 Welcome back to our lecture series Math 3130 Modern Geometries for students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Missildine. I have in front of me one of the most interesting lectures of the whole semester, lecture 34, particularly from a video recording perspective. It's one of the interesting things. Well, I guess you'd say the interesting thing about this lecture is you notice that it's blank. It's not that I didn't prepare anything ahead of time. But this is one of those type of lectures where I want to talk with you students and with you viewers about some important geometric issues here, right? But it's not going to really lead to any proof for today. We're going to talk about elliptic geometry as the name of this section suggests. And so what do we mean by elliptic geometry? So humor for me for a moment. Up to this point in this series, we have studied extensively two important geometric systems. So we we've always had in the background and sometimes in the forefront this idea of Euclidean geometry. This is a neutral geometry. It satisfies this Euclidean parallel postulate. Given any line L and any point P off of the line, there exists a unique parallel line passing through P. And then the other one, of course, is hyperbolic geometry, which we've been talking about that very recently in this lecture series. Hyperbolic geometry is a neutral geometry. That means it has the it has nice completeness continuity axioms. Locally, it looks like a real geometry like the Euclidean plane is what I mean by real, of course. But of course, with the hyperbolic parallel posture, we now have the situation that when we take a line L and a point P off of that, we can, in fact, get multiple parallel lines passing through that point. So these are these are the two main geometries we've been studying this semester, right? Much of what we did this semester was building up to Euclidean geometry, but we also prepared in such a way that we could go into a different path like hyperbolic. And what we want to do now is enter sort of like the third main real geometry that is of the eucl... the elliptic geometry. That is the geometry that satisfies LPP, that if you're given a line L and a point off the line, well, there are no parallel lines passing through that point. Or the way we usually like to think of the elliptic parallel postulate is that given any line and any two lines I should say they always intersect each other at some unique location P. So this is the type of geometry we want to construct. Well, what's so what's so weird about that that leads to this blank lecture screen you saw in front of us here is that neutral geometry and the elliptic parallel postulate are somewhat incompatible with each other. So let me kind of explain, right? The elliptic parallel postulates telling us that we want a real geometry, but we want no parallels. So what we mean by real geometry is we want these notions of measure, angle measure, segment measure that mimics the Euclidean plane. We want that completeness, that continuity. It's a manifold is what we're looking for. It's a real manifold. That's the type of geometry we want, but we want with no parallels in it whatsoever. So how do we how do we keep our notion of measure in continuity, but get rid of parallels? Well, in neutral geometry, remember, we have the following construction, neutral geometry, given a line L and a point P off of that line. We can construct the perpendicular dropped from P onto the line L for the moment, we'll call that perpendicular line T. And then given that perpendicular line T, we can erect out of the point P, a perpendicular line, which we'll call M. It's perpendicular to the line T at the point P. And so we have this line M. And this line M we often refer to as the guaranteed parallel line. This line M will be parallel to the original line L. So M will be parallel to L right there and it passes through P. So when we start counting the number of parallels with respect to a point in a line, we're because of this construction, we always get its greater equal to one. It's exactly one in elliptic or in Euclidean geometry and it's greater than one in hyperbolic geometry. So how can we construct a real geometry that has no parallels when we have issues like this? Well, let's let's scrutinize this situation right here. Hmm, okay. How did we construct this guaranteed parallel line? Well, the first step was we constructed a perpendicular dropped from the points that has given any point in a line. And the way the point is off the line, we want to guarantee a perpendicular line there. And so it's like, well, construction of lines shouldn't be in all jeopardy. That's something we should be able to do in a real geometry construct lines. And also the issue about this right angle, the issue is we're just trying to construct a line from P to L, but so that the respective angle is a right angle. All right, we could remove perpendicular lines, but that seems quite devastating. The second thing we did, the second thing we did was we constructed that perpendicular that erects out of the point P. And it's it's a slightly different problem, right? We want to find a line perpendicular to T where the point is on the line as opposed to the first situation where we're finding a line perpendicular L that passes through P, which is off the line. The constructions