 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 Misaline I should warn you before we jump into this lecture that this one's gonna be a long one It basically is one proof, but the proof is gonna be kind of long You'll understand when we get to it there. So I confess this movie isn't gonna be as long as like Avengers Endgame or anything like that. It's not gonna be as long as the year you had to wait between Avengers Infinity of War and Endgame But anyways, it's gonna be long for our geometry lectures right here We are gonna talk today about Projective geometry and before we get into the project of geometry part I want to kind of give you some idea of why we're talking about projective geometry So in the last couple lectures, we've been talking about Euclidean geometry and hyperbolic geometry Which are the two options one gets when we have a Neutral geometry right because neutral geometry as the name suggests is we stayed neutral about our Parallel postulate But then we did make a decision. Okay. What if we assume the Euclidean parallel postulate? What can we do? What if we assume the hyperbolic parallel postulate? What can we say about our geometry? And there's a lot of stuff that's similar between Euclidean and hyperbolic geometry because they're both neutral geometries But our final our final goal in this discussion of modern geometries is to talk about elliptic geometry Which elliptic geometry is a little bit more of a challenge, right? The elliptic parallel postulate, which you can see right here states that all lines intersect that is there is no parallel lines and so this this is somewhat of a problem because in Neutral geometry you have the following result If you have any line L and you have a point off of that line call it P Then we can construct The perpendicular drop from P onto L like so let's call it T and then we can also erect from the point P a perpendicular line to To it's a line T here, so we get this perpendicular angle right there We'll call that line M and then it's a consequence of the alternate to your angle theorem that This lines L and M are gonna be parallel this line M is the so-called Guaranteed parallel line that goes through P. So when you look at this argument right here this construction It's somewhat incompatible with the elliptic parallel postulate and the main culprit here is gonna be the alternate interior angle theorem And we'll talk about this more in the next lecture lecture 34 But we're gonna have to we're gonna have to deal with this this situation, right? How can we construct a geometry that kind of locally resembles Euclidean geometry? but satisfies the elliptic parallel postulate we have to be a little bit careful because How do you just throw out the alternate here angle theorem? We'll talk about how we'll do that, but we have one has to be sort of careful, right? We don't necessarily want to destroy all of the structure of geometry we've developed this semester And so one has to be a little bit careful on how do we approach this elliptic geometry conundrum that we have right now And so we'll see this will make much more sense next time as well But it turns out that this the solution the the way of reconciling these contradictory desires Notions of congruence with the elliptic parallel postulate is to go down the avenue of projective geometries So with regard to Wallace and West textbook, we can say that this section Can be found in section 7.2 about real projective geometry, but what we're gonna talk about right now really is not exactly What's talked about in that section we have to make the connection somewhere or another So earlier this in this lecture series. We had talked about the notion of projective geometry, but very very briefly You'll recall that we had talked about incidence geometry, which is a geometry which satisfies the four axioms That that what what Hilbert called the four incidence axioms. We have line determination secancy point existence and non-colinearity An affine geometry was then an incidence geometry Which we equipped the Euclidean parallel postulate to and then we also defined around that same time period a projective geometry Wasn't which was an incidence geometry which satisfies the elliptic parallel postulate and we strengthened the secancy axiom Secancy said that every line had at least two points on it for a projective geometry We require that every point had or every line had three points on it and it might seem a little bit weird Why why did we require three points on a line? Well, the issue is kind of like we didn't want things like fan geometry to be considered projective geometry and At the moment you just thought I was a bigot because it's like why why can't fan geometry be projective? You're just so rude, right? They have feelings to yes fan geometry does have feelings But it doesn't fan geometry doesn't follow the proper taxonomy of what ought to be Projective geometry and we're gonna talk about that sort of why in this lecture this try can't see axiom The why why try can't see will actually become very apparent because of that one massive proof We're gonna do it's gonna use try can see several times with Miller do it on fan geometry So let's remind ourselves exactly what the definition of projective geometry is you can see the axioms right here There are five axioms to projective geometry the first one the first four are essentially just the incidence axioms Axiom one our line determination Axiom for each two distinct points there exists a unique line