 Welcome back to our lecture series Math 3130, Modern Geometries for students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Misildine. It seems appropriate that for our geometry class, it's lecture 3 that we finally start talking about geometry. You might recall from the previous lectures of this series that we've been talking about the academic method, models, theorems, axioms, those type of things. And we've been developing this theory, which we called FIFO based upon an example from Wallace and West Roads to Geometry. And so we did this to only focus on the axiomatic method from a very abstract point of view for which we had no intuition of the things we were proving. We just proved them by the axiomatic method. So it turns out the example we've been using this entire time has been a geometry. We'll see exactly what that looks like in the next video. In this video, I first wanted to provide to us what are some of the essential definitions of a geometry. In other words, what is a geometry? Let's define that somewhat precisely. It's actually going to be a little ambiguous, honestly, but it's okay. It'll work for us here. So when we talk about a geometry, a geometry is an axiomatic system which contains, among other things, the three following undefined terms. We have some notion of what a point is. We have some notion of what a line is. And then there should be a relationship between points and lines that we call incidents. So we'd say something like a point is incident to a line or a point is on a line. We'd say something like that. And so a geometry is some type of axiomatic system that says something about points and lines. That's what we're requiring to be geometry. There could be other notions like angles. We could have shapes. We could have surfaces. We could have planes. There's a lot more to it, but really the bare bones of what is a geometry. A geometry talks about points and lines. There should be some notion of points and lines in there. And I'm saying this as an undefined term, but admittedly, there could be a structure for which other undefined terms are taken instead, like maybe points are undefined. But because of other definitions, maybe we can define a line. So it doesn't necessarily have to be an undefined term, but points lines and incidents should be in the dictionary of this axiomatic system somewhere. Most likely the points and lines are going to be these undefined terms. So again, this definition isn't exactly precise in that aspect, but we have some axiomatic system that talks about points and lines. Now, when we talk about a model of a geometry, which because geometry as we've defined it is a theory, it's an axiomatic system, but we often refer to models of the geometry as a geometry itself. We don't usually talk about geometry theory, like the same way we do to talk about group theory. So like an abstract algebra, a group is a model of group theory, right? It's a binary operation that's associative, has identity and has an inverse. That's what a group is. And so we often talk about the models of group theory as groups themselves. Well, we don't really talk about geometry theory. A geometry, the word is abused in meaning the theory itself, but also models of geometry are called geometries. And so in lecture three, and also the next two lectures in our series, I want to introduce to us some ideas of finite geometry. That is to say, we're going to take some geometries which are finite and develop the theories of those. And the reason we're doing this is again, well, as a toy, just to kind of help us practice getting used to geometric notions. But also I want to introduce many notions of geometry very early on, but not in our native language, which for most of us would be Euclidean geometry. As we have gone through primary and secondary school, Euclidean geometry is pretty much the exclusive geometry we've learned about. And so as we start to study abstract geometry, it's important that we divorce ourselves from that native geometry and understand it's only one of many geometries. And so we're going to deliberately start with finite geometries where many things are foreign to us, but at the same time, there's an intuition of what to expect here. And we can develop those notions. We're going to do those in the subsequent videos for Lecture 3 and also Lectures 4 and 5. But before we do that, I wanted to introduce some definitions, because after all, the axiomatic method, we start off with undefined terms. So for us, at the very beginning of our geometric study, we have three undefined terms, points, lines, and incidence, which is a relation between points and lines. Now we're going to come up with definitions so we can define things. And so these definitions, I'm going to say for the rest of this video, apply to all geometries we will study in this lecture series. So the first definition we're going to define is contains. So if a point is on a line, that is to say, if a point is incident to a line, we say that the line contains the point. So when it comes to points and lines, this incidence relation, you can go either way. You could say a point is incident to a line. You could say a line is incident to a point. That second one isn't used as common, but it is acceptable. But we do sort of think there's a direction to it. So we would say that a point is on a line. That means a points incident to a line. We rarely would say that a line is on a point. Now there are some geometries that do use that phrase. They use the word on an incident interchangeably. For me, person, that feels a little awkward on the language. And so that's why we're introducing this new term, this