are a little bit different, but basically if you have one, you're going to have to have both. Because if if one right angle exists, then by angle translation, we can basically move that right angle around, right? If we can drop perpendicular angle translation means that we're going to be able to erect perpendicular as well. And the converse is essentially true as well. If we can erect perpendicular, we're going to have to be able to drop them as well. So the existence of perpendicular lines is going to come down to angle translation. I should say if one if rolling right angle exists, we'll be able to move it around another place and construct perpendicular lines. So do we want to get rid of right angles? Again, that kind of seems in contrary to this idea of measure. We want to preserve the measure theory associated to the Euclidean and hyperbolic plane. We want things like that. We want like area and the metric axioms. We want those still to be true. And as the as the notion of measure comes from the congruent axioms of Hilbert, we basically want to preserve the congruent axioms. We want to preserve the congruent axioms with continuity. So we get this notion of measure. So I feel like the right angles have got to stick around. This first and second step has got to stay. But what was the third step? How did we conclude that that L was parallel to M? Well, the person to blame here would be the alternate interior angle theorem, which as a reminder had told us that if two alternate interior angles were congruent, then the associated lines would be parallel to each other. Right angles are congruent to each other. That's Euclid's fourth. That's derived from the congruence axioms. So if we want to preserve measure, we kind of have to accept that right angles are all going to equal 90 degrees. So it turns out that if we are going to get rid of parallel lines, we have to scrutinize this alternate interior angle theorem. Because after all, the alternate interior angle theorem guarantees the existence of parallel lines under certain circumstances. So let's scrutinize that. So parallel lines exist in neutral geometry because perpendiculars exist in right angles are congruent and because of the alternate interior angle theorem. I feel like if we want it to be a real geometry, we got to keep perpendicular lines. We got to keep congruence of right angles. So we're going to go after the alternate interior angle theorem here. So how does one bump off the alternate interior angle theorem? It's not an axiom. It's not like we can just pull it out and replace it with some other axiom. It's a theorem we prove from axioms. So if we want to get rid of the alternate interior angle theorem or modify it to construct elliptic geometry, what path can we do? So the proof of the alternate interior angle theorem, it kind of looks like the following, right? The proof of alternate interior angle theorem. Now, if you don't remember the proof and you want to see some more details, feel free to go back in our lecture series to when we did that. It was a couple of lectures ago and I don't remember off the top of my head, but you can find it in the script somewhere. Or the corresponding video would also be great here. But let's talk about the basic proof. We have two lines which we don't yet know are parallel. We're going to kind of assume they're parallel. So we have these two lines, L and M, and these two lines are cut by some transversal T. We have this idea here. And by assumption, we know that alternate interior angles are congruent. So we have basically this picture right here. We know that these two angles are congruent. Of course, the other angles are congruent as supplementary angles, but we don't need that here at the moment. We have these alternate interior angles. So we proceed to go by contradiction. So by way of contradiction, let's suppose that L intersects M. So that would mean that if we were to extend these lines, keep on continuing with the picture, there's going to be some point where these things intersect each other. Let's call the intersection P. Let's call this point A and this point B right here. And so then you see as weird as it is shaped, we have a triangle ABP that sits here on the right-hand side. So at the moment, we haven't done anything except that we have negated the thing we want to prove. So we got the point of intersection there. So what we're going to do is we're going to proceed to copy this triangle over here to the other side. That is, there should exist some point, some point along the line M. If we were to extend this over here, right? There's some point which we'll call P prime so that we have a triangle AB prime B, right? And how we're doing this, we get this triangle ABP prime and we're going to copy it so that these triangles are actually going to be congruent to each other. They're congruent to each other. And how do we know that? Well, it's going to be congruent using our technique of triangle translation. If we have a triangle, we can translate a copy of it, a congruent copy of it somewhere else. Well, what's the argument behind triangle translation? How does one copy a triangle? Well, in the argument, you have to use segment translation. That was an axiom of congruence. Next, you had to use, not necessarily in this order, but you use angle translation. I guess there's no period needed