 When looking at the axioms of Fano and Young Geometry, we see they differ in only one axiom, what we call axiom 5 for those two geometries. Which axiom 5 states something about the existence of parallel lines. Axiom 5 for Fano geometry said that parallel lines don't exist. Axiom 5 for Young Geometry says that given a line and a point off a line, there's a unique parallel line to the given line through the given point. So it doesn't just say that parallel lines exist. It says they exist up to some uniqueness criteria. Now because both are models of incidence geometry, we see that their respective parallel properties are independent of the incidence axioms. That is, the incident axes are not enough to determine the nature of parallel lines in geometry. So what we're going to do in this video is actually list three parallel alternatives. We call them parallel axioms, but it's also appropriate to call them parallel alternatives, meaning that their axioms are actually mutually exclusive. These three alternatives we put on the screen right now, you cannot have all three of them. In fact, you can't even have two of them because they're mutually exclusive. If one of these statements were true, the other statements would have to be false in that geometry. And that's why they're often called parallel alternatives. I should also mention that in axiomatic theory, the word axiom and postulate are really synonyms of each other. You can use them interchangeably. But when one talks about parallel alternatives, the word postulate is typically more used than the word axiom. Why is that? Well, that's really just tradition. And it really comes down to Euclid's book on the elements where he essentially established the modern idea of the axiomatic system studying geometry. With Euclid's geometry, his fifth postulate was a statement about parallel lines. And that postulate has been scrutinized so much that the word postulate is always used when describing parallel axioms. And also it's a literative in English. That's kind of fun, parallel postulate. So you'll probably hear me call that, and they're all listed as parallel postulates, but postulate and axiom, the words mean the same thing. So what are the three parallel postulates that we're going to consider in this lecture series? The first one is known as the elliptic parallel postulate. These names will make a little bit more sense in the future, because in particular the elliptic parallel postulate, the Euclidean parallel postulate, and the hyperbolic parallel postulate, will be sort of the designator for elliptic, Euclidean, and hyperbolic geometries, something we'll define later on. But for the moment, just imagine that this is an axiom, a fifth axiom we could attach to the four incidence axioms and then get a different, a more restrictive geometric theory. The elliptic parallel postulate says that all lines intersect, and another way of saying that is that parallel lines don't exist. Axiom 5 of phano geometry was exactly this, the elliptic parallel postulate. We had the first four axioms, phano geometry, plus this elliptic parallel postulate. The Euclidean parallel postulate says that for every line and every point not on that line, there exists an exactly one parallel line 2L, which contains P. This was exactly axiom 5 of young geometry that we explored previously in this lecture series. This is known as the Euclidean parallel postulate because this is equivalent to Euclid's famous fifth postulate of geometry. In the elements, Euclid gave five postulates for geometry. These are the five rules of geometry, and everything else can be derived from that. Euclid's work in the elements on axiomatic geometry was breakthrough. There's no other contemporary or predecessor to Euclid that we know of that approached geometry in this axiomatic method. I confess that Euclid's methods were not perfect. There were assumptions that Euclid used that were not spelled out in the axiom, so his axiomatic system was incomplete. He didn't have all the axioms on paper. He had five of them, but there were some other implicit axioms he was using in his proofs that weren't stated. That's a big no-no for geometry. Euclid did this thousands of years ago, and so for his age, that was amazing. In modern standards, it's not good enough, but I should mention that his approach to geometry was groundbreaking. His fifth postulate in particular has to do with parallel lines. His fifth postulate is what makes Euclidean geometry Euclidean. It's named after him for a reason. Non-Euclidean geometries can have Euclid's first four postulates, but they don't have the fifth one. Long history of geometry that led into the modern era actually was people believed the fifth postulate to be a theorem of the other four postulates. And, of course, also the invisible postulates he didn't mention. But no one could ever do it because, in fact, Euclid's fifth postulate was independent of the others. You could take Euclid's first four postulates and attach the elliptic parallel postulate or the hyperbolic parallel postulate. Again, Euclid had some gaps in his lodge. You could have to fill in those gaps, but you could