 Welcome back for one short segment to finish up lecture 36 here. What I wanna actually talk about is the limit of triangles in hyperbolic and elliptic geometry and specifically how this relates to the angle measure of those triangles. We know that in Euclidean geometry, every triangle has an angle sum that adds up to be 180 degrees. And so it really does, since this angle sum is constant, it doesn't matter how we distort the triangles, we're always gonna get this 180 degrees. So as we start like pushing vertices together or pulling them apart, that is we're taking the extreme of what could happen with a triangle. This angle sum never really changes whatsoever. So let's think about this in the hyperbolic realm just for a second. What happens in hyperbolic geometry as we distort our triangles? Well, so let's think of this maybe with the disk model at hand and we have some hyperbolic triangle like so, A, B, and C, like so. Well, what we know about the angle sum of this triangle, the sum of ABC, since it's hyperbolic, it'll be strictly less than 180 degrees, but it'll be greater than zero degrees. Can these limits actually be obtained? We say we're less than 180 and greater than zero, how close can we get to these things? And we can get arbitrarily close and I kinda wanna illustrate how this would occur. So to obtain the 180 figure right here, cause the idea is the smaller, so to speak, the triangle is, the closer it'll be to a Euclidean triangle, that is the difference between them will be harder to tell. So the smaller the hyperbolic triangle is, the closer it'll be to 180 degrees. And so what one can accomplish is if you start to let this, you keep angle vertices A and B fixed, so don't move them. But if you were to continuously deform this triangle, sliding angle C closer and closer and closer to the line AB, what's gonna happen to angle C right here? Well, angle C would continue to get bigger and bigger and bigger. Angles A and B would continue to get smaller and smaller until that limit is obtained where essentially you're just gonna get a line segment, A, B, with C sitting between it. In that situation, the measure of angle A would actually be the same as the measure of angle B and those would both become zero degree angles. But on the other hand, your angle C, it's basically turned into two flat angles. You have angle ACA and angle BC. Yeah, I mean it's a kinda weird what's happening but essentially angle C has become a 180 degree because it's become a flat angle in that situation. Again, it's kinda weird describing this thing but that's the idea is we're taking the limit as the thing converts to each other. And then of course as you continue to push these points towards each other, this thing will eventually squish it just to a single point, right? That's another way of thinking of continuously shrinking your hyperbolic triangle. So this 180 degree measure can be obtained when your triangle degenerates into a segment or to a point, a point right there. Well, is there a way of doing this for the zero bound below? How does a triangle get a measure of zero? Well again, a triangle's not gonna do that but the limit of a triangle can. So again, consider the disc model right here and if we have our triangle like we did before, ABC, something like this, A, B, and C. Well, we can distort the triangle by actually sending the vertex C far away, send it away from A and B. Well, what kinda happened in that situation? Well, because of the finite area of hyperbolic geometry, we would end up with a picture kinda like the following, B. So we left A and B fixed but then we allow C to go off towards infinity like so. And so this right here isn't exactly a triangle anymore. It's not a triangle because there aren't three vertices. Notice with the hyperbolic model here, these points on the boundary circle right here are not part of the geometry, these are points at infinity. So if we think of this as a geometry that lives inside of Euclidean geometry, although the measure and such is completely different here, we can see this point at infinity as Euclidean point, but notice it's not a hyperbolic point. This right here is what's commonly referred to as an asymptotic triangle. In many ways it behaves still like a hyperbolic triangle. I'm not gonna, I won't sort of verify that right here, but things like the exterior angle theorem, the secarioligional theorem still apply in this situation. So the angle sum's still less than 180 degrees and things like that. And some really interesting things can happen for these asymptotic triangles, but this is starting to get closer. We still haven't quite hit the zero degree measure yet, but as you send a point to infinity, some really funky things can start happening that it kind of still behaves like a triangle, but there's still again some funky things that are going on with this thing. You'll have an opportunity to explore these ideas of an asymptotic hyperbolic triangle very soon. But we can also kind of repeat this process for what happens if we send A and B towards infinity here. Let me clean this up real quick. If you allow both A and B to go to infinity as well, C stays at infinity, you'll get a picture looks something like the following maybe, where you now have three points at infinity, all the vertices of this so-called triangle are at infinity now. We might call this an infinite triangle or we'll call it an ideal triangle. It's not really a triangle because there are no vertices. All of these vertices are at infinity, right? But in terms of the measure, one can make sense of the measure. And when your vertices goes towards infinity, the measure became zero. That was true for that asymptotic triangle before. Although an asymptotic triangle could still be, it could still have an angle sum that's positive because angles A and angle B could be positive angles, but for this ideal triangle, you can actually get that all three angles go to zero. And so this would be the limit of the bounds of what a hyperbolic angle sum could be. So this bottom side here, you get at this ideal triangle. And notice this picture represents sort of the largest possible area that the hyperbolic triangle could get. It's not actually a triangle, but your area of a hyperbolic triangle is bounded. And so this ideal triangle is what happens as we allow this thing to go towards infinity. All right, well, that takes care of hyperbolic geometry. What kind of happens in elliptic geometry? Can the same things happen here? And so let us clean up this picture one more time. And so think of what happens as we push to the limits, so to speak, in elliptic geometry as well. So we have elliptic geometry here. And for the sake of drawing, we'll think of this as the spherical model. Although analogous, things could happen in the projective model of hyperbolic geometry. So imagine we have a triangle and for ease of drawing, we'll make it into a double right triangle. Anyways, fun things happen, of