 In this video, I wanna conclude our discussion of non Euclidean geometry. So we've explored some manifolds, elliptic and hyperbolic to give some examples of things that locally behave like Euclidean geometry but globally can look a little bit different. All right, and the main difference between these different manifolds is their parallel postulate, Euclidean, elliptic and hyperbolic. Are parallel lines unique? Are there no parallel lines? So parallel lines don't exist. Or maybe parallel lines are multiple, right? There could be a plurality of multiple parallel lines in your geometry here. So the existence, and I don't, so the existence or non-existence, the multiplicity of parallel lines is a direct consequence of the parallel postulates. But I should also mention that there are other consequences of the parallel postulates. I want you to consider the following three diagrams you see right here. So it turns out that in, and it's kind of, might seem surprising at first, but it turns out that of these three geometries, Euclidean, hyperbolic and elliptic geometry, only one of them has a rectangle. That is a quadrilateral with four right angles because that's actually what the word rectangle means itself coming from the Latin. It just means right angle, okay? Rectangle there. And so rectangles are actually, they seem quite ubiquitous in Euclidean geometry but actually rectangles are a Euclidean feature. If you go to an alien planet where the geometry is now hyperbolic or even elliptic, you're not gonna find rectangles. They don't exist, which is a very strange thing. Now I'll give you some more explanation of it in a moment, but when you try to construct things that are similar to rectangles, this is a topic we'll study a little bit later, the idea of a security quadrilateral. When you try to construct security quadrilaterals, which these are quadrilaterals which have two right angles on the base, right? You get things that look like the following in hyperbolic geometry. The curvature goes inward, right? And then when it comes to elliptic geometry, you see this curvature going outward. There is curbing going on, but the direction is different. These angles right here are called the summit angles and Euclidean geometry, the summit angles are measured at 90 degrees. In hyperbolic geometry, the summit angles are actually acute. They have an angle measure less than 90 degrees. In elliptic geometry, these summit angles are actually obtuse. They have an angle sum greater than, excuse me, these angles are obtuse, they're greater than 90 degrees. And this is significant because if I take two right angles and I take two obtuse angles, I want you to notice here that the angle sum of this quadrilateral will exceed 360 degrees. You don't see that in Euclidean geometry. If you look at the hyperbolic quadrilateral here, you have two angles that are acute, two angles that are right, the angle sum there is actually going to be less than 360 degrees. That's kind of strange. It's actually in the Euclidean geometry where we have the principle that the angle sum of a quadrilateral is 360 degrees. And so it turns out that rectangles only exist in Euclidean geometry because the angle sum theorem, angles add up to 360 degrees in the quadrilateral, that is actually a Euclidean result that's proven by the Euclidean parallel postulate. And if you don't have the Euclidean parallel postulate because maybe you are hyperbolic or maybe because you're elliptic, then you don't get it. And it seems really strange if you're not used to these things. But after all, let me kind of change the direction a little bit here. A rectangle is a special type of parallelogram. Ah, a parallelogram is a quadrilateral or opposite sides are parallel to each other. So if you have an axiom about parallelism, different axioms of parallelism could affect parallelograms. Could they not? Yes, they could. And that then starts to leave like, okay, I can start to see now maybe rectangles do have something to do with parallel lines because after all, these lines are parallel to each other. These lines are parallel to each other, right? And the least that's how it is in Euclidean geometry. Are these two lines parallel to each other? In hyperbolic geometry? Well, I can tell you that in elliptic geometry these two lines are not parallel to each other. And the idea is if I were to extend these lines with the direction we have here, the curvature seems to suggest that, yeah, eventually they're gonna intersect each other. On a sphere, they absolutely would. In hyperbolic geometry, these two lines will be parallel to each other. Notice how the curvature, it bows away, it curves away from this line. These lines will never intersect each other because of the curvature. And so that's significant. But it turns out there's gonna be another parallel line we could come up with, but that's a topic for another day. Talk about this issue of angle measures. I want to tell you another property of non-Euclidean geometry. In Euclidean geometry, we are familiar with the principle of the sum of all three angles of a triangle is equal to 180 degrees. That's what I was alluding to before when I said that the quadrilateral angle sum is 360. A quadrilateral is just two triangles glued together. So the angle sums become double that of a triangle. Two times 180 is 360. A Euclidean triangle, always the angle sum is always 360, excuse me, 180 degrees. That is a Euclidean property. In hyperbolic geometry, because of the curvature, you actually see the curving, it curves inward towards the middle of the triangle here. And as such, these angles will actually give you something strictly less than 180 degrees. So this is an example of what looks like an equilateral Euclidean triangle. This is an example of an equilateral hyperbolic triangle. These angles are not 60 degrees. They're smaller than that. And a weird phenomenon of hyperbolic geometry is I can make the angle sum get smaller and smaller and smaller. I can make a triangle in hyperbolic geometry