 Welcome back to our lecture series math 3130, Modern Geometries for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. So I want to point out that if you've been following along with this lecture series. In lecture eight, we talked about the idea of affine geometry. So we saw last time that the Euclidean parallel posture was in fact a very, very powerful axiom and that the inclusion of it, even into the most broad axiomatic system such as incidence geometry, because that's what an affine geometry is. It's just the four incidence geometry axioms equipped with the Euclidean parallel postulate. So just those five axioms, affine geometry is enough to lead to some very powerful results. We listed some combinatorial results about how all lines have the same number of points. And there's just a lot of uniformity that happens, a lot of symmetry that happens with affine geometry. This is a fact, this fact, this observation about affine geometry, the Euclidean parallel postulate, is the reason why geometers have been fascinated with Euclid's fifth postulate for centuries. Because in the book, The Elements, when Euclid began the work of axiomatic geometry, he had his four postulates, which admittedly, in Euclid's work, his axiomization was incomplete, but given he's probably the first, at least from historical records, the first mathematician to really axiomize geometry. I'd say he did pretty good for thousands of years ago. But yeah, in the modern sense, we see there are a lot of unstated assumptions that Euclid used in his proofs. And so because of that, it made studying Euclid's work very problematic. It was cutting edge when it was first developed, but nowadays we're much better about many of these things. And so for many centuries, geometers believed that Euclid's fifth postulate, which is equivalent to what we call the Euclidean parallel postulate, about uniqueness of parallel lines. The Euclidean parallel postulate that we have been using is actually due, not to Euclid himself, but actually due to play fare. But anyways, Euclid's fifth postulate, which is logically equivalent to the play fare version of the Euclidean parallel postulate we're using, it was believed to be a theorem of the other axioms of Euclidean geometry. And it's such a powerful result that we want Euclidean's parallel postulate, but they couldn't prove it because it turns out it's actually not equivalent. This is actually one of the, one could argue, one of the starts of modern geometries, the discovery of non-Euclidean geometries. There are other geometries besides the geometry studied by Euclid in the book, The Elements. In particular, we're only going to focus on Plainer, that is two-dimensional geometries. And so our discussion in this video will be two-dimensional non-Euclidean geometries. But this lecture, number nine, will focus on a couple of non-Euclidean geometries. That is geometries where the Euclidean parallel postulate fails. In particular, our main focus is going to be on two geometries. We call elliptic geometry and hyperbolic geometry, which we're going to study in the future. And they get their names from the elliptic and hyperbolic parallel postulates that we've studied before. So in this first video, we're going to give an example of a elliptic geometry, which is commonly known as spherical geometry. This is honestly, assuming you're watching this video from Earth, this is the geometry of our world. Earth is a sphere and the geometry of Earth is therefore spherical geometry. So what is spherical geometry? Well, spherical geometry is the geometry of a sphere for which to borrow topology terms, we're going to refer to the sphere as S2. Just so you're aware from topology, S1 is a circle. That is, it's a one-dimensional circle, one-dimensional sphere, I should say. The sphere, as we usually think about it, is a two-dimensional sphere. And topology, they do consider higher-dimensional spheres, like three-spheres, four-spheres, five-spheres. These spheres can always be described using distance. For example, the two-sphere, which is what we're talking about right now, we're only focusing on two-dimensional geometry in this lecture series. The two-sphere is the set of all points of unit distance from the origin in R3. So if you take the intersection of the x, y, z axis, the two-sphere is all of those points which are one unit away, but in any direction that's forming the sphere. Just like the unit circle in the plane is all those points which are one unit away from the origin and makes a circle. You can do the same thing that in four-dimensional space R4, so this will be four-dimensional Euclidean space, we could take the set of all points which are one unit away using the Euclidean distance formula, one unit of distance away from the origin that would make the three-sphere. We can do this. So as long as we can talk about higher-dimensional Euclidean space, then we can talk about higher-dimensional spheres as well. But like I said, this video will focus just on the two-dimensional geometry right there. So how do we make sense of this as a geometry, right? A geometry, like I said, it needs to have points and these have lines and these have incidents. Well, the lines are just gonna, excuse me, the points are just gonna be the elements of the set. So