course, in that situation. So we have our triangle A, we have vertices A, B, and C. We'll say C is the North Pole and it's a double right. So what would happen if we allow C to converge towards the segment down below? Well, you can imagine kind of what would happen is you're just gonna end up with this line segment with C sitting between them, A and B, like so. And again, kind of like we saw before, the limit, if we look at the limited angle sums, that'll converge towards 180 degrees. And so that's much like we saw with the hyperbolic setting as well. If you take the sum of this elliptic triangle, ABC, we know that it sits between 360 degrees and 180 degrees. And much like the hyperbolic setting, this 180 degree measure will be obtained when this thing approaches a segment. Oh, of course, you can also squish it down to a point if you want to. And so this is gonna represent a small, a small elliptic triangle will have a angle sum close to 180 degrees because its excess will be super small in that situation as well. That's very similar to hyperbolic geometry. The smaller you get, the closer you are to a Euclidean triangle. But what about the other limit, 360? Well, we can see by using the excess function, this is gonna happen when you have a large triangle. What does a large triangle mean in hyperbolic or in elliptic geometry right here? So because of the polar distance idea that exists inside of elliptic geometry, in some essence, we can't really shove C any farther away than it already is with this double right triangle right here. So what we kind of want to see next is what happens if we allow B kind of, we'll keep C, C fixed and A fixed for this discussion. But if we allow C or B to kind of move to the right, you can move it so you get something like a picture going on over here, over here, over here. And eventually, eventually you're gonna reach the opposite side of the sphere and you get basically angle C, right now it's like, who knows, maybe 60 degrees or something like that. But as we move B around, that all allows C to get bigger, bigger, bigger, bigger, bigger until eventually it reaches 180 degrees. And in which case, if I try to sketch that picture, well, we'll just think of it this way. We'll change our orientation a little bit, but if we have our sphere again, what can eventually happen is you stretch out angle C bigger, bigger, bigger, is you can get something that looks like the following. You're gonna get these two points which aren't, they're antipodes of the sphere. We'll call them P and negative P right here. And so we could take two lines that go through P and negative P, right? Because we're in spherical or elliptic geometry, it turns out, well, in spherical geometry, two antipodes, there's actually not a single line that connects them. There's infinitely many lines. Think about the globe. We have all these different longitudinal lines that connect the north and south pole. If you take any two of those and think of the region that is colored between this, right? This is what's commonly referred to as a loon or sometimes it's called a bi-angle because there's only two angles that determine this thing right here. And so a loon is essentially a two-sided polygon that exists in elliptic geometry. And this thing is well designed, well crafted, right? This is a legitimate polygon, right? It's made by straight lines, although those straight lines are elliptic in this case, but it's two straight lines and two distinct points that form the vertices of these things. This idea of a loon also makes sense in the projective model of elliptic geometry, although in that situation, you wouldn't necessarily call it a bi-angle because there's only one vertex because the P and P, negative P get identified together. That's why I'm drawing this as an elliptic, as a spherical model, it's a little bit easier to see. And it also kind of explains why we call this a loon here because if you think of the phases of the moon, these loons represent sort of different phases you could see on the moon here. And so the angle sum of this loon is gonna correspond to be the sum of these two angles right here. Although admittedly your third angle of the triangle actually became a straight line, which is 180 degrees. So the angle sum of this loon is gonna be 180 degrees plus the measure of angle P and the negative P here. But as these angles are vertical angles to each other, try to convince yourself of that, you're gonna get that the measure of angle P is always equal to the measure of angle negative P. And so the angle sum of a loon is gonna be 180 degrees plus two times the measure of angle P. And so if you allow, we'll draw, I'll draw one more picture of this. If you allow the angle P to go off towards a right angle, so basically you take off a whole octant of our sphere right here. If you allow a right angle right here, then this would be something that actually obtains a angle sum of exactly at 360 degrees. Cause the idea is when you go past the right angle, you're making bigger loons, but you're also making smaller loons. I mean, so you can actually get, so a loon is the limit of an elliptic triangle. I actually can hit this 360 measure, but be aware that the loon itself, actually its angle sums will sit between 540 and 180, you can get loons that are pretty small, but you can get some loons that are pretty big. They can go all the way at the 540 degrees right there as well. And so that kind of just ends this quick little discussion about limits of triangles in hyperbolic and elliptic geometry. When you cross the triangle down, it becomes a segment or a point. That kind of is clear. That's fairly simple for all of these geometries. That'd be true for Euclidean geometry as well, but as we start stretching these triangles out to their limits, so to speak, I just wanna sort of show you this idea that in hyperbolic geometry, you can create these asymptotic and ideal triangles. In elliptic geometry, you can create these loons, which in some ways still behave like triangles because they're limits of triangles, but in many ways, they break the rules and give you something that's a little bit different. All right, everyone. That actually I think concludes our series. I suppose there's always the possibility in the future I could add more lectures to this series if time allows for it, but for our standard semester at Southern Utah University, this would be our final lecture. It was great having everyone participate in this lecture series. If you do like what you've seen here, as always, please, please, please like these videos, share them with friends who might be interested, subscribe, post any questions you have below in this video or any videos you see here. And let me know. And if there's any extra content you wanna see in the future, let me know and I can try to create those videos for you. Have a great, have a great future, everyone. I will hopefully see you in another lecture series sometimes, bye.