there. Angle translation, which is also another axiom of congruence. And then you basically go through a construction where you copy an angle, then you copy a segment. And then you invoke a side angle side argument that the constructor triangle is congruent to the original one. So these three, the segment translation, angle translation, and side angle side, these are three of Hilbert's axioms of congruence. So if we want to preserve measure, which includes the congruence axioms, we kind of have to keep Hilbert's congruence axioms here. So we're going to get triangle translation. Because notice with this triangle, the triangle in question, right? They share a common side. The A-B side is shared by both of them. The angle is congruent. Angle P prime A-B is congruent to angle P-B-A. And so the issue is you just have to choose so that P prime is chosen in such a way that the segment A-P prime is congruent to the segment B-P. And so again, this translated triangle here is congruent. That's the basic idea of the argument there. So those are all congruence axioms. What's the next step here of the argument? Well, you have to do a little bit about uniqueness, uniqueness of angle translation and things. By looking at corresponding angles, you can see that this segment over here, this ray associated to the line L, actually has to agree with this one right here, P prime B. So it's kind of like, oh, JK, we were wrong in our assumption. These things are actually one in the same line, something like that. Again, that's going to come from uniqueness of angle translation, which we already used for triangle translation. So using it again is no other consequence. We get something like this, and then you'll notice that if P, so we get that A sits between P and P prime. And we also are going to get from this that B sits between P and P prime, right? So we have two lines that intersect at two distinct points. So we get that our lines L and M intersect at the point P and P prime. And so we get this double intersection, which seems very disconcerting with when it comes to our geometry, right? Why can't lines intersect at two places? Well, this actually comes from the line determination axiom of incidents. So Hilbert's line determination axiom that we have that given any two points. There's a unique line determined by the two points. Well, in the case of P and P prime, there's actually two lines that connect the two. And so this uniqueness of intersection is a consequence of line determination. It's actually logically equivalent to line determination there. Okay, so line determination gives us that there's a problem there if we get a contradiction. But the other issue that came up in the proof is how do we know how do we know that P prime is actually different from P, right? There's an argument. I mean, what if this intersection is one of the same thing? If it's unique intersection, it could be the same point. Well, the argument there is that you're supposed to show that P and P prime are on opposite sides of the line determined by A and B. That's how you choose the point P prime in the in the first place. So the fact that we have two sides of the line is why we can guarantee that P prime and B are distinct points, thus giving us an intersection contradiction. So then it comes down to line, plane separation, excuse me. Plane separation is the next culprit there. In order to get the alternative angle theorem, we need to have the axioms of congruence. We need line determination. We need plane separation. Which line determination, remember, was was equivalent to uniqueness of intersections, plane separation. It was a theorem of betweenness, but it was logically equivalent to Posh's axioms, Posh's axiom about intersections of lines and triangles. And so there we have it. Again, this is the rough proof here, but these these are the these are the ones we have to consider. If we want to remove the alternative angle theorem, we either have to modify the congruence axioms, which given the directive we had the charge we had was to construct a complete real geometry, which has measure. That's what the real part means, but has no parallels. So this really isn't an option to us. We have to keep the congruence axioms if we're going to preserve our measure. So now it's like, okay, which axiom do we want to go Posh's axiom or line determination. And this of course is sort of like a horrible situation we placed ourselves in because it's essentially we have a terrorist in front of us right now pointing a gun at our family and now I have to choose which child of mine is the terrorist going to kill so that the other ones can survive. Well, that's a wonderful choice there. So do we do we knock off line determination or do we knock off plane separation aka Posh's axiom. So when it comes to elliptic geometry, it turns out there are sort of two, two approaches to two, two, two models to elliptic geometry, which are actually not isomorphic. And it turns out because of these two decisions that has to be made right here, do we bump off line determination or do we bump off plane separation. So the first option is if we take line determination what happens if we mix it. That is, we don't get we don't get unique lines between between points anymore. And it turns out that we don't have to necessarily get rid of all of line determination. What we have to do is