S2 is a subset of a geometry. So the elements of S2 are elements of R3, so we can actually think of those as points. But as we restrict from the larger set to the smaller set, the notion of lines doesn't necessarily have to stay the same anymore. And in fact, when we think of lines on the sphere, we really are talking about great circles. Well, that's a great circle, right? A great circle is actually just a diameter, excuse me, it's a circle that goes around the surface of the sphere, but the diameter of this circle is identical to the diameter of the whole sphere. So if you think of a planet Earth, for example, one such great circle would be the equator, right? Those are, that's a great circle using the usual topographic meanings of the equator. You can also take any line of longitude as they go from the North Pole to the South Pole. That also would be a great circle. I guess technically I have to put two lines of, excuse me, two lines of longitude together to form a great circle, but a great circle is just the maximum circle you can go around here. And so if you take any two points on your, if you take any two points on your sphere surface, because again, when we're talking about the sphere here, we're just talking about the points on the sphere, not in the sphere or anything like that. There is a, excuse me, take any two points on your sphere, there will be a great circle that contains it. And so that's what we're gonna consider a line in this situation. After all, line is an undefined term. We can give it different interpretations. So line doesn't have to mean line, at least not in the Euclidean sense, right? So you're like, how can circles be lines? Well, lines are just Euclidean lines the way you usually think about it. But in spherical geometry, a line is actually a Euclidean circle, but it's a spherical line. This is that paradigm shift that's necessary, what many jobs are struggled for such a long time. And an incidence means what you usually expect it to mean that as a line is a set of points, we say that a point is incidental line, if that point belongs to the set of points which determines the line in a set theoretic sense, all right? So we have incidents, we have lines, we have points. We can talk about this as a geometry. And so it's important to mention that this geometry won't be Euclidean because you don't have unique pair of lines. In fact, all lines are gonna intersect each other. Like if you take these two lines, for example, clearly by construction, they intersect each other. But just imagine as you try to think of any, if you take the equator, for example, because after all, depending on your orientation of the sphere, the equator is an arbitrary line right there. If you take any other line, any other great circle, it's gonna have to intersect the equator somewhere. You can't avoid it. You can't try to do something like this, aha, right? I just have a smaller circle. That's not a great circle. The diameter of that circle is not the diameter of the sphere. So that would not be a great circle, thus it's not a line for this geometry. All lines will intersect the equator somewhere. So all lines intersect each other in this geometry. It's kind of curious that the Euclidean parallel postulate isn't satisfied. In fact, we actually get the elliptic parallel postulate. This is sort of like the poster child of elliptic geometry that all lines intersect on the surface of a sphere. So the spherical geometry, it satisfies the elliptic parallel postulate, like I said, since all great circles intersect each other. In fact, spherical geometry is probably the best understood and most studied version of elliptic geometry, at least historically speaking. And particularly this is because we live on planet Earth, which is a sphere. In fact, when you look at the word geometry itself, that we use in English here, this comes from the Greek word gematria. I apologize for my pronunciation here. But see Gamma is the Greek letter for G, Epsilon is for an E, Omega is for an O. So you get Geo right there, for which Geo is the Greek word for Earth. All right. Then the next word, Mu, is the Greek equivalent of M. Epsilon is an E, like I already said, that tau is a T, rho is an R, iota is like the I sound, and then you have an alpha at the end. So it's metria, right? So meter, metria here means like a distance, it means a measure. So gematria is the measurement of Earth. So geometry, the word itself, this study of mathematics with shapes and curvature, is comes from the Greek word for Earth measure. Some of those fundamental geometric principles came about because we were trying to study the geometry of Earth, because as the legend goes, ancient man believed the Earth to be flat, and it came from scientists and mathematicians and explorers and cartographers and what have you, that eventually we're all approved to the world that the Earth was in fact round, it had curvature, it was a sphere. And so that's why we get this word geometry. It comes from spherical geometry, because geometry, as the word translates from the Greek, is the measurement of Earth, the first spherical geometry we understood very well. Now, while spherical geometry is an example, it's sort of like the prototype for elliptical geometry, it's not without some problems. Let