 You have this vertical line and you want to invert it about some semi circle, some semi circle. So, let us say again this one, so assume I mean if you look at the semi circle, look at a vertical line, one will be, one phase would be if you invert it about a semi circle centered around its base, then you get this, then the line back, the top will go to the, it is tough inside, the inside will go to the outside, you will not get a new line. Okay, then the point of a circle which is good, but not as good as the center is a point of tendency, a preferred point. Let us try that out, again you are playing these games, just trying out various things. So, you have this semi circle, change this vertical line here, this is your L, this is your sigma and I want to compute the inversion, so L invert it about the semi circle. What is that? That is my question. The answer is that this is what was known to spherical people working in spherical geometry, it is actually the semi circle centered at half, zero, half the radius and of radius equal to, so the radius is equal to half of the original guy, the center is shifted in this direction. Why is that? So, in order to prove this what we have to do, we have to basically prove that this is x and look at this, this is x. So, this is our sigma, take little x, invert x about the sigma and you get this new point y, so distance, so this is o, so length of o x is equal to capital R, given, want to show, so to show that length of o y is equal to 1 by R. That is what I want to show. Then I will know that that is the inversion, that is the point that I get under inversion, so then the locus of the inversion as we move along the cell is going to be about the same sense. So, now this is a similar triangle thing, actually what we will prove is that this is, so this is a b, this is a, this is b, this point is c and there is this point which is b. So, the only thing is just draw this here, this is an angle in a semi circle, so that is the right angle, this is the point of tendency, this is the right angle. The angles, give this the name a, b, e, a, b, e and a, b, c are similar, the right angle and there is an angle commons so the third angle is also constant. So, now apply that, so what you get, the hypotenuse is a, b divided by what is the what we use it is AC this is equal to angle this I want to take this side so point this is this angle is equal to this angle towards the side opposite this this is AB I want the side opposite this is AE so AE square equal to AB times AC what is AE that is R R is the radius so R square is equal to AB times AC that is our inversion formula the product of this length with the big length is equal to the square of the radius which means what which means that if you have this vertical line and you invert it about a semicircle which is tangential to its base then using this little bit of Euclidian geometry you get that the image is another semicircle which is also centered on the real line whose radius is half the radius of the semicircle you started off with good now have we gotten so yeah this is this is really the upshot of this computation length of AB times length of AC is the length of the radius square means the radius square so we have got a whole new bunch of gear assets semicircle centered at the origin in the the real axis actually we have got a all semicircle centered on the real axis based centered on the real axis in the process why now invert the problem the reverse engineer start off with a semicircle this is the semicircle which contains AB take a semicircle which is double that size centered at A and draw that semicircle yeah now take this vertical line at the end so this is just turning the logic around and then you know that the semicircle you started off with the AB semicircle is the image of this new vertical that you constructed therefore the semicircle you start off with this AB semicircle is at your asset just your problem what so now what you have what have you proven you have proven that all semicircles centered on the real line are also geodesics correct and now now do you have enough lines to enough parts to join any pair of points in BDS take any pair of points what we want to do now what we want to do is take any pair of points on the on the aqa half plane what do you want to do given any pair of points here if they are on the same vertical turn them by a straight line vertical line if they are not on the same line what do you want to do you want to have a semicircle centered on the real line which passes me through these two points that simple join these two points take a perpendicular face actor in the euclidean sense and my points of intersection are always off so I've made that point a little thick and so that is that's your new center and to draw your semicircle so this semicircle is centered on the real line passes through both these points so any two points can be joined by a geodesic but the vertical lines by unique geodesics so run this isometry machinery once more and just a logic once more and you'll see that this semicircle is the unique geodesic to any addition right so we will form all geodesics in this new geometry okay what are their vertical lines is the semicircle centered on the real line so we've got all the bi-infinite lines yeah Euclid's fifth postulate deals with lines it has nothing to do with circles or angles right so now we can attack the fifth postulate question now it's a question what is the question given given any line so now what is the question so now choose one of these lines this is our line L and you have a point P outside it how many geodesics are there through P not meeting L what are geodesics semicircles centered on the real line or vertical lines so certainly this line is there there is one chop already that is L prime question is are there any semicircles other semicircles that you can drop passing through P centered on the real line not meeting L oh indeed yes all that you need to do is assume that it passes through yeah I wanted to pass through this so again do the same thing so here you look at this guy right so for every point between this and at the point that you start off with this whole interval for every point you can take that thing join it to P it was perpendicular bisector all the same right so what does it mean you have an intervals worth of lines parallel to the first one all passing through P interval is uncountable yeah so it's the most dramatic breakdown of the uniqueness part of the parallel postulate what does it say what have we proven