fill in those gaps and adopt a different fifth postulate, and you can get a consistent geometry there. Now, I should mention that this version of Euclid's, the Euclidean parallel postulate, it was not actually how Euclid wrote it. This version is actually due to Playfair. Euclid's fifth postulate is expressed a little bit differently, and we'll talk about that in a future lecture. For our purposes in the language of incidence geometry, Playfair's version is a little bit better. And therefore, because, again, you could have had some implications that weren't stated and assumptions that weren't stated. So Playfair's version is a much simpler, cleaner version for the realm of incidence geometry. Then the third parallel postulate that we're going to take on is the hyperbolic parallel postulate, which sounds a lot like the Euclidean parallel postulate, which says that for every line and every point not on L, there exists more than one line parallel to it. So the hyperbolic parallel postulate guarantees the multiplicity of parallel lines. You could have a line with a point off of it, and you could have multiple parallel lines that go through that point. That's what the hyperbolic parallel postulate guarantees. It doesn't necessarily guarantee how many parallel lines there are, but there are more than one parallel line given any point. The elliptic parallel postulate could be rewritten to be in harmony with the structure of these statements here. It could be written to say that for every line and every point not on L, there exists less than one line parallel to it. Well, if you're less than one line, there's zero lines parallel to it, so there exists no parallel lines. I wrote it this way just for the sake of simplicity, but it also is in this situation. So given any line and one point off of the line, how many parallel lines do you have? Do you have less than one? Exactly one are more than one parallel line. Those are these three alternatives. And when you think of that way, you can see why we have these three parallel alternatives. And they are mutually exclusive. You can't have less than one and exactly one parallel lines. You can't have less than one and more than one parallel lines. So these are mutually exclusive. You can't have more than one of them. Now, let me give you some examples of geometries. We have talked about that, that satisfy these parallel alternatives. Three-point geometry that looks something like this. Three-point geometry is an example of, well, it satisfies the elliptic parallel postulate. All lines intersect each other. That was actually one of the axioms of three-point geometry. Phano geometry, which looks a little bit more sophisticated than, of course, three-point geometry. We get the picture looks like this. Remember, it's the deathly hollows with three elder ones. Not my best drawing of it because these lines should be, we should be going through this part right here. We're going to bend my elder one for the sake of this diagram. But this is also an example of a geometry satisfying the elliptic parallel postulate. All lines intersect each other. That was axiom five. When it comes to the Euclidean parallel postulate, we've seen some examples of geometries that satisfy that. So if we take what we call four-point geometry, we get a picture that looked like this. For which if you take a line and you take a point off the line, there was exactly one parallel to it. If I take this point, there's your parallel line there. If you take a different line, take a point, there was exactly one parallel line. Same thing here. If you take this line and this point, you're going to get that parallel line. Remember, because even on the cross of the diagram, there's no point of intersection there. They are parallel. So four-point geometry is, in fact, it satisfies the Euclidean parallel postulate. It actually was axiom two for the four-point geometry. For young geometry, we also took axiom five for young geometry. That is the Euclidean parallel postulate. And hopefully I can do my best at reconstructing young's geometry on the fly right here. Let's see how well I can do. So we had our horizontal lines. We had our vertical lines. We had these slant lines. One right here. One like this. One like this. And then we had the other slant lines. One like this. One like this. And then, oh, now that I'm almost complete. Oops, sorry, that's not supposed to go there. It's supposed to go here. And then this one is supposed to go here. Nope, not there. Try that again. This is why color coding can help a lot. So, okay, now we get young's geometry. It's just a mess. It's on the screen right now. If you pick any line and a point off the line, there's a unique parallel line to that line. Take a different point, like this one. There's a unique parallel line. If you take, for example, this line and you take a point, you take this line and take a point off this line, there was a unique parallel line to it. No other lines going through this point are parallel to it. So the uniqueness is there. It was one of the axioms. Now, an example that we haven't yet talked about is, well, what's an example of a hyperbolic