new term contains, which is just the inverse relation of incidents here. So if we say something like, oh, a point is on a line, right? We often use the set symbol right here. So we think of, and this is something we'll get to a little bit later, but we might think of a line as a set of points. In many geometries, that is actually sufficient to describe the lines. And that's why I kind of was saying earlier that it turns out lines actually could potentially, depending on your framework here, could be a defined term. But we're leaving it as undefined right now. It's common to use this set theoretic notation to say that a point is on a line because we think of the lines as sets of points. We would never really go the other way around. We would never say that a line is a member is on a point in that set theoretic sense. That doesn't really make any sense. Therefore, this notation is kind of weird. And we'll stick with things like this. And so you could read this as P is on L, but you could also say that L contains the point, L contains P, again, thinking kind of as a set, but that gets a little ahead of ourselves. Lines are not necessarily sets of points, but oftentimes they are, we will lose the examples where that can happen. Our second definition. We say that two lines intersect. If there exists a point that's on both lines. So if there is a point that's on both lines, we say that the lines intersect and it's part of the intersection. And that's often what we'll do here. If we have two lines, m and n will again borrow notation from set theory here, maybe look a little bit more rounded here. So if we think of L and m as sets, then the symbol we draw here in between is then the set intersection. And so borrowing that notation, we're going to take the definition of this L intersect m to be the set of all points P such that P is on L and I should emphasize here and here P is on m. Alright, we say that two lines intersect if this is non empty. Alright, related to that, we say that two lines are parallel if they do not intersect. So if this if this set here is the empty set, excuse me, if the intersection is the empty set, then we say that two lines are parallel. Now some comments I want to make about this. So lines are undefined terms, but we're defining the intersection of lines to be sets of points. Alright, that's that's what this is. The intersection of two lines is a set. And therefore it could be the empty set potentially. Now I'm not saying that L the lines L and m are sets, they could be sets of points, but they don't have to be it depends on the interpretation of the undefined terms to determine whether lines are actually sets of points or not, we don't need that. But we can still define the intersection of two lines to be a set. And that set could be the empty set it might not be. If it is the empty set, we call that we call those two lines parallel. Now, many of us have notions of parallel. And honestly, our notions of parallel that we have generally come from Euclidean geometry when we think of parallel lines, we often think of as lines that run side by side, you know, kind of like train tracks. That is we often think of parallel lines with regard to equidistance that no matter where we are along the lines, the two points corresponding to each other on the two lines here are always the same distance apart. That idea of equidistance would require we have first of all a notion of distance for which in this basic framework, the only notions we have are points, lines, and incidents, plus the things that we're defining presently. So for a general geometric framework, we don't have distance. So we can't define lines, parallel lines using equidistance because we don't even have distance. What about slope? It's common to say that two lines are parallel if and only if they have the same slope. Again, that's something true in Euclidean geometry, the geometry probably most used to, but how do you define slope in this situation rise over run, rise over run, you know, you have some change of y, some change of x, I want you to be convinced, I want you to convince yourself that's still a statement about distance, right? We're saying that the amount that the y coordinate changes and the amount that the compared to how how the x coordinate is changing, we're saying that's the same. We don't have a notion of slope at this moment here. So that also we we couldn't introduce that effect thing and things like x and y coordinates, we don't even have developed yet, at least not for our most general framework. So how can we define parallelism using just the fundamental principles of points, lines, and incidents? Well, we can define intersection using just points, lines, and incidents like we just did. And then parallelism is just one step beyond intersection. Oh, when there's no intersection, we say that things are parallel. Now I should mention that when it comes to parallel lines, we'll often use the symbol, you have a line L, you'll use this double line as this relation between us, we actually think of parallelism as a relation. We say that L is parallel to M if their intersection is empty. I should mention that this symbol is derived from this idea before. So like in Euclidian geometry, we say two lines are parallel, of course, via the equidistance, the train track like picture. So our symbol for parallelism is actually looks like that. I'm not saying that's what it's going to look like in general geometries, but just from a cultural perspective, that parallel symbol is commonly used here. And for those who like to type things in latex, just so you know, this symbol is created by backslash parallel. This