actually so in the hyperbolic plane vertical straight lines and semicircle arcs with center on the real axis are geodesics this gives rise to the following new parallel postulate given a geodesic L in H and a point x okay so this x has gotten called P here flying outside L there exists infinitely many what are in what is infinite uncountably infinite here lying on H passing through X and parallel okay so the unique axis is broken down what have we proven therefore that the first four axioms of Euclidean geometry are consistent with unique line they are also consistent with infinitely many lines these two fellows don't talk to each other to a point a single line passes through a point infinitely many lines pass those guys can't talk to each other but you can take any of these contradictory guys put them there and you'll still have a consistent system of time in fact what Riemann did is that you can remove the fifth postulate all together throw it out of the window it's up no use basically if you put it in and modify it in some form there is no line is also possible 0 is also possible existence can be contradicted it's called spherical geometry a little tweaking has to be done but anyway still possible no line is possible one unique line is possible infinitely many lines are possible and nothing is also possible what is nothing nothing means sometimes there is one sometimes there is 0 sometimes there are infinitely okay so that means that the fifth postulate has absolutely no special status if you can discover geometry where the postulate is needed okay thank you very much what about hyperbolic triangle oh I should say one thing so maybe this is a question I'm asking why why it's called hyperbolic and that's a question that everybody should ask what does it have to do with hyperbola and actually that came much later so it does not have anything to do with the hyperbola it has something to do with the hyperbola idea so if you look at I mean and this is where it ties up with physics so if you look at so when Einstein put space and time together the metric does not it does not remain this positive definite metric which was given here it becomes ds square equal to dx square plus d y square minus dd square plus ds square whatever so if you have this three dimensional space two space dimensions one time dimension and you have this minkowski metric ds square equal to dx square minus dd square and you take the unit sphere there what are the units here x square plus y square minus d square mathematicians do not give any particular special status to type so x square plus y square minus x square equal to one maybe minus one that's the unit sphere so that's the hyperboloid right so you look at the minkowski metric restricted to the hyperboloid you are going to get exactly something isometric so that's why it's called the hyperboloid in fact actually yeah I think that model using the hyperboloid is using the hyperboloid was used by Klein extensively so it's called the Klein model also yeah so okay so about triangles what can I say some of the angles is strictly less than 180 and the area of a triangle is equal to pi minus the sound of the triangle so what do they look like okay I can draw some draw an example they all look very similar this is what they look like you have a semi-circle here yeah you have another semi-circle here you have a third this is what triangle is sure this is the thing is vertical and horizontal is again our Euclidean perspective yeah that is that is I mean the high quality geometry does not say that this vertical lines are special and you want to understand high quality geometry in terms of Euclidean geometry that is why there is a vertical direction and a non vertical direction from the you see if we were hyperbolic which is living inside hyperbolic plane you would not have this choice of something is I mean because of gravity there is a vertical horizontal yeah I mean that the mathematical axis nothing to do with gravity so if you kid I mean if this this verb was stated and there were no ends where is the horizontal line so there is no I mean there is nothing special here so if you had this one you have this this is an example okay the high geometry and this you can flip it over I mean so this triangle and this triangle may be this Euclidean geometry then come as a special case of no I mean it can't right so because there are infinitely many lines here and there's one and only one line there what happens is that there is a certain sense in which you can take a limiting case of hyperbolic what do you do basically I had written this metric here right ds square equal to dx square plus dy square by y square you start blowing up this metric so you start multiplying this by say epsilon n so what does it mean you are passing more and more so I mean distances capital N square let N tend to infinity yeah what is happening is that so there is a motion of curvature yeah so the sum of the angles of the triangle here if you take the largest yeah okay so there is a what is happening is that this triangle which was our certain diameter in the original hyperbolic in this guy it's going to have diameter in times and so which means what all the whole space is running away from you if you have if you choose your tools to be at a particular base point in the limit you will be standing on the so in that particular sense the Euclidean plate is a limit of that it's also the limit of oh the notion of angle is okay so that's all yeah that's all there so what is angle I mean angle is if you want to look at an angle so you can complete I mean physically compute angles between tangent directions right so what are we really doing we are taking a point we are looking at tangents to that the collection of tangents actually forms a Euclidean space then we can look to our inner product compute the angle that's all that stuff so what is going to happen you take this point of intersection you take this tangent direction this tangent direction and compute the angle between them in the normal Euclidean sense that will turn out to be exactly the angle in the hyperbolic matrix also why because actually basically if you take two lines to a point and you scale the angle between them does not change right so this is called a conformal change basically every point every point the x direction and the y direction are being multiplied by the same number look at this function this y is scaling both the Dx direction and the dy direction by the same amount so conformally in terms of angles the angle in the hyperbolic