containing both of them like I mentioned before we're gonna upgrade Axiom two of incidence the secancy axiom. We're gonna upgrade it to try can't see and I confess Try can't see is probably one of the most made-up words. You'll ever see in this course It's it's got a fun sound to it, right? It's an amalgamation between triple and in secancy, right? And it kind of makes you think of like Geomancy or like necromancy, you know these fun little black magic spells We could do but anyways try can't see it's gonna say that for every line there exists at least three Three points on the line. So that's a little bit stronger than we usually require For incidence geometry and the main issue is that we don't want the fan geometry to be considered a projective geometry The other axioms are as we've seen before axiom three point existence there exists at least three three points In the geometry, that's the axiom for incidence. We can actually prove stronger statements than that And we'll see forthcoming Non-colinearity not all points lie on the same line and then for our parallel postulate for projective geometry We do take the elliptic parallel postulate, which is to say that all lines intersect or there are no parallel lines Or another way of saying it is that given a line and a point off the line. There's no Lines that go through the point which are parallel, you know, those type of statements are all equivalent to each other and we have seen some examples of Projective geometry throughout the semester back at the near the beginning of this course When we had talked about finite geometries phano geometry was an example of a projective model And the one that we're mostly interested in is our P2 here the so-called real projective The real projective geometry we've talked about this thing before but as reminder what we're gonna do is we're gonna take the unit Cirque or the unit the unit disc This is the closed disc so that that the interior is included part of our geometry But also the points on the circumference of the circle are part of our geometry now This geometry had the this is the set of points But then in terms of the points we have this wraparound feature that When you when you go from one side of the circle, it'll map to the other side So these two antipodal points here are actually identified as the same point on the geometry So those two points are one of the same thing and you often see this wraparound feature Denoted in the following way you draw these little arrows on Antipodes of the sphere but you point them in opposite directions This is to somewhat suggest that there's a twist that happens when you go along here So you can think of as this unit circle some other models of the real projective plane some people like in topology They really like to draw it as a square but with the square here you have these This reflective property that when you go to the opposite when you hit the boundary You're gonna wrap around to the other side So if you're chugging along and you hit the boundary here you wrap around from the other side But you there's a reflection that happens when you when you hit the boundary right there So that's another example a model that I really like and we'll probably use this a lot in the future is that we can We can describe our geometry as a hemisphere So we think of like this upper hemisphere model This back here is a dash line because we don't see it so we have this hemisphere model which It's a hemisphere, but we'll wrap around the boundary so there's that same wraparound on the boundary and so a line in that geometry would look like a great semicircle That goes around so that's though these are these three are all isomorphic copies of the real projective plane So we have a finite a finite model and just as a reminder the fan of plane Looks like our harry potter Model from before When of course, maybe grindenwald's not defeated. He still has the elder wand But then Dumbledore has one for example, and then Voldemort has the third one So we have those deathly hollows and the three elder one. So this is our phano geometry From the past there. Let's see. I'm missing some points, right? We need a point right here One two three four five six seven there they are so we have our seven point seven lines So we want to top on general these phano not with these projective geometries So here's now the mega theorem for this lecture here This one's a little bit difficult to digest But it's a really really interesting theorem and a really elegant results when it comes to projective geometries So let's start off with the projective geometry Where the points of that geometry we're going to call that set P and the lines in that geometry We're going to call those set L As it's described here So we have a projective geometry which it comes with points and it comes with lines And there's an incidence relationship between the points and lines there So what I want you to do is that with this geometry we're going to fix a specific line L Just pick your favorite line in the geometry and we're going to call this this line is going to get a special name We're going to refer to this line as the line at infinity and For a projective geometry it doesn't really matter which line becomes the line at infinity But for this construction we have to we have to be specific and we have to choose one So we have this line at infinity or sometimes it's called the ideal line The ideal line all right So fix that line and then do the following construction We're going to we're