing competition, one side, on this side only, because he was always doing it with Pankai. And anyway, so I mean, I think what happened was that he was trying to find out, discover certain groups. Pankai found them out, and he called them Phuxian groups. Klein was very annoyed. He said, Phux had nothing to do with it, which is perfectly true. And then, so then I think Pankai went up one more dimension, these were two-dimensional objects. He went up one more dimension, discovered some groups in three dimensions. OK, I'll call them Kleinian groups. So Klein had nothing to do with Kleinian groups. Phux had nothing to do with Phuxian groups. Pankai had everything to do with Phux. So anyway, so this tying up various things, I mean, things which are, even today, applicable in number three things called modular forms. All of it, I mean, has origins there. Certainly geometry, certainly topology, but also things which look far like numbers there. All right, so now let's get back to the mathematics from the gossip, and let's restate the parallel postulate in terms which are more precise. So let's, what is the parallel postulate? It says that given a straight line, L, name them now. But crucially, we define the context in a plane P, in a flat Euclidean plane P. See, if you define two straight lines as parallel if they don't meet, then in three dimensional space, you have infinitely many lines, right? So you have one line running on this flat on the floor. You take this point on the top, you take a straight line, and you just rotate it in a plane parallel to that line. You're going to get infinitely many lines, right? So you're talking about two-dimensional flat plane Euclidean geometry. So that has to be set in a plane P. And a point X on P outside L, there exists a unique L-prime line on P passing through X and not meeting L, which we call parallelism. And now the Euclidean problem becomes prove the parallel postulate from other axioms of Euclidean geometry. But the thing is, look, I mean, this is how a lot of mathematics is really done. There's a problem that is posed. And there is an unstated assumption, which is obvious to everybody. But we don't spell it out. So the problem was being posed in the context of Euclidean geometry, which means all of it is being done in a plane P. Yeah, yeah, everybody knows that. You spell it out. And then you really start understanding, once you spell it out, you understand that really this unstated assumption is the only place where you can attack the problem. So if you've not stated it out, if you've not exposed the vulnerable part of the problem, there is no point of attack. And so what the reason it took 2,000 years is that a large part of the time went in trying to understand what is the point of attack. There were several things, apparently this poet, Omar Hayam, he attempted this problem for a long time. He attempted this problem with a bunch of others. Yeah, it's not a who's who. You go through the history of these problems, I forgot on the list. And there's a very impressive list of mathematicians who attempted this problem. So basically, once you stated the assumption, the basic question that we need to ask now is what is the context of the problem? So that means the problem is deep. This unstated assumption, which is really at the background of this entire problem, is the place where you need to attack it. The question is, what is a pain? That which was undefined, some points on it, et cetera, et cetera, we don't know what it is. Some axiomatic thing is a piece of paper. That's not a definition. So what is a pain? That was the question. We don't know the context. Basically, the question was asked because there was a common shared context, but it was not formally clear. It was not something you could manipulate. You could manipulate the players, you could manipulate lines, angles, circles, et cetera, et cetera, et cetera. But you were not allowed to manipulate the underlying structure, which is a pain. So what exactly is the pain? Can we understand it in some more fundamental terms so that we can manipulate? That was the question. And that's one of the main reasons why this problem took such a long time. And the answer to this question actually comes from a very different area of mathematics. That's why it took a long time. And once we have defined that plane peak, just note that in the Euclid axioms, you have this line segment, and then the straight line is defined in terms of that, angle is defined in terms of that. So what we would like to do ideally is start off with the motion of a plane peak from that, derive the motion of a straight line. So a straight line now becomes an emergent mathematical object, not something intrinsic, not something defined, but something that emerges from the definition of a plane peak as an object which has some characteristic properties. So that is our goal. That is the goal of these three