 So in this video, I'm going to keep my promise and talk about hyperbolic geometry and our previous videos for lecture nine We've been talking about non euclidean geometries in particular. We've been focused on this idea of a manifold a smooth manifold I should say it's a geometry for which locally is euclidean, but globally could be something different Can have some type of curvature to it. Maybe it's not flat. We've explored some options of elliptic Geometry spherical geometry and the projective geometry real projector geometry as examples Those both satisfy the elliptic parallel posh it all lines intersect each other globally Even if they're do exist neighborhoods for which they look parallel on So I want to explore the other parallel Other parallel alternative the so-called hyperbolic parallel posh it The hyperbolic parallel posh it tells us that given a line and a point off the line. There's at least two parallel lines going through that point Can we construct a manifold that satisfies that condition? Because after all like handshake geometry or finite geometries that satisfy the hyperbolic parallel posh it that's great But what people really care about are their manifold smooth manifolds that satisfy the hyperbolic parallel posh it So that's really what we mean by a hyperbolic geometry a Manifold with the hyperbolic parallel posh it satisfied. Well Let me give you such an example. So we're going to call h2 That'll be short for the hyperbolic plane We're going to do a euclidean model to make it a little bit easier to see here So our lines are going to look a little janky But that's because these aren't going to be euclidean lines. There's going to be hyperbolic lines We can define h2 to be the set as a set of points It's going to be the upper half plane That is we're going to take things that are above the x-axis in the euclidean plane We do not include anything below the x-axis nor do we include the x-axis itself It doesn't belong to our geometry just the upper half plane So we want to collect all of those euclidean points such that the y-coordinate is positive for this model. We call this the um We call this the upper half plane model. Okay So those are the points in our geometry. What are the lines in our geometry? Well, it turns out we have to just we have to define two different lines If we have a point which has if we have two different points Which had the same x-coordinate, but their y-coordinates different Then we're going to define the line between them to be the vertical ray that emanates from the x-axis It's perpendicular to the x-axis and contains those line those two points right there So vertical lines are considered lines in this geometry But it's really a vertical ray because we don't get anything below and actually this point on the x-axis is a point at infinity We don't have that point as part of our geometry So if two points are on the have the same x-coordinate, there's a vertical line that connects them We consider that a hyperbolic line now for any other point Right if we have any other point in our geometry, then we define the line To be the semicircle whose diameter Rest on the x-axis that the arc the semicircle arc that goes through these So you get something looks something like this, right? So this semicircle is then Considered a hyperbolic line the hyperbolic line determined by these two points It's a semicircle So we only get the upper semicircle and in fact these two points on the x-axis They are not part of the geometry. So we get this circular arc right here It's an open arc. We don't include the end points. So this gives us two types of hyperbolic lines And then incidence is exactly you would expect points are just elements of a set Lines are collections of points. So incidence is satisfied in the usual The usual aspect that you would expect right there All right, so this this set of points with this idea of lines and incidence gives us the real hyperbolic plane and this is in fact a It's going to be a hyperbolic two manifold because locally Things look like euclidean space if you were to take the neighborhood of a the neighborhood of any point You could distort it in such a way that it looks like the usual xy Axis the euclidean plane like so and again, we're not going to go into all the details there But this does in fact it was a hyperbolic two manifold It satisfies all the incidence axioms I told you how to determine too long a determine a line between two points line determination satisfied We have three points every line has two points and no line has all the points It's an incidence geometry and it also satisfies the hyperbolic parallel postulate. How do we see that? Well Consider the following Here's a line l actually take a vertical line for example, but this model could easily be done Uh, excuse me this example could also be done when using a semicircle as a line No big deal But we have a line here We have a point off of the line and look there are multiple you actually see three Lines passing through p that are parallel to l because parallel just means they don't intersect each other So this point has at least three There's actually Uncontrollably many because I can make these semicircles get larger and larger and larger As long as they pass through p and in fact, there's also the vertical line that goes through it as well Right. These are all lines that pass through p. They're perpendicular parallel to l excuse me Um, and then if we wanted to find lines like here's another point over here some lines that are parallel to this one right here Oh, well, we have this vertical line. That's one. We have another semicircle. We have another semicircle We have another semicircle You have uncountably many parallel lines going on here. So this is a hyperbolic Model and so this model right here is commonly referred to as the pancreas Half plane model kind of like the real projected plane There are other models of hyperbolic geometry and we'll explore some more in the future, of course But this half plane model is a good one to use as we go forward as we think about hyperbolic geometry Think about this model right here