we have to stop the scenario that appeared in the alternate here angle theorem. That is, on two, we really need that on the two opposite sides of the line. We have these points that we need to allow for double lines that go between them or multiple lines between just two specific points. And so our first model. Sorry about that. Our first model of elliptic geometry where we kill off line determination. This is what we're going to this is what we've talked about before as spherical geometry. Spherical geometry is actually a pretty good model that takes this direction right here. And so the idea is we take our points to be the points of the unit sphere S2 that lives inside of our three. So our points are points on this sphere and how do we define lines in that scenario if we take two points two points on the sphere. Then we think the segment that connects that are well we'll start with the line the line. It's going to be the great circle. Kind of like the great pumpkin there. It's the great circle. I guess that should wrap around a little bit more something like that. So we take this great circle like it's an equator or a line of longitude on our earth right here that we're going to take the great circle that connects these two points together, and we call that the line that's between them. So that is we call that the line the line determined by the two the two points there. Well, what can we do about segments then so if we go back in time a little bit. We get this picture right here. This sort of naturally describes what we would mean by the segment between the points. So we have the point A and the point B. We could say that the distance between the two points would be the length of this segment right here. Now this the thing is lines in the on the sphere actually circles so there's actually two ways to get around this thing. If you have this this segment right here you could take the back door that we had on the screen just a moment ago, like so. And so which is these is the is the segment between A and B right when so you have issues like what is what is the segment A to B right what what is between this mean is C between A and B, or is D between A and B and some respect. They both are both C and D or between A and B but in other respects, neither one is and we'll kind of talk about that a little bit more when we talk about the real project to play in just a moment on the spherical geometry because we don't have line determination. We do have the sort of the following issue that there's actually two paths to straight paths mind you because what is straight even mean after all straight just means it's the shape of a line. A linear shape. That's all that straight means. And so if we interpret if we interpret great circles to be lines then straight means it's a circle. And so our lines are circles and so there actually does make sort of two paths between A and B. Now in this situation you can see that there's a shorter path, the way that A and B, the path that goes through C is in fact a shorter path. And so we can call that the distance or the geodesic the geodesic distance between A and B. There is this alternative path but it is much longer and it's going to be somewhat less preferred. So we can kind of get around this this issue of betweenness we might try to argue that C is between A and B as opposed to the point D there delta dog because it's closer right you try to make that argument but there's some other issues we have to consider with this spherical model. You have issues like the following. Well what if you have three points, which are perfectly distance from each other so they're exactly one third away around the circle so that this the segment, the measure of a B is equal to the measure of BC which is equal to the measure of AC. So a B equals BC, which equals CA. In that situation, do we get that B is between C, do we get that C is between B, or do we get that like a is between B and C. So in retrospect, it's all of them, but also none of them at the same time. We don't have a notion of betweenness when we take lines to be great circles. We try to fix that in a second and again I'll come back to that issue in just a moment, but we do we do have some issues going on right there. What how does this violate line determination, as a reminder when it comes to the spherical model becomes the spherical model. So let's take two antipodal points that is two points on opposite sides of the sphere. So think of like the north and south pole. In this situation, there are multiple, multiple geodesic paths that go from the north to the south right. And so for these antipodal points, there is not a unique line between them there's actually infinitely many lines between them. And so that's how line determination is weakened in this situation. Line determines normally would give us that there is exactly there is a unique line between the two points and spherical geometry, we get a different version of line determination we're going to get that there's a greater than equal to one line. And in fact we get greater than one only only here at these antipodes. So only the antipodal points are going to get this so and so it's not it's not full blown line determination we lose a lot when we do something like that it's not even an incidence geometry anymore. But this is one way of trying to define elliptic geometry. And