going to define a new set of points. We're going to call that set P prime P prime here where P prime is all the points that were in the projective geometry Except for the points on the line at infinity. All right, so I want to mention here that the points The points on L are going to be called the points at infinity points at infinity or They're sometimes called ideal points All right, and and for this lecture here I will typically use the color blue to describe describe points at infinity or the line at infinity That is the ideal. I'll describe using the color blue All right, and so getting back to the to the set P prime here If you think of the line at infinity It is a set of points if you take away all those points at infinity And you take all the remaining points you get what we're going to call the ordinary points You're going to get the ordinary points and then Related to that is we're going to take every line that exists inside this geometry Other than the line in infinity and we are going to subtract the line in infinity away from M That is if you have a if you have a projective line Subtract from it all of the ideal points and take the subset that only consists of ordinary points So we had the ordinary points right before and so these right here are going to represent our ordinary lines We got our ordinary lines right there So we've constructed this new geometry right here. And so let me kind of clear out this Let me clear out the screen a little bit because it's kind of getting messy kind of summarize What we what we're trying to describe right now So what we're saying is we have this set of points Curly P curly L. This is our projective geometry This is the projective geometry Then let's remove from All the let's remove all the ideal points So we're categorizing the the types of points in this projective geometry We have ideal points and we have Ordinary points if we only gather the ordinary points That's what we mean by P prime And if we we can also segregate the line at infinity from all the ordinary lines Which admittedly the ordinary lines you take away their point at infinity So this P prime represents all the ordinary points this set L prime represents all the ordinary lines And so this is our ordinary Geometry there is no infinity. There's no ideal going on in this ordinary geometry Did you see an ideal point there in the ordinary geometry? No, you didn't There are no ideal points in the ordinary geometry So we're assuming that we started off with the projective geometry And then as we dissect this geometry the projective one to make an ordinary geometry We claim that the ordinary geometry is going to be an affine geometry affine geometry member It has the um, it's an incidence geometry which satisfies the elliptic parallel pulse. Excuse me the euclidean parallel postulate And so I just also so that's going to be our theorem here if we have a projective geometry We can remove a line at infinity. That is we pick a line We say it's the line of infinity. We remove it and then the consequential geometry formed by removing that line And all the points on that line that will become an affine or an affine geometry So when I come back to these axioms of projective geometry I want to mention that in the literature these five axioms are not typically how one defines a projective geometry typically you take point existence tip Sorry line determination you'll take the elliptic parallel postulate But these three axioms right here are typically consolidated together into one axiom And that axiom will say something like there exist four points. No three of which are collinear That one you can kind of argue that that statement I think I said it correctly if I didn't Maybe I can correct it later on in the comments, but I'm pretty sure that's the statement as is as given there And so that statement that one axiom kind of is equivalent to these other three The reason I'm opting for these five axioms is that I want these five axioms to resemble the axioms of affine geometry Which affine geometry? It has line determination. It has secant C Which is a stronger or which is weaker version of tricant C The fact that we actually take tricant C is that when you remove the points at infinity you had three points But now you maybe only have two so we need an extra point So then we remove the ideal points. We still have two points point existence non collinear are the same And then it turns out when you remove the ideal points things that intersected in the projective geometry Will actually be parallel in the ordinary geometry So let's let's talk about these axioms one by one by one So let's first look at axiom i1 the line determinant axiom So imagine we and so we're trying to prove line determination for the ordinary geometry We know that line determination holds for the projective geometry So there's often this is going back and forth between the ordinary with the projective So imagine we have two ordinary points a and b like so. So these are ordinary points Well, as they're ordinary points, there does exist a projective line We'll do projective stuff in white. There's a projective line We'll call it m that harbors the two ordinary points because every ordinary point is a projective point But by tricancy Uh, let's see. Well, I guess I don't need tricancy yet. Um, we have the line at infinity, which we're going to do at blue I'm the line because of the elliptic parallel postulate. We do know that the line at infinity intersects Uh, well, I guess I guess