plane and the angle in the Euclidean sense so the notion of angle certainly makes sense and through every point I mean all right angles angles are right angles I mean they are all equal to each other that's even all right that is what I mean oh here from the formula it's very simple why goes to zero you can't have that right if y is going to zero right that matrix is blowing up so there is a single line then there are two pieces then you will be doing geometry on the top geometry at the bottom in a parallel universe and the two will not communicate so there is no there is no geodesic that goes from the top right to the bottom no no that is not an accurate statement then you will get the same geometry on the other side yeah my question is like in the last slide like you have mentioned that all the geodesics are running you mentioned the by finite geodesics by infinite geodesics yeah but I can understand vertical geodesic being infinite but a semi-circular geodesic why is it very good question very good question no you can imagine that the vertical line is infinite why because it is going off see this is an example of again our Euclidean schooling this is going off to infinity in the top direction where is it going in the bottom direction complete the length there log of it you start off at the point one yeah and you start going you take your point going off to zero what is the length log of one by epsilon that length is going off to infinity right and now what has happened here with the semi-circle is that the semi-circle is a much better fellow than the real life why because it treats this step and this step equally they are both going off to infinity and they are not not looking as if they are going off to infinity okay so I have another question like how does this look like if I project it into another dimension like imagine that so yeah there is a hyperbolic free space all that you need to do is put dx square plus dy square plus dz square and then you got then xyz is a restriction of the human alphabet you put x1 x2 xn that's n-dimensional hyperbolic space divided by xn square is there some way to visualize it like sure there's somebody called Bill Thurston who is a mathematician who actually after Pankare there's a new person who sort of brought it back in this very tactile way by which you can manipulate so it's been done I mean the field has actually reached in some sense and all the major problems got resolved over the last four or five years just just now thank you that's only dimension three I think we are still waiting for a four-dimensional test yeah so you don't have the problems are very different okay so three-dimensional have you heard of somebody called Rishabh Peralman for instance okay so this is a mathematician called Rishabh Peralman who won the fields medal but refused it and he was offered a million dollar prize refused it because he is well that's the shortest and dishonest way of saying but yeah so he thought perhaps I basically said that you take a generic three-dimensional structure you're called three-man books it has a hyperbolic structure so with probability one the universe we live in is not equally so if there is no other extra assumption yeah so suppose we bring this here I'm by three-dimensional space here by him but says that you cannot take a complete hyperbolic object and embedded in three-dimensional flat Euclidean space there are from 1901 and suppose the number of parallel lines is reduced to zero not a hyperbolic like suppose from generating manipulate stuff what do you mean by reducing number of lines to zero suppose suppose there's a line and we have a point there is no such line to that point which is parallel to this line are you imposing this as an axiom no we want we want a plane which holds this property like yeah yeah we wanted so basically what you're saying is what is an example of a geometry where the number infinitely many number one is replaced by zero is that your question no I want to say that can that type of plane be brought into the space yeah there's the sphere but this cannot be called no this is the theorem of him but in fact actually all planes follow great circle roots right why because the great circles are the shortest and two great circles certainly meet at two points so they meet at two points not one point but then but it's certainly true that given any great circle given a point outside the great circle there is no great circle through that point which does not intersect the first kind of like a primary question in the second theorem you prove that all vertical lines and semicircles centered on the real axis are yeah they are isometric sorry they are geodesic so sorry in the second theorem you prove that all inversions using such semicircles are geodesic sorry I saw what is this way these are your objects and these are your maps so objects get mapped to other objects what goes in between them is the relation that's the function so my question is your second theorem was that all translations and all inversions using such semicircles at isometry sorry then that doesn't how will you prove the converse that any general isometry I did not say the converse the converse is false any state but does not have necessarily mean that the converse has to be true okay so what is to I tell you what the what is any isometry can be described as a composition of three of these three at most this is another fact it's also a nice nice thing you can work it out all it involved is euclidean geometry so have you heard of momius transformation that goes to a z plus b by c z plus d yeah so what are the so composition of two of these inversions are going to give you z going to a z plus b by c z plus b where a b c d are here yeah and modulo this one reflection in the plane so if you if you are given a line and you say that the two sides are preserved then the isometry of the hyperbolic plane are precisely equal to the set of mobius transformations and place of this most transformations is really the context from which that metric comes up okay so the motivation for picking that metric out of this uncomfortable act is basically because conquering was looking at solutions to differential equations in complex numbers complex and so he was trying to find complex analytic transformation that preserve various and he found apparently when he was getting on to a bus that the hyperbolic isometry is all this hyperbolic geometry that he was thinking of and this complex analytical stuff that was coming from differential equation there are the same things