this is sometimes referred to as the as the two point or the double point, the double point model of elliptic geometry. And that's because you have these two points on that is every line that contains one point will always contain its antipodal point as well. But I should mention that in spherical geometry, we do have plain separation, right if you think of like the equator of the sphere it cuts the sphere into two regions the northern hemisphere and the southern hemisphere. There is plain separation. The only way to go from one one hemisphere to the other is to cross the equator that's the only that's the only possibility. So we don't have line determination we have a weaker form of line determination. There are allowed certain exceptions to line determination but it usually works. But the trade off as we get, we do get plain separation. But in the process we kind of lost something about betweenness. What does between even mean anymore? Again, we're going to have to come back to this because it turns out the other model that we're going to talk about for elliptic geometry doesn't fix this problem of betweenness either it's a trade off between line determination and plain separation. So the next approach is what if we want to get rid of plain separation? What type of model for elliptic geometry could we have there? What if plain separation is the one that gets nicked? We get we still have lines determination but we can't necessarily separate planes that is there's only one half plane. I know that seems weird to say but that's exactly what we're going to require. And it turns out we've also already discussed a model for this. This is going to be the real project of plane, RP2. And so we'll think of it. I'm going to bring here the the spherical model, the hemispherical model excuse me of RP2. So we take the upper hemisphere but the equator which is included inside of our geometry. We have this antipodal identification to it so that when you hit the boundary like a video game you wrap to the other side. Basically this is from a topological one of you RP2 there's a lot of ways interpreting it but we're going to take the unit or the unit sphere and we're going to quotient out so that antipodal points are actually considered the same point. So a point in the real project of plane is actually a pair of antipodal points on the sphere. There are other equivalent definitions and models we could use but this is the one we're going to stick for right now. And so if we are given any two points right we can construct the unique the unique circle or in this case it's a semi circle we can find the unique circle that connects those two points together and there's no there's no missing lines and there's no duplicated lines as well because if you were to pick two points which are antipodal to each other that's actually considered the same point right. And so one point is insufficient to form a line so there's no violation to plane separation but I was sorry no no violation to line determination there definitely is a violation to plane separation that's sort of the trade off here because we live in this like video game world. If we're flying our little our little spaceship here playing the game asteroid and there's this big asteroid over here what we can do is we can just fly to the other side of the screen we would wrap around like over here. And we can avoid our obstacle by using the wraparound feature of our video game. And so you actually can get from one side you can get from one side of the line to the other side without actually crossing the line. And so this would actually show us that there is no two sides of the line there's only one side of the object. And again this can be someone kind of weird for students to understand here but as a metaphor, sort of a similar phenomenon what I want you to do is take a strip of paper take like a really really really long strip of paper. And if you were to glue the two sides together of this paper or tape them together, you would make like a cylinder, right. You get a band that looks like that that's a fun little construction that people can do. So what if we were to try that again, but instead of when we glue the sides of the paper together what if we put a half twist to it, when you do it. In that situation, oh boy, I should have prepared this at a time I don't know if I can draw a Mobius band, but you're going to get something like the following boy this is the worst drawing that's ever existed. Oh boy, you should look it up on Wikipedia sometime, but you're going to get this band, which has a twist in it, and it has the property that this Mobius band you should again look this up on on the internet sometime look up the Mobius band or the Mobius strip right. This is actually a one sided object. It's a one sided geometry. If you were to take a normal vector because this thing does live inside of our three if you take a normal vector and you were to follow along the edge of this thing because of the half twist. When you get back to the other side, your vector is actually going to be pointing the opposite direction. So your your your Mobius band is one sided. And this is very similar to the idea we're seeing right here that the real project of plane. It doesn't have two half planes, it only has one half plane. And that's because as a manifold the real project of plane