questions. And then motion of parallelism, once you have a motion of straight line, the motion of parallelism is clear. Two straight lines are parallel and they do not meet. It's just a one. So basically what we'd like is an a priori definition of a plane, and then from that, define what straight lines are. OK, good. So the answer for Euclidian geometry is R2 equipped with a certain metric. Metric means a way of measuring distances. So what is that? It's ds squared equal to dx squared plus dy squared. What is this? This is just an infinitesimal way of saying what to Pythagoras theorem is. a squared plus b squared equal to c squared. But if a is very small and b is very small and you want to sum, this is the only place where I need calculus. So basically what is this infinitesimal way of using the Pythagoras theorem? And it's packaged in this one neat formula. What's the meaning of this formula? There's something infinitesimally small. It's square equal to sum of two other infinitesimal squares apart from some vague philosophical content. How are we going to translate that into maths? So if you open that package, and that's a formula, it's a package, what does it signify? It means that if you have a smooth curve on a plane, r2, then the length of that curve can be calculated. And this is where integral calculus or differential calculus comes in. Calculus comes in. This is computed by a standard formula that we know from high school that integral of ds is, well, you parameterize your curve x of t, y of t, take the derivative x prime t, y prime t. And then you look at x prime t square plus y prime t square. So basically, at every point you're looking at the two derivatives, you're applying Pythagoras. That is giving you values to a certain length of a tangent vector at that point. And then you're multiplying it by dt. So that's the infinitesimal amount that you're traveling along the curve, then you sum over t. So basically, behind all this, there is a fairly sophisticated amount of thinking that has gone on. Calculus came 1,500 years or 1,580. So basically, it's 70 plus. So maybe some 1,700 years already of that 2,000 years has been captured. So in order to manipulate the plane, we want something, some more fundamental structure. So this structure came with Newton with my life minutes, much later, 1,400 something or the other. And this is a fundamental advance. I mean, we start doing our calculus in maybe class 11 or so. But we don't realize that today it's become part and parcel of our toolkit. But it was not there before that. So people were not able to solve this Euclidean geometry problem before that basically because there was no fundamental model. OK, so how many people are familiar with this thing, this formula A on the board? Very good, enough. So that means I can carry on with that. So yeah, so this is the length of this curve. And then what is what's going on behind this is that you use some parameterization of the curve. Now, if you are given this formula for, I mean, if you define a Euclidean plane this way, do straight lines have a certain property, have a certain special property? Yes, students, yeah? With respect to this distance? Yeah, yeah, yeah. Go ahead, go ahead. Yes, yes. Come up with whatever answers you know. I mean, you took a, yeah, go ahead. The ratio of dy and dx is constant. The ratio of dy and dx is constant, OK. But I have told you just this one formula, dx square where is dy, dx and du, yeah? The length of path is very good. The length of path is very good. So in terms of this, this L of sigma, the straight line is the unique line which minimizes distances between points, yeah? All right, where length is some object calculated according to this formula? Fine. So now we have, once you have this formula, yeah? Once you have a way of computing lengths, shortest lengths are exactly what defines straight lines. So now the straight line has become an emergent object, right? It is defined in terms of this metric, in terms of the plane with the metric, yeah? It's no longer an a priori thing, which you have to introduce as an axiom. Good. Let's proceed. So the straight line is the shortest distance as per the previous formula between them, right? Parallelism now is just as before. To buy infinite straight lines are parallel if they don't be. Good. So now the whole thing boils down to what is a plane P? Now if you have a plane P with that formula, yeah? Then prove that two parallel lines, I mean that there is one and only one line parallel to the first one. So if you take, so what does it mean? If you take your definition of a plane which was not stated in Euclid's axiom, to be ought to equip with that metric, then the parallel postulate does hold, right? Which means in order to address the question, can you prove the fifth postulate from the remaining four axioms? What will you have to do? This is basically trying to expose the point