what I'm trying to say right here is that we we have a projective line Is there an ordinary line that'll contain a and b right here? well We can say for a fact that this line m is not the same thing as the line at infinity Because the line at infinity only contains the line infinity only contains ideal points every point Or at a point is ideal if and only if it's on the line at infinity So although there could be some there is going to be some Ideal point on m because after all the line at infinity And the line m necessarily intersect Uh, they necessarily intersect at some ideal point, which we call p, right? So the line at infinity is this blue one right here So m and l must intersect as projective lines at the point p and because of line determination The intersection between these two lines will be unique. So there's only one ideal point on This line m right here. So let's construct Let's construct the ordinary line m prime by taking m subtract l So if we only take the points except for If we only take the points except for p we take all the points except for p. Oops There's no point there anymore. Sorry line at infinity. You're gone If we take away the line at infinity then there is no other There's no ideal points on here. We now have an ordinary line This tells us that there is an ordinary line that contains these points a and b But what if we have a second ordinary line that does this? Like what if there's some line in prime that is a second ordinary line Different from l of different from m. Excuse me that contains a and b Well, this ordinary line in prime can be extended into a projective line It can be extended into a projective line call it n And then this projective line Which is not m It's not the line m I mean because if it were it would have to be I mean, I guess I take it back, right This line in prime would have to equal Sorry, the line n would have to be m because there's only one projective line that contains a and b By line determination so n and m would have to be the same thing Which would imply that if n and m are the same thing when you subtract the line in infinity you would actually Because n is just n take away l and m prime is just m Take away l if you take away l from both of these things that forces that m prime equals m prime Sorry, that was kind of a messy way of arguing this one But the thing is if you have two ordinary points there exists a projective line that's between them By line determination of the projective geometry remove the point and infinity that gives you an ordinary line And it has to be unique. Otherwise we get non unique lines on the projective plane So line determination works for this ordinary geometry Let's talk about the next axiom, which is tricancy. Sorry about secancy For for the projective geometry we have secancy for tricancy, but for the project for the ordinary geometry We want secancy. So that's our goal in mind. We want to show that every line has at least two points So what we're going to do is we're going to start off with an ordinary line So we have some ordinary line. We call it m prime But an ordinary line is just a projective line That's a little bit longer because there's this extra point at infinity That's on the or are on the projective line. So m is the white one But p if it's it has to be a point at infinity because those are the points in the geometry that aren't ordinary So this is going to be the intersection between the line at infinity and the line m And so in particular m prime doesn't contain the point p Um now here here you go. Here's the point. Um by the tricancy axiom There are going to be three points At least three points on the projective line m Now one of those points Is p but the other two points Have to be ordinary points. There has to be two points. Maybe call them a and b, right? Why can't because so tricancy gives us Tricancy is going to give us three points a b and p are some of those points we could do but because of line determination the I mean by the elliptic parallel posh we know the line at infinity does intersect the line m at some ideal point p But by line determination the intersection between m and l is unique. So there can only be one Ideal point on an ordinary projective line And so since three points are guaranteed the other two points have to necessarily be ordinary points And therefore if we go back to the line m prime m prime contains at least two points So we needed tricancy on projective lines to guarantee secancy for ordinary lines And so that's that that right there is the main reason why we want to make sure That projective geometries have at least three points per line So that when we remove the line in infinity the the resulting ordinary lines will have at least two points still on them All right. Let's talk about point existence A point existence Why are there at least three points in the geometry? Well, let's start off with three projective points. Whoops so in The projective geometry which lives above the ordinary geometry. We have at least three points a b and c But we can also choose these points in such a way that they're non-colinear because after all Projective geometry is a incidence geometry and we proved in incidence geometry. There exists a set of three non-colinear points So we have three We have three Non-colinear Projective points now the reason for non-colinearity is there's no one line that can contain all three of these points In particular the line at infinity if it contained any of these points It could only contain