me mention that although it's a very natural elliptic geometry, that is it satisfies the elliptic parallel postulate, it turns out that spherical geometry is not an incidence geometry as we've defined it earlier. So spherical geometry is not a model of the incidence axioms plus the elliptic parallel postulate. And the issue comes down to line determination. So remember line determination says that given any pair of points, there exists a unique line that's on both of those. And most of the time, we're perfectly fine. Like if you take this point A and B right here, there is a unique line that contains both of them. It's the equator of the globe I have drawn right here. Or if you take the point C right here with A still, there's going to be a unique line that takes it. Between B and C, there's a unique line determined between A and B. For which the curious thing about incidence, curious thing about spherical geometry, this is part of the issue, is that there's actually sort of like two paths, two straight lines between A and B, between any two points. In this case, it's B and C right here. So like if we could walk straight from B to C, there's a short one and there's a long one, which clearly if you want to go from B to C, you're going to take the short one. So we kind of ignore that one. We look at the short one. And so we can talk about triangles. We can talk about distances. We can talk about angles. So you see, for example, on this diagram I'm coloring on, this is an example of a spherical triangle. All right. But there is one bigger important counter example. What if you're at the North Pole on your planet here and you want to go to the South Pole? What is the fastest way to go? And honestly, the answer is any path. You go any path. You go any direction, right? If you take, if as long as you go straight, you're going to reach the South Pole, right? There's no worries about that. It doesn't matter where you go. And so this diagram over here gives us a better example here. If we have a point P, there is a point on the opposite side of the sphere. These are referred, these two points are referred to as antipodes. The idea here is they're on opposite sides, on opposite poles of the Earth, the North Pole versus the South Pole. And so if you have an antipodal pair, P, and often it's called negative P, because as spherical geometry is a subset of our cube tier, this antipode is actually, it's additive inverse with respect to the vector addition that is present on the vector space R3 right here. So these antipodes are actually, they're additive inverses of each other with respect to the algebra. But geometrically, we call these antipodes. The problem with spherical geometry is that if you have an antipodal pair, there do exist, there does exist a great line, a great circle from one to the other, but there's actually infinitely many, there's multiple ones. And so there is, and a very important counter example to the line determination axiom that you do have that every pair of points has at least one line between it, but antipodes have more than one line connecting the two together. So that's a violation of line determination. But with regard to the incident axioms, that's the only violation. Point existence, yeah, we have at least three points, we have uncounted linear points. There is no great circle that contains all of the points, so we get non-collinearity. We get secancy, every great circle has at least two points. It has uncounted linear points, mind you. And this does also satisfy the elliptic parallel postulate. So what stops us from having an elliptic incidence geometry is that line determination isn't satisfied because of these antipodal pairs, but it is somewhat isolated that even though it's not, even though it's not an incidence geometry, it's really close to being an epi-activity careful. As we see in this diagram right here, two different circles intersect each other at two different places. That's not a phenomenon we saw in incidence geometry. Intersections of lines was unique. But in this situation, lines intersect at two different points. And so because of this problem, since it's not actually an incidence geometry, some people introduce alternative models for elliptic geometry. We'll explore those in subsequent videos here. But this spherical geometry is sometimes called two-point elliptic geometry. And the reason behind that basically comes down to lines can intersect each other at two different points because you have these two antipodes. That if you choose antipodes, lines will intersect at two different places, like so, of course. But that's of course only, you'll notice like with my other diagram over here, if I take any two lines, these two lines, this one and this one, they intersect at A, but they also intersect over here at negative A. So when lines intersect in spherical geometry, they always intersect at two points, but those two points are always antipodes. So because of that double intersection, this is sometimes, spherical geometry is sometimes referred to as two-point elliptic geometry. Now if you really want an incidence model of the elliptic parallel postulate, then we're going to proceed forward with that in the next video. We'll introduce projective geometry, aka one-point elliptic geometry.