friend right here, just so you know, it's very obvious that when we define parallelism using this symbol here, this gives us a symmetric relationship. That is, if L is, if L is parallel to M, that means M is parallel to L, because after all, L is parallel to M exactly when L intersect M is empty. But as a set, L intersect M is the same thing as M intersect L, which then must likewise be empty as a set. And therefore we would get that M is parallel to L. So the parallel relationship is always going to be a symmetric relationship. Now for reasons that might seem strange at this venture, we also are going to find the parallel relationship to be reflexive. Okay, so that means that L is considered parallel to itself. Now again, that might seem a little bit weird at this moment, because if two lines being parallel are means their intersections empty, if you take L intersect L by construction, that should be all of the points on L, which, but interesting enough, could actually be empty at this venture because we have no no axioms that guarantee that lines actually contain points. But besides that little issue, if a line contains a point, then the intersection of L with L should be those the set of those points. So that intersections on empty. So what's going on here? So it turns out that this this relationship is typically defined between two distinct lines. So we say that to notice in the definition to distinct lines are parallel, if they do not intersect. But what about if your two lines are not distinct? That is, it's the same line. Our definition of parallel actually doesn't apply there. And so in all reality, we can extend the definition of parallel to that setting. So when two lines are distinct, we call them parallel, if they don't intersect, but if it's the same two lines, that is, if the two lines are actually coincide, we can still call them parallel. So it's actually not in contrast with the with the definition we have previously. So we've actually constructed the parallel relationship to be reflexive and symmetric. The reflexive one is sort of silly, but we're allowing it. The symmetric one is the usual idea where two lines are parallel to different lines are parallel if they don't intersect. Now you might wonder reflex is symmetric. If I had one more property called transitivity, parallel lines would form an equivalence relation. At this venture, we can't do that. Because the type of geometries we can construct, there are some types of geometries for which parallelism does form a transitive relation. This is called affine geometry, which is a broader category than Euclidean geometry. But there are going to be some geometries like phano geometry, elliptic geometry, projective geometry, and some other ones we're going to define for which the parallel relationship is actually not, it's not transitive. And I won't say so much about that right now, but just so you're aware, we define parallel, the parallel relationship to be reflexive and symmetric, but is not in general transitive. There will be some cases where it is, but in most cases that we explore it won't be. So be aware of that going forward. The rest of our definitions here, these next few will be able to get through a lot faster. We say that a set of points, notice this is a set of points, not necessarily a line, but we say a set of points is collinear, which is spelled with two L's just so you know. A set of points is collinear if there exists a single line incidence to all of the points. So if our set of points, if there's one, if there's a, if there's is a line, maybe there can be multiple lines, but if there's at least one line that's incident to all of the points in the set, then we would say that the points are collinear. So if L intersect L contains the set of points, that set of points is collinear. Related to that, if we have a set of lines, we say they're concurrent. If there exists a, at least a point, you know, when I say single here, I don't mean there's only one, there's at least a point incidence to all the lines in the set. So in other words, if we look at the intersection of all of these lines, it's non-empty. That's what you get a concurrent set of lines. And then this is a term I used earlier here. What does it mean for a geometry to be finite? Because after all, lecture three is supposed to be, and four and five, mind you, it's supposed to be about finite geometries. A finite geometry, we say a geometry is finite if the number of points and the number of lines is collectively finite. So we have a finite number of points and we have a finite number of lines. That's what we call a finite geometry. If you have a finite number of points and a finite number of lines, then of course you're going to have to have a finite number of incidence relations as well. But we don't have to worry so much about that. Count the points, count the lines. If that is finite, then it's a finite geometry. If it's not a finite geometry, then it's otherwise an infinite geometry. Now that could mean that you have an infinite number of points and an infinite number of lines, but it could also mean you have an infinite number of points and a finite number of lines, right? It could also mean that you have a finite number of points and an infinite number of lines. So if you have an infinite amount of something, we call it an infinite geometry, otherwise it's a finite geometry. So these are going to be some basic geometric terms we're going to use to describe every geometry in this lecture series, but in the next video we're going to take a look at a finite geometry that we've actually already been studying, but maybe just didn't know it.