 In the previous video, we introduced the idea of spherical geometry illustrated right here where our set of points is the unit sphere inside of three-dimensional Euclidean space. Points are just elements of the sphere, the surface of the sphere, not the ball, but just the surface, just the crust of the earth there. Lines are defined to be great circles, that is circles on the surface of the sphere whose diameter is the same as the diameter of the sphere. And then incidence just means a point is incident to a line if, as a set, the point belongs to the set, which is the great circle. So this is the circle's a collection of points right there. Now, when we study the spherical geometry, it does satisfy the elliptic parallel postulate and it satisfies all of the incidence axis except for line determination. It is possible, like in this diagram, that two lines can intersect at more than one point. In fact, if you take any two lines, they always intersect, the elliptic parallel postulate is satisfied. But in spherical geometry, the intersection will always, always, always be two points. This violates a theorem of incidence geometry, which that theorem is essentially just the cause for positive of line determination. If you take a pair of antipodes, points that are on opposite sides of the sphere like the North Pole versus the South Pole here on planet Earth, antipodes always have more than one great circle that connects them together. Like again, if you think of Earth, you take any line of longitude, that's a line from the North Pole to the South Pole on Earth, that violates line determination. So, spherical geometry sometimes called two point elliptic geometry because these antipodes, two lines always intersect at a pair of antipodes there. Because it's not an incidence geometry, if you wanna come up with a geometry, well, if you want an incidence geometry that's infinite, that satisfies the elliptic parallel postulate, it's tempting to instead use the so-called one point elliptic geometry as opposed to spherical geometry, which is the two point elliptic geometry. The one point elliptic geometry, the model we're gonna construct is what's referred to as the real project of plane, RP2. So how do we get this? Well, the idea is we need, we want the incidence axis, line determination is the only one we're missing. So is there a way that we can capture back line determination? And the patch that we're gonna take here, because actually there's two possible patches one could take to try to fix spherical geometry. The one patch we're gonna discuss in this video is what if points aren't actually elements of R3? What if points are actually pairs of antipodes? So we actually consider these two points to be the same thing. Now, this might seem counterintuitive at first because you're saying points are actually pairs of points. So a point actually has the form P minus P. So this is a point in the geometry we're constructing right now. And yes, that's exactly what I'm trying to say. Well, how can I get away with that? Well, I'm not saying that P is a projective point. I'm saying P is a Euclidean point because after all this sphere exists inside of R3. And so P is a point of R3. I'm saying that P is not a point of the projective plane I'm trying to construct right now. Instead, you take the two different Euclidean points, P and negative P, you actually glue them together, you identify them as a single point and you no longer distinguish between them. The way this works is that line determination's fixed because the only violation to line determination on the sphere were by antipodes. If antipodes are now considered the same point, then it turns out that there's no longer repeated intersections. It turns out these different lines are actually no longer in violation of line determination. So basically what we're doing here is the real projective plane is we take the sphere and we glue opposite sides of the sphere together. And so let me try to draw you a picture of the real projective plane in a way that will make life a little bit easier for us. What I'm gonna draw on the screen right now is the so-called hemisphere model for the real projective plane. So basically you're gonna take the upper hemisphere of your sphere right there and you're going to then just take all the points above the hemisphere, so you just have the northern hemisphere here and you're also gonna take the equator here but you're gonna have a wraparound feature on the equator. So in other words, this point right here is equal to its point right over there. So the indipodes are the same. So by dropping the southern hemisphere, we no longer have to worry about identifying points with the north, with the south because we just took the north representation here but on the equator we do have this wraparound feature. So an example of a line would actually be a semicircle that looks something like this where these two points on the boundary are actually considered the same point. So they're glued together. So this line is still a circle in sort of a Euclidean sense but the real projective plane then gives us this model, this hemisphere model here that this then satisfies the incidence axioms now. In terms of three points, yeah, we have three points. There's kind of many points, no big deal. Lines will of course have more than two points. That's pretty easy. There is no line that takes all of the points because an arbitrary line would look something like this and you're missing so many points. You do still get the elliptic parallel postulate because if I take any two points, excuse me, take any two lines, excuse me, they're gonna intersect each other. That's always happens, you can't avoid that. What about line determination though? If I take any two points, is there a unique line between them? So something like this. And the answer of course is gonna be yes. You're gonna get a unique line that's determined by those two points. This whole antipodal problem is no longer available to us because we basically killed off the antipodes by identifying them together. So as a geometry, I should say that the real projectile plane can be difficult to visualize, difficult to see because we want to see it like we see the sphere. But the problem is we really can't do that because RP2 cannot be embedded inside of R3 without some type of self-intersection or some type of wraparound feature. So like this model has this wraparound feature. It's kind of like you're playing some classic video game, like consider the game Asteroids. That's a great game, you know, back in the day, right? You had this little high tech spaceship that looked like in the Saucy's Triangle. You had these Asteroids that you had to blow up. You blow them up, they make