is this non orientable manifold much like the Mobius band, because as you try to apply a normal vector fields to it you can't do it. Same thing happens here with the project of plane. And that's because we lost that plane separation. And so this is sort of the trade off we get we can get line determination we lose plane separation we can take plane separation or we can lose line determination. And so we have to pick one of our one of our children to keep right. And so this right here the project of the project of plane is often referred to as the single point, the single point model for elliptic geometry. And in the literature both are commonly used to describe the elliptic geometry, you have, you have the spherical model or the double point model, or you have the real project of plane which is the single point model. We call the single point model versus the double point model because on the sphere antipodal points are considered different, but on the project of plane, the antipodal points are considered the same so do we have two points for antipodes are one point. Hence the name here. So we have to make a decision for our purposes in this in this lecture series. So our choice is, you know which which child do we do we choose. We're going to define elliptic geometry to be the project of cover of the Euclidean geometry. And so we talked about this in the previous lecture right that if you have an affine geometry. I don't have to draw it all wiggly but if you have some affine geometry. You see something like this affine geometry. What we can do is we can add this line at infinity. Right, kind of looks like the the the nexus from Star Trek generations doesn't it we get this we get this line at infinity right here, and this line and infinity has the property that if we have any any parallel lines that lived in the affine geometry. If we extend them into the project of cover these lines actually become intersecting lines at some ideal point at infinity. Or another way think about drawing art in perspective right if you have like the horizon line and you have these train tracks that are running towards the horizon right. Well, they are parallel. But we look as they go towards the horizon that the two rails on the train tracks seem to be coming together and they touch at some point and infinity. That's this relationship between the affine geometry and its project of cover. We define elliptic geometry to be the project of cover of the Euclidean plane. And so locally speaking the project of elliptic geometry will it'll be a project of geometry, but it will look locally like the Euclidean plane because the project of geometry on a local scale looks like Euclidean geometry. But because of this extra line and infinity we actually got we removed all of the we removed all of the parallelism that existed in the previous model. Now this is going to affect things like distance right because if you look at these like ruts right here, these ruts that are on the train tracks you have one here these crossbars in terms of the train tracks these are all equidistant along the rail. But as we go towards infinity, the distance between them appears to get closer and closer and closer so as we go from as we go from the the affine geometry to the project of geometry notions of measure are changing infinite distances that exist in Euclidean geometry. Are going to become finite distances. So like if you think of like the distance from here to infinity on this line, that's an infinite distance but but when we switch over to the project of geometry, it actually becomes a finite distance. And so there's there's some oddities that are going to exist between this but this is sort of the the cleanest way to preserve the theory that we've developed for neutral geometry into the elliptic geometry setting. So be aware that the real project of playing because the real project of playing is the project of cover of Euclidean geometry. What we're going to retain is we retain incidents. We retained all the axioms of incidents so we get all the theorems of incidents we're going to keep those. We are going to keep all of the congruent axioms the way they are so angle translation segment translation side angle side, we're able to keep all of those congruent axioms and the theorems that come from that. We're going to keep our notion of measure and more specifically completeness continuity. We're going to keep all of those. So we retain our notion of measure that was our goal we want to find a geometry with no parallels. There's no parallels. But it also has measured that was the goal we wanted here we are we retained incidents but we lost something with between this right so we have to be careful when we talk about between this with respect to this geometry. Some things to note here is we don't have posh axiom for an elliptic geometry here in this project of playing we don't get posh axiom. We're also lacking the trichotomy axiom. One of the reasons that we talked about earlier with the spherical geometry, because in the real project of playing. If we were to go back to that picture we had one right here, the same the same situations happening like we had above. It's exact same picture with the ABC, ABC, because in project of geometry lines are still circles. Same thing in spherical geometry and so you have this issue of trichotomy we don't have