where you can attack the problem. It means you have to go back to that formula, ds square equal to dx square plus dy square and modify that. With that formula, the fifth postulate will have to hold, right? So if you want to have a hope of saying that what you want to do, you want to change that formula, see that the first four axioms are satisfied and the fifth axiom is viling. And if you want to prove it from those, then what you'll have to do is that for all possible formulae that you put in here, that satisfy the first four axioms, the fifth axiom is all the same. So now the point of attack is very different. So we'll have to manipulate this one simple-looking formula. So we'll have to basically, I mean, this is really okay. So hindsight is where it's 2020. I mean, the person who really brought in this perspective was somebody who in whose second thesis examination is called the Hamiltonian Schiff. Gauss was the examiner. Bernard Riemann. So that's how this, yeah. I have stories to say about that. Okay. So let's try to modify that formula. So how do we want to modify it? Take some subset of R2. And now it's kind of, so basically, okay. So the other thing to notice is that we wrote dx2 plus dy2. Again, there was something that was unstated. The coefficient of that dx2 is 1. We did not write 1. Oh, everybody understands it's 1, right? So because everybody had to understand, nobody writes it. And 1 is not treated as a constant, but 1 treated as a constant function over R2. So basically understanding what is unstated in the problem is what probing this problem is about. So what do you do? You replace, that's why you can hit the problem. So you replace 1 as a function over all of R2 by arbitrary functions now. Say f of xy plus g of xy. So now what is the general geometry? dx2 equal to f of xy dx2 plus g of xy dy2. What is happening? f and g are positive. And the length is computed according to the formula. You again parameterize x of dy of t, x prime t square plus y prime t square, type earlier it was 1, 1. Now you replace the first one by f, the second one by g. And now you again have a generalization of that previous formula. Now the problem from having absolutely nowhere to attack now become is given you a huge amount of choice. All positive smooth functions f, all positive smooth functions g. How are you going to try it out for all f and all g? So there are sort of two basic things that go on in mathematics. One is you want to take a problem and work very hard once you know that there is a and find some way point of attack. That's what this discovery of this theory of calculus gave us. Once that calculus was discovered, now there is some huge amount of choice. Now you want to find one particular metric and test it out for them. But now it's like hunting for a needle in a haystack with cardinality 2 bar c. So what is the cardinality of the collection of functions? That's some huge thing. Real line has cardinality c. And you want to choose one metric out of these. So now there is basically really a reaction to this story. There was no technique available prior to calculus. Now you have so many metrics available, you don't know which one to choose. So the motivation to choose the one, so there are two. So the second thing, the previous thing only accounted for some 1700 years or maybe something like that. The next thing, choosing the right metric has to come from again some kind of motivation. You have to have some clue, some intuition that if you look at this direction, maybe you're going to get a metric which is going to give you something, some new interesting kind of geometry. And then you have to play, exercise that part. So before we get into that, so the notion of a straight line in this metric is replaced by a notion of a geodesic. So it's a path which minimizes the distances but according to this revised formula. And all our concurrences that we studied in high school, angle to angle, side to side, SSS, side angle, side, etc. All of those are examples of what are called isometries. So what's an isometry in this context? It's something that preserves that infinitesimal form of the formula. If you preserve distances in the infinitesimal level, then you integrate out something, then the length of paths will be the same. So an isometry is something that preserves the infinitesimal form, hence preserves all lengths, hence preserves minimum length. Hence preserves the distance that we started off from in Euclid environment. So we have these two notions. So what are these? An isometry is a map which means that you take xy going to x1, y1 under this map I, infinitesimal thing of satisfied means you take f of xy dx square plus g xy dy square that is the first ds square. And after you've transformed it's x1, y1 dx1 square plus x1, y1 g of x1, y1 dy1 square. These two infinitesimal lengths must be the same. So an isometry, if you, now you just set it out