smaller Asteroids. You blow those, they make smaller Asteroids. You blow them up, you know, you keep on breaking them until, well, they're always repeating, they just keep on going from high score over and over and over again. The longer you survive, the better. Your spaceship had a propulsion system. You could shoot fire from the back that sets you forward. You could then also rotate your angle. So you combining thrust and angle, you could then move anywhere. Of course, you like hit the space bar and you shoot a laser to break the Asteroids, what have you. The reason I bring this up though is that when you flew off the edge of the screen, you would wrap around from the other side, right? Or if you flew here, you would wrap around here as well. I guess I messed up. If you were to wrap around the screen, you would actually come from this side over here because Asteroid forms a torus, not a projectile plane. That'd be kind of interesting if it did though, but when you go off the screen, you're gonna come back the other side, all right? So in other words, when it comes to a game like Asteroids, Mario Brothers has a similar feature if you play the original one, there's this wrap around feature that when you go off one side of the screen, you're gonna come back from the other side. So you wrap arounds like this. With this one here, you have a wrap around feature like this. So you go and you're gonna wrap around from here, like so. And so the reason why this is relevant, of course, is that if we try to think of this object, it looks like a flat object. If we try to fold it together, we could glue these two sides together and that would form a cylinder, like so, where now the wrap around really is a circle. We traveled like that, a line is a circle in that situation. But then when we glue these two ends together, we end up with actually what's called in topology as a torus, it's the shape of a donut, right? You glue the cylinder together. And so this flat object right here with the wrap around feature really is a donut in three dimensional space in R3. Now, you might try to do that same thing with the real project of playing. Let me back up a little bit. But the wrap around feature, if you're playing that, it looks a little bit different. The wrap around feature actually looks something like this. Notice how my arrows have reverse orientation. What this means is as I go on this side of the screen, it actually wraps around with a reflection. If there's a wrap around with a twist, you don't come out from the top like you did on the torus, you actually come out down here. And then when it comes to the top of the screen, the same thing, you wrap around like this, you're gonna get something like this. And so this square with these type of wrap arounds where there's a wrap around with a twist to it, this actually gives you another model of the real project of playing. I want you to think of this model right here where we took just the upper hemisphere with this gluing feature. If you were to basically take the top of the hemisphere and you flatten it, what you would do is you would fill in a circle that looks like this, for which you now have a ball that's filled in, the disc, I should say it's filled in, you have all the points in the middle, but you still have this wrap around feature, okay? For which that when you go off the screen, when you leave the boundaries here, you wrap around to the antipode like so. Now this boundary is a circle, but if we were to straighten it, kind of straighten it a little bit, we could turn that into a square like so, because in topology, we don't really care about sharp corners versus round corners or anything like that. Smooth doesn't really make any sense in topology, we just care about continuous. So you can continuously deform a sphere into a square. So these give us different models of the project of playing, none of these models allow us to draw it without this wrap around feature, because essentially, projected geometry is a little bit, it's kind of weird compared to some of the other models we have here. In this model we have right here, there is this wrap around line, we keep on seeing this wrap around line in this model as well. This line is often referred to as the line at infinity, the line at infinity, because once you hit this horizon, you actually wrap around from the other side. And so we're gonna study elliptic geometry much more in the future. This line at infinity business is actually the reason why it's called projective geometry. This is actually a statement coming from art, where if you wanted to like draw parallel lines as they go off towards infinity, like you have like train tracks, well actually the train tracks that you look at, they're parallel lines, right? Otherwise the trains don't work very well, but as they go off towards infinity, they appear to intersect each other at the horizon, right? This horizon line we think of as the line at infinity and all parallel lines intersect themselves at infinity. And so while these lines are genuinely parallel in the Euclidean sense, as they go off towards infinity, they appear to intersect. And so when one draws a picture of a horizon, you have to draw things in perspective there. It's that projective nature where you take the three dimensional curvature of the earth and project it to two dimensional drawing, it looks like the parallel lines intersect at infinity. And so there is a lot of history to the projective plane with regard to, of course, geometry, art as well, many applications of the projective plane here. I want to introduce it to you as this one point elliptic geometry. This is in fact an incidence model, an incidence geometry that satisfies the elliptic parallel postulate. And so that's pretty nice. And as we did, we did actually study while we introduced the idea of a projective geometry earlier. Remember, a projective geometry was an incidence geometry that satisfies the elliptic parallel postulate and all lines have at least three points. The lines here have uncountably many points. So this is in fact a projective geometry. It's a projective geometry that locally behaves like the Euclidean plane R2. And that local behavior is a really important thing. I said there's two patches to ellipt, to spherical geometry. One is to turn it into an incidence geometry by identifying antipodes together. They gave us the real projective plane. I gave you a couple models of the real projective plane. There are so many others that I'm not gonna introduce, I'm not gonna introduce right now. In the next video, I wanna introduce an alternative patch to spherical geometry that'll make more sense, of course, in that video. But we've alluded to many of those topics already in this video.