two of them. It can't contain all three of them So we are guaranteed that one of these points is ordinary And so without the loss of generality, let's assume that a is an ordinary point that kind of looks sloppy Let me try this again. We're going to recolored a as an ordinary point Are b and c ordinary? We don't necessarily know that but what we do know is the following that if we take the projective line So the point a and the point b are projective points if we take the line Determined by a and b that's given by projective geometry We get we get a projective line and that projective line by tricancy contains a third point which we'll call d And likewise the the projective line determined by a and c It exists and it has to contain a third point which we call e as well and so these two lines the line a b Let's call it m And call the line determined by a c n These two have to be ordinary lines because they're not the line at infinity Because after all a is an ordinary point. It's not on l And so because these because m is an ordinary projective line What we know is that it contains only one Point at infinity the other points on this The other points on this a line have to be ordinary. So when you look between when you look between d and b One of those points is necessarily Um ordinary and so let's say without the loss of generality b was an ordinary point It could be that d is two but we can only guarantee that b is an ordinary point And now when we and so then when we when we switch over to the ordinary line m prime We know that a and b are on that line Now if we look at the projective line n that line, it's also can only contain It only contains exactly one ideal point. So when you look at these two points right here c and e One of them has to be ordinary. Um, they could both be ordinary, but one of them must be ordinary So without the loss of generality, we could say c is an ordinary point right here And so then if we think of the ordinary line In prime then we see that we have we've now found three ordinary points a b and c So our ordinary geometry has at least three points. Uh, so we get the point existence Uh, next on the docket we want to talk about non-colinearity axiom So that so what we want to show is that there is no ordinary line that contains all the ordinary points So for the sake of contradiction suppose there was suppose there was an ordinary line m prime That contains each and every one of the ordinary points. It has all of them Well, as it's an ordinary line, it could be extended into a projective line So think of that projective line m and since it was an ordinary projective line That means it contains one extra point a point at infinity, which we'll call p Uh, it's an ideal point and that's just the one extra point we have there Well, because the projective plane here is non-colinear, right there exists a point In the projective geometry, we're going to call it q There exists a projective or the projective point that is not on the the projective line m Because the projective geometry has the non-colinearity axiom So what I want you to do is consider the line that's determined by q And an ordinary point that lives on m So let's say we take the line between the points q and a All right, there exist at there exist a ordinary Uh, there there exists an ordinary point a and there's this point q right here And so then by tricancy, you notice how we use it again by tricancy Actually, you can see in the proof over here if you're reading along you can see tricancy keeps on popping up over and over again There's gotta be a third point on this on this line right here b Right. Um, this is a projective point now. There's a couple things that could happen We know that a is we know that a is an ordinary point. Um, we know that this line right here Let's call this line n. I don't know if I gave it a name in the script there. I did Let's call this line n. Whoops. We don't Oh, yeah, sure. We can call it n. It's a projective line Uh We have this line n right here. So look at the line n it contains b a and q a is an ordinary point So this tells us that the line n is not the line n infinity. Likewise n Doesn't contain p. So also it's not the line infinity. So this is an ordinary line an ordinary projective line Which means only one point on this line is ideal Well, it could be that b Is an ideal point which means every other point on the line is ordinary which would include the point q But this would show that q is not on m prime which would contradict m prime holding all the ordinary points And then the other The other option is also true that if q was an ideal point that would force b to be An ordinary point and that would show that m prime doesn't contain it So if we assume there is an ordinary line that contains all the ordinary points, we get a contradiction Which shows us that the the ordinary geometry It satisfies non-colinearity and now that we've shown Line determination try can't see point existence and non-colinearity Did I say all those ones right line determination c can't see point existence and non-colinearity We've now shown that the ordinary geometry is an incidence geometry There's one more axiom to go to show that it is an affine geometry We want to show that it satisfies the euclidean parallel postulate So to prove to prove epp for this geometry what we're going to do is the following We start off with just an arbitrary Ordinary point an ordinary line call it m prime Take a point that is not on take an ordinary point that's not on