geometry? An incident's geometry that satisfies the hyperbolic parallel postulate. Now, I haven't mentioned this in the lecture series up to this moment, but in the homework for my students, you have been exploring this geometry, which we call five point geometry. Five point geometry satisfies the hyperbolic parallel postulate. We could draw a picture of a five point geometry by taking points of a pentagram, something like this maybe, a pentagram and a pentagon right there. The star forms a pentagram, the pentagon. This is five point geometry. This is an example of a hyperbolic geometry in that it satisfies the hyperbolic parallel postulate. That's what I mean here. If you take any line like so, and you take a point off the line, I can actually find two lines that are parallel to it. And by the symmetry of the diagram, every point I consider will look something like this. So there are two lines parallel to the given line through that point. So this is not Euclidean. This satisfies the hyperbolic parallel postulate. Now, given five point geometry in three point, four point geometry, I really should clarify that when I say five point, four point, et cetera, right? Well, this is the only three point, this is the only three point incidence geometry. This is not the only four point. This is not the only four point incidence geometry. There's another one which we'll talk about in a later slide. And there's two distinct four point geometries there. And this is not the only five point geometry. There's a couple of those. I'm not going to list all of them right now. But they do belong to a family, which is known as the handshake geometry. Handshake geometry, what it means is a line that only contains two points, because secant C says lines have at least two points. If a line only has two points, we call that a handshake because you can think of it as like as you go around in a meeting and you shake everyone's hand, there's a line between two points if they shake hands. When people shake hands, you can have three people shaking hands at once. That would be weird even without COVID-19. That would be a weird thing to do. And so a handshake is a line with exactly two points on it. Handshake geometry is when you take your points, every line is a handshake. So you take every possible pair, right? And so in handshake geometry, if you take the set HN to denote the handshake geometry within vertices, there's going to be in vertices. And since every line is exactly two points and you have to have a line between two points, the number of lines in handshake geometry is the same as the number of pairs. And so the number of lines in handshake geometry is equal to N choose two, where that's your binomial coefficient N choose two. So for this one, handshake geometry H3, this is what we call three-point geometry. And three choose two is going to give you three of that situation. You have three points, three lines. H4, that is the handshake geometry on four points, that's what we've been calling four-point geometry. This is really the four-point handshake geometry. It has four points and it has four choose too many lines, which would be six. And then lastly, H5, this five-point geometry we've been talking about, this is really five-point handshake geometry. It has five points and it'll have five choose too many lines, which is 10. On the screen, I've now illustrated the six-point handshake geometry. It has six points and it has six choose too many lines there. With six choose two, you're of course going to take six times five divided by two. So you should have 15 lines in that geometry. Handshake geometry with the exception of three-point and four-point handshake geometry. Handshake geometry always satisfies the hyperbolic parallel postulate. Take six-point handshake geometry, for example. Take a line and take a point off the line. Notice that this line right here has two points on it since it's a handshake. And this point right here, I can form a handshake with the N minus two other lines here. Okay, so I have two points in play on the line. Then there's a third point here. So there's three points that are not part of the original line nor part of this point. Let me change the color scheme to make this a little bit better. So we have two points here, one point here. So if I take N minus three, then these three other points can be formed handshakes with the point in question. So I'm going to get three lines parallel to the original line that contained this point. And so I can actually tell you how many parallel lines there's going to be. So in HN, you're going to have N minus three parallel lines to parallel lines to this situation. And so, of course, when you take N minus three, when is that greater than one? Well, that happens when N is greater than four. So five, six, seven, eight. Handshake geometry is going to be hyperbolic when N is greater than four. When it was four, you had Euclidean. When it was less than four, you get elliptic. And you can't have a handshake geometry with only two points because point existence says you need at least three points and handshake H2 would look like this. That's not an incidence geometry.