whose angle sum is less than one degree. I can make a triangle whose angle sum is less than one one-hundredth of a degree. I can make a triangle whose angle sum is one one-billionth of a degree in its size and you can make it arbitrarily small. Never equal to zero, but you can make it really, really small. And in terms, it's always less than 180. I can also get arbitrarily close to 180. And this is what's kind of weird in hyperbolic geometry. A small triangle will have an angle sum that's close to 180. But a big triangle will have an angle sum that's really close to zero. And so when your triangle is small, it looks like a Euclidean triangle. This is the business with manifolds we said earlier. Locally, it looks like Euclidean space. But globally is where you can see the difference. And so if we're just a small speck in the universe, it's very hard to tell the difference between a hyperbolic and a Euclidean triangle. Because in that situation, your angle sum is approximately 180 degrees. But it is less than 180, but it'll be close to 180. And the smaller you are, the closer it is. In which case, it might be so close that our instruments can't measure the difference. Yikes. What would an elliptic triangle look like? My diagram doesn't have this, but it bows outward. So here, your points are right here. Your angles will be these ones. These are gonna be up two singles. In this situation for this elliptic triangle, it kinda looks like it's a bloated triangle, right? Too much Thanksgiving. In this situation, your angle sum will be greater than 180 degrees, okay? But again, if you have a large elliptic triangle, its angle sum will be very, very far away from 180 degrees. But when you're a really small elliptic triangle, its angle sum will be very close to 180 degrees. So if you're looking at a triangle that's only just a speck compared to the size of the whole geometry, it'll be very difficult to tell whether it's elliptic euclidean or hyperbolic because they're all approximately equal to 180 degrees. But an interesting thing you can see here, take spherical geometry, for example. Take the equator, take the North Pole, and let's take some points here so that we get a right angle. So let's take, for example, we take two longitudes from the North Pole that form a right angle. It turns out that you have a right angle right here. These angles will be larger than nine, well, what I'm trying to say here is you can actually form a triangle with three right angles in spherical geometry. You can actually make these angles can be bigger than right angles if you wanted to. They can get really, really big. But in particular, in elliptic geometry, you have a three right angle triangle. Don't right angles have to be unique, right? Something like this. That's why you think about trigonometry, but oh wait. The trigonometry you're used to is probably Euclidean trigonometry where you use things like sine, cosine, tangent. So Catoa. Well, turns out those objects are inherent to Euclidean geometry. Circular trigonometry as they're sometimes called. But in particular, this is Euclidean trigonometry. And hyperbolic geometry, trigonometry gets a little bit weirder. You actually use something called cinch, hyperbolic sine, hyperbolic cosine, cosh, and hyperbolic tangent. It turns out that an elliptic geometry, you can also use some of the... I won't go into the trigonometry of elliptics, at least not right now, but I want to be aware that the trigonometry you study like in an SUU, that's the math class math 1060 trigonometry. That is Euclidean trigonometry. If you did trigonometry in hyperbolic space, it would be different. If you did in elliptic space, it would be different. Things like the Pythagorean theorem don't hold anymore in hyperbolic or elliptic geometry. Turns out that the Pythagorean theorem is actually logically equivalent to the Euclidean parallel posh. And you can't have the Pythagorean equation without the Euclidean parallel posh and vice versa. They're a package deal. You get one, you get both. And so, with that, I want to end lecture nine right here. And this was meant just as an exploratory lecture. If you watched all these videos, you will have noticed that we didn't prove a single thing about any of these geometries. It was just exposure. I want you to get used to these ideas of non-Euclidean geometry, like elliptic and hyperbolic manifolds and such. You don't have to be experts of all of them, at least not at this point, but I want you to be aware of these things and realize that things in hyperbolic and elliptic geometry behave very different from each other. So in the next lecture, lecture 10, we're going to continue to develop Hilbert's axioms of Euclidean geometry. We've already done the first group, which is the incidence geometries. We took a detour talking about parallelism, but in the next lecture, we're going to introduce the ideas of the betweenness axioms and in the future, we'll talk about congruence, we'll talk about continuity, what those things mean, until we eventually get to Euclidean geometry. But the thing is, once we've developed Euclidean geometry, all we have to do is remove the Euclidean parallel postulate and replace it with the hyperbolic parallel postulate. And then we get a brand new geometry by switching one axiom. That was the beauty of Hilbert's axioms, they're independent of each other, so that all the theory developed up to Euclidean can be modified to hyperbolic in a very interesting way. And the discovery of hyperbolic geometry is probably one of the most important discoveries ever made in the realm of mathematics, at least in the realm of geometry for sure. So if you learned something about Euclidean geometry today, please give this video and all the videos a like. Subscribe to the channel to see more videos on geometry or other mathematics topics in the future. And if you have any questions or just have something great to say, post them in the comments below and I will reply as soon as I can. Have a great day everyone, bye.