the ordinary line We're going to call it a and we want to show there's only one parallel line Through there's only one ordinary parallel line through a with respect to m prime right here So let's view these things as A projective geometries now. So what if we think of m prime? As a projective line m Well, because it's a projective line. There's ordinary points mind you, but there's also this extra ideal point p And let's take the let's take the projective line determined By p and a Like here, right? We'll call that line N I think so what I saw it. Yes n so n is the line determined by p and a and m was the line that was given Well, if we think of if we think of n prime Right m prime is just n takeaway l Likewise, remember m prime was m takeaway l like so then The i the ordinary line in prime doesn't contain p Right just like how m prime doesn't contain p as well And so the intersection that existed between m So the intersection that the exists between m and n is now removed And so we don't have an intersection between m prime and n prime lines intersect at unique locations Because we have line determination. So therefore we get that these lines are now parallel m prime is parallel to n prime So it's kind of interesting. We took two intersecting Lines and because we moved the point at infinity they actually become parallel ordinary lines But that's only half of the battle, right? We now shown that parallel lines exist in this ordinary geometry But are they unique? Let's suppose there was a second parallel line So there was a second line that goes through a and it's parallel to m prime We're going to call this line in double prime So m prime is parallel to n double prime Why why can't this happen? Well in prime Is just in double prime is equal to some line Take away The line and infinity that what I guess I'm trying to say is this this line in double prime has to somewhere Intersect an ideal. It has to contain an ideal point, right? If we think of this if we think of n prime as a If we think of n prime as a Projective line then by the elliptic parallel postulate This this project of n double prime must intersect m somewhere and it's going to have to be at an ideal point Well, because it intersect in this this extended n double prime Intersects m at an ideal point It's going to have to be p because that's the only ideal point that m has and now you can see this is a problem with Line determination on the projective geometry Uh in prime the extended version contains p and a and likewise the extended n double prime also contains p and a as well So this contradicts the line determination axiom of the projective geometry And so we actually get that parallel lines are unique in this geometry So that was a big hunker wasn't it right took us about 20 minutes. I think to do that one But this shows us that this ordinary geometry be constructed by removing the line and infinity Of the projective geometry it forms an affine geometry So I think it's useful to look at some examples before we end this video right here Um, let's take for example the fano plane, which is a finite projective geometry We have points like the following. We have our three elder wands We have the resurrection stone And so we have the fano plane right here Well, who who's going to be your line and infinity right in terms of the proof It just said fix a line It doesn't matter which line you choose because of the symmetries of the projective geometries any line can act as the line and infinity Now when people talk about the fano plane, they often pick the resurrection stone as the Line and infinity because when you draw the typical diagram for fano geometry this configuration All the lines look like straight edges except for that resurrection stone It looks like a circle. So sometimes people think that somehow that makes that surfable exceptional. It really doesn't It's just the way we don't draw it But if we treat that as the line and infinity what we want to do is Consider what happens if we remove the line and infinity. So if we draw the picture again Get your cloak of invisibility No resurrection stone. We only get elder wands now Uh, we get the following. So I didn't draw the circle that time But we also can't draw any of the points and infinity. We take away any ideal point So like for example, these three points That were on The line and infinity they have to go as well And so you lose this one this one and this one these points aren't there anymore And so as I try to fix this picture, we have an ordinary point right here We have an ordinary point right here and we have an ordinary point right here and also right here So we took away the three ideal points and we're left with four ordinary points And if we try to salvage this thing, it's like, okay, we could draw some ordinary lines We could reconstruct the cloak of invisibility but in terms of Those elder wands they all got broken half harry potter got on top of all three of them Like so and so let me kind of erase a little bit of the clutter there But when you redraw this geometry When you redraw that geometry you get the following Residue right here, and I'm going to try this once more without any of the extra clutter But when you draw this geometry When you remove the line and infinity you get these four points like this And what does this geometry look like? This might not be how we drew it originally, but this geometry right here is the four point affine geometry Or using language we used before this is the order two Affine