 In a previous video, we introduced the idea of the three parallel alternatives. We had LPP, the elliptic parallel postulate, we had EPP, the Euclidean parallel postulate, and we had HPP, the hyperbolic parallel postulate. The elliptic parallel postulate says that there are no parallel lines. The Euclidean parallel postulate says that parallel lines exist and are unique. And the hyperbolic parallel postulate says that parallel lines exist and are multiple. There's more than one parallel line. And play here. Now it turns out that these three parallel postulates, they're alternatives, you can't have more than one of them. You don't even have to have any of them. But if you take the four actions of incidence geometry and you equip one of these parallel alternatives, then you get a consistent geometry. The Euclidean parallel postulate plus the incidence axioms gives you a consistent geometry. There exist geometries that we've seen in this lecture series to have their incidence geometry with Euclidean parallel postulate. The elliptic parallel postulate, if you equip it to the incident axioms, you get a consistent system. There exist incidence geometries that satisfy the elliptic parallel postulate. We have listed some ourselves. And then finally, if you take the hyperbolic parallel postulate and equip it to the four incidence axioms, you do get incidence geometries with satisfy the hyperbolic parallel postulate. We've seen some examples of that as well. Hand shaked geometry is a good example of that. And so what this tells us is that these parallel postulates with the incidence axioms is a consistent logical system. It also tells us that parallelism is independent of the incidence axioms. The incidence axioms are insufficient to tell us what is true about parallel lines. Maybe there's no parallel lines. Maybe there's one, maybe there's more than one. Maybe you have a hybrid of these things like we saw in a different example of five point geometry. But importantly, the parallel axioms are independent of the incidence axioms. Therefore, when you want to further develop your theory, because we've basically gone as far as we can with incidence theory, if you wanna develop the theory using parallelism, you have to accept one of these parallel alternatives or again, maybe some hybrid of them and then start approving the theory with that. And so that's what then leads to our definitions right now. So we say that an incidence geometry is affine if it's additionally satisfies the Euclidean parallel postulate. So if you take the four axioms of incidence, line determination, secancy, point existence, and non-colinearity, if you also give it EPP, the Euclidean parallel postulate, those five axioms together give us affine geometry. And affine geometry is gonna be the titular topic for lecture eight. Another important geometry that's an incidence geometry we get is projective geometry. An incidence geometry is projective if it satisfies the elliptic parallel postulate and, that's a big and, all lines contain at least three points. So if we take incidence geometry equipped LPP and this other strengthening of secancy, because secancy says you only need two points on a line, why not to be a projective geometry, we need at least three. We'll talk about projective geometry later on in this lecture series. Take a look for that lecture, but it's gonna be a while from now, so sorry. We're gonna do projective geometry in tandem when we talk about elliptic geometry because projective geometry, elliptic geometry are projective, projective is more broad. But again, that'll be stuff we talk about later. Why not, why do we need at least three points? Now I should mention that phano geometry is a projective geometry. So if we draw our deathly hollows right here, real quick, like so. So we remember we have our seven points, our seven lines, and we have our resurrection stone, cloak of invisibility and three elder wands. Phano geometry is an example of a projective geometry. It does satisfy the elliptic parallel posture that was actually actually five of phano geometry. It's an incidence geometry, we've talked about that. And all lines contain at least three points and phano geometry, all lines have exactly three points. But you could have a projective geometry with more than three points per line. You have to have at least three. Now it sort of begs the question, why do we need at least three? Now let me kind of give you an example here. What if we looked at our three-point geometry before? This is the only three-point incidence geometry. And notice here that it is an incidence geometry, like I said, it also satisfies the elliptic parallel posture because all lines intersect each other. There's no parallel lines here. So this would be a projective geometry except, you know, there's at least three points, right? The lines in three-point geometry, they're only handshakes, they only have two points each, not three points. And it feels like projective geometry is defined in the following way. It feels like a projective geometry is an incidence geometry that satisfies the elliptic parallel posture that's not three-point geometry. It's like, what? Why can't I join the club? It's kind of like, you know, no girl's allowed sign on the clubhouse or something. It's like, that's really rude. There's no reason why we shouldn't include this. Well, it's not just some arbitrary, sexist, racist distinction. No, no, no, we're not intentionally segregating our geometries just for superficial