trichotomy. We don't have posh axioms with those those are the sacrifices we had to make in order to have no parallel lines. Now the good news is we can we can retain some notion of between this, not not the complete notion between this but we can get some notion of between this. So let's take a an elliptic line which is essentially a circle. Now, not all not all circles are elliptic lines but all elliptic lines will be circles. If we have points A, B and C, which one is between the other? Well, although we can make a definite statement about who is between who what we can do is we can make a relative statement here. What if we what if we pick a point D, which is on this line that this point D, which we can think of as a point at infinity, right? D is some point at infinity here. Because after all our elliptic geometry sits above the Euclidean plane. And so, with the exception of the points at infinity, there is notion of betweenness, A, B and C. If there are three ordinary points, we should be able to order them by betweenness, right? So A, B and C, if we think of them as ordinary points, we should be able to between them. So if we think of D as this point at infinity, what we're going to do is we're going to bring our scissors over here and we're going to cut the line at this moment at this location. So we sever the line into a thing that looks like the following. When we cut the line by removing that point at infinity, we can flatten the elliptic line and have now look an ordinary line, an affine line, that is to say. And so then we get that A, C and B, we get that C is between A and B. So we could say that C is between A and B. But like we saw in the previous lecture, how we cut the line, that is, it didn't matter which line is the line at infinity, which therefore means on any given line, any point could be a point at infinity if we just change perspective. So what if we take a point over here, call it E, and what if we cut along the point E? Well, in that situation, if we flatten the elliptic line, like we did this time, then the line would flatten in the following way. We would get B right here, A right here, and C right here, which then in terms of betweenness, we're saying that A is between B and C. So which one is it? C between A and B, or is A between B and C, or is that the third possibility? And so what we're going to do here is what we're describing is the notion of relative betweenness. That is, we can describe betweenness of points relative to a point at infinity. So this first one where we said C was between A and B, this was relative to the point D. And the second situation where A is between B and C, this is relative to the point E. And if we were to slice the line at some point F like this, then we could say that B was between A and C. So although we don't have the usual notion of betweenness, we can get this notion of relative betweenness. And so that way, when we talk about betweenness in the usual setting in elliptic geometry, if we think of it in terms of a relative manner, we can preserve a lot of the main theory. So that's sort of like the trade-offs we're going to get here, that in order to have no parallel lines, but we have a real geometry, we're going to sacrifice betweenness for the weaker notion of relative betweenness. And that's the trade-off we can make in elliptic geometry, a.k.a. the real project of plane. So for my students here, you're going to be given one exercise for this lecture here. And that exercise is to go through all of the theorems of congruence and neutral, basically go through all the theorems of neutral geometry and tell me which of them are still theorems for elliptic geometry and which ones are not. I'll give you a hint, the alternate-tier angle theorem is not a theorem anymore for elliptic geometry. Now be careful that when there's theorems that talk about betweenness like the crossbar theorem, right, you have an angle and you have some crossbar and you have some interior point. You can guarantee that there's an intersection between them. That would have to be switched to relative betweenness, but you can make a statement about the crossbar theorem in elliptic geometry. And also be cautious that although some proofs we've provided, they maybe were proven using the alternate-tier angle theorem. Is there perhaps an alternative proof that doesn't use alternate-tier angle theorem? So that's actually a quite daunting task, which is why it's only one question. I want you to think about that for a while. And even for the people viewing who aren't actually my students, I think it's a good exercise for you to understand what elliptic geometry is all about. And so that actually brings us to the end of our lecture today. Like I said, it was kind of a blank lecture, not a lot of proof, but tons and tons of drawings. That was pretty fun. I'm glad you came along for the ride today. We'll talk some more about elliptic geometry next time now that we have a specific model in mind when we talk about elliptic geometry, the real project of plane. And I hope you'll tune in then. If you like these videos, feel free to subscribe. Please click the like button, the subscribe button, leave some comments. If you have things to say, particularly if you have any questions, post them below and I'll be glad to answer them in the future. I will see you next time, everyone. Bye.