geometry Or what the book originally called four point geometry And it's not the only four point geometry Because there's the four point fan geometry, but this is in fact an affine geometry So this kind of is evidence of what the theorem was proving that this the the theorem proved that this works in general That if you remove a line from a projected geometry you always get an affine geometry All right, and so I also want to kind of mention it if you think about The projective model the real project of playing we saw before if you take this disc Which has the wraparound feature Right here. Um, well, let's treat Let's treat the wraparound line because it's kind of exceptional here, right? Let's take this to be our line and infinity And so if we draw the geometry without the line and infinity what we are going to do Is we get an open disc Uh, that is we get all points whose distance from the origin is less than One we don't get the boundary. We just get the open disc itself and that as a geometry is Um isomorphic to the euclidean plane if you want to you could use like an arc tangent map to make the finite distances extend to be Of a infinite distance, but this does in fact become a euclidean This this actually becomes euclidean geometry in this situation And so this right here is the main reason why we're trying to connect uh, that's we we brought back projective geometries here because um The so what this construction what the theorem has done here has proven that given any projective geometry You can move a line and you form a euclidean geometry, but this construction is also reversible Given any euclidean geometry you can add a line at infinity Right so if we think about our four point geometry here We can always add a line and infinity into the geometry and thus form a Projective geometry from that this is often called the projective cover of the affine geometry And so the significance of this space rp2 Is that rp2 is the projective cover of the euclidean plane? r2 And so this right here is going to be our official definition of elliptic geometry But i'm poaching the next lecture a little bit. We'll talk more about that next time Why that's so we'll talk about next time why this is going to be our definition of elliptic geometry But we'll wait until then Um an immediate consequence of what we have from the previous uh from that connection That is there's this one to one connection between affine geometries and projective geometries Since affine geometries have an order, which is the number of points on a line Projective geometries also inherit an order. Um, so like phano geometry is the order two It's the order two Projective geometry now you'll notice that the order two projective geometry actually has three points per line So when it comes to projective geometry Each line if your order n each line will contain n plus one points It's also true that all points are instant to n plus one lines In projective geometry, there'll be n squared plus n plus one points and n squared plus n plus one lines This is also true when you use infinite cardinals, although it becomes far less interesting Because it's all infinity But for finite geometries finite projective geometries We can count the number of points the number of lines the number of incidence relationships much like we did for affine geometry Basically, like how do we get this n squared plus n plus one? Count here. Well, if you recall that for the affine geometry affine geometry contains n squared points and then we add And and and every line contained in points Well, we basically add to every line A new point a point and a point at infinity So the number the number actually gets extended up to one You get one extra one right here I should actually be coloring this one here Because points were already entered incidents to n plus one lines It's the it's the number of lines that now contain n plus one points And so because every line gains a new point Uh, well, well, let's take this one for example. We had n squared plus n many lines We add one new line and that gives us the line in infinity So that's n squared plus n plus one many lines Um, but then you had n squared points to begin with and then the n plus Um, the in we add extra points and so that's where this extra n plus one is coming from All these ideal points because the line in infinity contains n plus one points So you have the ordinary points plus the line in infinity which contains n plus one ideal points as well And so I told you that was a big one I hope you brought some popcorn or maybe went to the bathroom before this video got started here I guess you could pause it, um, whatever But that that draws us to the end of this this sort of mammoth Proof here today showing you the connection between projective geometries and affine geometries And as I've already alluded to uh, this connection between, um, affine geometry with a projected geometry Is how we're going to proceed to develop elliptic geometry because elliptic geometry is going to be the projective cover Of the euclidean plane, which is affine If you like this video, please please please like it subscribe if you want to see some more Videos like this one if you have any questions post them in the comments below Or you can send me an email directly that would be appreciative as always the script to this lecture Can be found a link to it can be found in the description And I will see you all next time Have a fantastic day everyone. 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