into a formula which you do not have any other information except calculus. That's something that you can manipulate. So this is what an isometry is. Now, so the first part of the talk I think we have been able to say that okay, why was, why were mathematicians stuck because Imani's geometry was not there. But then we'll have to choose, we'll have to choose a particular f, particular g. Simplifying assumption, at least choose f equal to g. Mathematics is always trying to find, right? Plus simple thing if it works then why should we complicate our lives unnecessarily. So choose f equal to g, for instance okay, even then, even once you realize there's one function f, smooth positive function, there's so many other. So how are we going to do that? That choice was guided by something else in a completely different part of mathematics. This is what was that Poincare plant story was about. And that's the exposition that I'll be taking. You choose the Euclidean plane but not all of it. You choose the upper half plane, yeah? Vibrator than zero. And equip it with a certain metric where you take the Euclidean metric and scale it by 1 by y squared, yeah? So f of x, y equal to g of x, y equal to 1 by y squared. And you're taking y greater than z. Why am I made this particular choice? There's a short answer because Poincare turns out to be the problem. But the question is why did Poincare come up with this formula? That's a completely different story. It comes from a different part of mathematics which will define, I mean, it could be the topic of a completely different topic. So I won't tell you. So there is a little bit of a rabbit out of a hat here. So from this uncountable collection, we picked out one metric. So once you have, you are given, now let's try to compute what the state lines are in this geometry. And then we'll ask, we'll find out whether the first four axioms of Euclidean are satisfied in this geometry and whether the fifth postulate is also satisfied or not. And then we'll have an answer to Euclid's question. So the first thing is, now we have this one simple formula to manipulate with. First theorem says that vertical straight lines. So what is the space you're looking at? The space you're looking at is x, y. So this is y axis, the x axis. H is the half plane. That's called the hyperbolic plane for a certain reason. And now you're computing. So you want to prove that any vertical line is a geodesic, which means what? Which means you take any two points here. So that's x0, y1, and x0, y2. And you take any path between them. You want to show that the length of this path calculated according to this new formula is less than equal to the length of any other path, sigma. So you parameterize sigma as sigma of t is x of t, y of t. So what is l of sigma? So sigma of 0 is equal to x0, y1. So sigma of 1 is equal to x0, y2. What is l of sigma? This is equal to x prime t square. So f of xy, g of xy, both is 1 by y. So 1 by y of t square x prime t square plus y prime t square. And then this is equal to ds square. So you want to integrate from 0 to 1 to the length of this whole thing. In utility, in case you would have just taken this, but now you have to divide it all by this. Now this guy is always positive. So this is always going to be greater than equal to reduce x prime t to 0. So what does that become? This is integral 0 to 1, y prime t by y of t dt. What's that? That's log of y of t, integrated from 0 to 1. This is the log of mod y2 by y1. But what is this? x prime t with equality if and only if x prime t is identically 0. So anything where x prime t has non-zero values somewhere will happen greater. So what does x prime t identically equal to 0 mean? It means that the x coordinate is constant. That means what is the vertical line. So therefore what have we established? We have established that the vertical line from here to here is a huge unit of distance minimizing path. Not only that, it is the distance minimizing path because any other path has greater length. So there is a unique guide to any pair of points which minimizes distances. So first axiom of you could satisfy it. Or at least for pairs of points which are above each other. Now from this collection of geodesics, we want to collect compute new geodesics. We want to join any two points. The two points which don't lie on the same vertical, cannot be joined by a vertical line. So we would still like to join them by geodesics. So what is a geodesic joining two other two points? To do that, we like to get new isometries. So what are examples of isometries? One simple exercise that you can do is you look at f of xy equal to x plus a comma y. So then you look at d of this one. This is x1 is equal to x plus a. So dx1 is equal to dx. That shows you that the ds square and ds1 square are both equal. So these guys are these translations in the horizontal direction. They are isometries. So if you have