reasons. There is a reason why we don't want to consider three-point geometry as a projective geometry because in so many ways, it doesn't behave like a projective geometry. It behaves differently and so it does need to be excluded and those reasons will be more apparent when we talk about projective geometry in the future. So then if this point, if three-point geometry is not a projective geometry, where does it belong? It is a handshake geometry, but the parallel alternatives for handshake geometry can be a little confusing. For three-point geometry, it's actually satisfied the elliptic parallel postulate. All right, handshake geometry pretty much always is hyperbolic. The only exception is three-point geometry and then the four-point geometry we defined earlier. Right, with four-point geometry, it actually satisfies the Euclidean parallel postulate. So this is actually an affine geometry and so four-point geometry really is the start of affine geometry, which we'll talk about in the next one. So it is a handshake geometry, but we really want to think of it as an affine geometry. And then all of the handshake geometries, five and above, they're hyperbolic, those all belong together. This three-point geometry, we could consider it a handshake, but kind of like four-point geometry, it's somewhat degenerate in that regard. It should be something else. The family of geometries that three-point geometry should belong to is actually what we call fan geometry. Now, fan geometry, which we'll call the fan geometry with endpoints fn, this is the geometry that's constructed by the following scheme. Okay, five point, excuse me, fan geometry looks something like this. You have a line, whoops, you have this line. And this line contains every single point in the geometry, except for one, because if all the points were on this line, it would violate non-colonarity, but there's exactly one point that is not on this line. And then by line determination, there must be a line between all these other, there has to be a line between this point off the mega line, the mother line, and all the points on the mother line. But to avoid an intersection more than one, these all have to be handshakes. And so fan geometry looks something like this. You have this line that hosts all the points, except for one, and then the one point off of the line, there's handshakes from that point to all the other points. And so three-point geometry is really just f3. It's the fan geometry with three points. F4 would look like this picture right here. And it turns out that up to isomorphism, there are only four, there's only two four-point geometries. One is the fan geometry F4, and the other one's what we've been calling four-point geometry. It's the four-point handshake geometry, but really it's the four-point affine geometry, which we'll talk about more of those in the future. And then also for a little bit more exposure, this is a diagram for the five-point fan geometry. This would be F5. So I want you to note here that in an electric series, now I've seen three different five-point geometries. There's the handshake geometry, H5. There's fan geometry right here, which of course the reason why this is called fan geometry is because of the shape. It looks like this hand fan that you wave at yourself if you wanna cool yourself down. We also had this other five-point geometry that looked like a diamond that we saw in a previous video. It looks something like this. Right, these are all distinct. We get five, so far we have three different. Classifying all of the five-point, six-point, seven-point geometries turns into a very, very fascinating question, which can get a little bit technical the more and more points you get. And so that's not something we're gonna dive too much into this lecture, but I just wanna point out here that fan geometry is an incidence geometry, right? When you look at your picture here, non-coloniality was sort of like the motivating factor here. Point existence, right? As long as n is greater than three, you'll have three points or more. Line determination also kinda gave us all these handshakes here. That's why they are there. And then the last one, see, can't see. All the handshakes have exactly two lines and then the master line, the mother line there, it has more than two points in general. It has at least two though. So it is an incidence geometry. And also by construction, fan geometry satisfies the elliptic parallel postulate, okay? And for which all lines intersect each other because if you're two handshakes, if you're two handshakes like these ones right here that intersect at this point off the mother line, but if you're the mother line and a handshake, one of these children, then they intersect at that point that you see right there. So we do have the elliptic parallel postulate, no lines are parallel in fan geometry. So the reason why projective geometry has this rule that lines need at least three points is that projective geometry is different. It doesn't include fan geometry. And the three point geometry is really F3, the three point fan geometry. That's a distinction that I wanted to point out to you as we in this lecture talking about the parallel alternatives. I do appreciate you watching this video. If you liked what you saw or learned something, give these videos a like, post any questions in the comments below if you do have any questions. I'd love to see that and I'll answer them as soon as I can. 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