proven this whole thing for a particular vertical, the y axis for instance, and you translate it around, you're going to get all vertical lines, all of them are geodesics. Good. Now we come to another here. So this was one computation. Now I'll get to the second computation in the stock. And this is actually something that came from the vertical optics, the magnetic optics. So here's a fact that you look at, again, you have your unpoly plane. And this is your y axis. This is your x axis. And you look at, say, the safety circle centered at 00, radius 1. And look at the inversion, the spherical inversion. Assume this is a mirror, and you invert in that mirror. So there's a thing. Oh, why am I getting this in my mind? So what is spherical inversion? There's a picture way of looking at it. Suppose this distance is capital R, then this is 1 by R. And this theta is the same. This is exactly what you get pictorially. If you invert in a spherical mirror of radius 1, the theta quartet remains the same. The radial quartet becomes a recycle, 1 by R. So that's the next claim. Inversions about semicircles. So basically you can do this about a circle of radius capital R. But for simplicity of computation, we do it for radius 1. So there's a way. There's a nice way, a formula for writing this in terms of complex coordinates. So z equal to x plus i times y. This is x, and this is the imaginary axis. And what happens is that you are sending this guy. So the right coordinate for this is the spherical coordinate or the circular coordinate, polar coordinate. For that the complex numbers are better suited. But this is just a computational tool, nothing more. So this inversion is an isometry. That is the claim. What was our formula? ds square equal to dx square plus dy square by y square. And then the complex conjugate is becomes x minus i y. So can you write this in complex coordinates in dds? So this is dx plus i dy, dx minus i dy. What is that? dz times dz bar. And what is y? y is z minus z bar by twice i. So x plus i y minus x plus i y, which is 2 i y, divided by 2 i dy, y square. So you've now expressed this always in dds terms. And what is this inversion in this setup? It's become very simple. So that is our f. z goes to 1 by z bar. z goes to 1 by z bar if the radius of the circle was r. But there's a certain constant, which I keep messing up. Every time I do this computation, so I'll choose chemical r equal to 1. So if z goes to 1 by z bar, what does dz go to? This goes to minus 1 by z bar square d z bar. What does z bar go to? It goes to 1 by z. So dz bar goes to minus 1 by z square d z. Now you plug it all back. So dz, dz bar by z minus z bar by 2 i whole square. This was the ds square on the left side. What does it transform to? Minus 1 by z square d z bar times minus 1 by z square d z divided by 1 by z bar minus 1 by z divided by 2 i whole square. Which is the same thing as this is z z bar. This is z minus z bar. That's what it becomes. This is a multiplication. So what happens? This is minus. This is minus. This gets cancelled. This z square z bar square cancels off with this z bar and this z. So this is equal to dz dz bar by z minus z bar by 2 i whole square. Which means the formula is preserved under this inversion. Good. That means inversions about semi-circles centered at the origin. If you put r square, this is the same computation. What's up? These guys are isometries. But we know that translations in this direction are isometries. You can compose all of this. So actually you can shift this origin. Fixing an origin is something that we attribute to the plane. The plane itself does not care which fellow is there. The plane is much more democratic than us. It has its origin could be shifted anywhere on the real life. So this translation is what tells you that all these points are interchangeable. So really it's not, this is a formula because you set up these coordinates. There's a difference between affine, euclidean geometry and coordinate geometry. Quadrants are something we put in order to get maps, to do computations. But the isometries basically is a way of looking at it physically from the point of view of the plane itself. So inversions about semicircles centered anywhere along the real life because those are things that you can get by translating by isometries are going to be isometries. Using that, now we want to get, so the last thing is that we want to get new geodesics. We know that vertical lines are geodesics. Image of a geodesic under an isometry is another geodesic. What have we got? We've got these vertical lines as geodesics. We've got these inversions about semicircles as isometries. So images of vertical lines under inversions about semicircles centered on the real line are going to give us new geodesics. That's the last thing that we want to do. There's no computation but a simple euclidean geometry problem. This is our picture. So if you...