 Welcome back to our lecture series math 3130 modern geometries for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misaline. In lecture six, we're going to introduce Hilbert's incidence axioms. We've explored now four distinct models of finance geometry in previous lectures. We have our three-point geometry, our four-point geometry, we have phano geometry, and young geometry, and my students would have been exploring other finite geometries in their homework assignments, like five-point geometry, for example. So although these different geometries express some very different properties, like phano geometry, young geometry demonstrates some very different properties when it came to parallel lines, the existence of them and such, there were many theorems and properties that remain true for all of these finite geometries. These common properties give rise to the so-called incidence axioms, and those common theorems were in fact the consequences of the four axioms of incidence. Remember, incidence here is the relationship between a point and a line. We say a point is incident to a line. It's a non-defined term, but intuitively trying to think that the point is on the line, things like that. So these four axioms of incidence are going to be due to David Hilbert. They're part of the Hilbert axioms of Euclidean geometry. We'll talk about that in much more detail in the future, but for the current moment, Hilbert categorized his axioms of Euclidean geometry into various families. We have the incidence axioms, which we're going to talk about in this video. There's the betweenness axioms, sometimes called the order axioms, which we will talk about later in this lecture series. The congruence axioms and the continuity axiom, plus also the parallel axioms. The parallel axioms we'll talk about later. All of the axioms of Hilbert we'll talk about later as we build towards Euclidean geometry. And so in this lecture, I want to establish the four axioms of incidence and essentially every geometry we talked about in the future will be an incidence geometry. The only exception to that we will provide will basically just be counter examples for statements about incidence geometry. That is to say we want to provide examples that, oh, this is a theorem or whatever. We'll use them as these counter examples. We'll just we'll keep it simple like that. So there's four axioms of incidence geometry. I believe Hilbert just labeled them I1, I2, I3, I4. That feels a little impersonal and sterile. So instead we're going to add some colorful names to them. And as we refer to these axioms in the future, we'll always refer to them as these names. And this is not a standard naming system. So it's so unique to this lecture series. So as you get deeper, deeper into lecture series, these names will be you'll be quite affectionate to them, I'm sure. So the first axiom of incidence, we're going to refer to it as line determination. For each two distinct points, there exists a unique line containing both of them. Nearly every finite geometry we've discovered in the lecture series, I can't think of an of an example this, had this as an axiom. For phano geometry and young geometry, this was verbatim axiom 4. For 3-point geometry and 4-point geometry, while I don't remember the exact number off the top of my head, these were verbatim or at least something phrased slightly different but equivalent to what we have right here. These were an axiom given to those geometries. This is a very important axiom for geometry because it really tells us the relationship between points and lines. And the reason I call line determination is the following idea. If we have two points p and q, there's often a common phrase, at least when one talks about like elementary geometry in the United States here, we often say the phrase that two points determine a line. That is, if you have two points, there's only one line on both of them. There's at least one line but it's also unique. There's not a second line that passes through these points. You don't get something like this. And so this is line determination, two points determine a line. That's how we want to think of this. And an alternative geometry without this axiom, that could be a possibility that there could be multiple lines between points. Or maybe there's no line between point, not the case here. Because of line determination, any two points have a unique line incident to both of them. Then our second axiom, we're going to call it secant C. Secant C says that every, for every line, there exist at least two points on it. Every line has at least two points. So that this, I don't want you to confuse this with axiom one here, because axiom one says you start off with two points. There's exactly one line between them, for which that line will have two points. But what about a line that exists but its existence was not determined per se by points? Could there be a line with only one point on it? For which axiom one wouldn't apply to that line because there's only one line on it. Or one point on it. Or could there be a pointless line, a line without any points on it whatsoever? Well the secant C axiom says that all lines have at least two points. There could be more. There could be infinite for all we know. But secant C guarantees that every line has at least two points. So you don't have a one point line. You don't have a pointless line. And the reason we call it secant C, this is just a geometric metaphor. Remember a secant line is a line that intersects a circle at two distinct points. There's at least two points of intersection. I mean for a circle that's the maximum there. But that's where this term secant C coming from. That the points are going to cut the line. All right. Axiom three tells us, well we call it point existence. It says there exist at least three points in the geometry. There exists at least three points. So we have at least three points, something like this. There could be more. There could be infinite. There could be four. There could be 17 points, whatever. There's at least three points. Every geometry really needs some type of axiom of existence. When we looked at three point geometry and four point geometry, there were axioms that said things like there's exactly three, there's exactly four. Which of course means there's at least three in that situation. Those are stronger conditions than the point existence we take on now. When we came to phano geometry and young geometry, we didn't say that points existed. We said lines existed. But we had statements similar to secant C. In phano geometry and in young geometry, we had an axiom that said all lines have exactly three points, which is a strengthening of this statement. If you have exactly three, then you have at least two. It's just more specific. And so because lines exist and lines have points, points exist. And we can get point existence from that situation. So my point is that with all these geometries, there's some statement of existence. If you don't have a guarantee that points or lines exist, then you might just have the empty geometry, which is not super exciting because there's no points or no lines. You also could have, could you have points without lines? Well, because you have points and because you have line determination, lines have to exist as well. Or the other way around, if you have lines and secancy, then point has to exist as well. So secancy and line determination tell us there's a relationship between lines and points. Line determination gets us from points to lines. Secancy gets us from lines to points. And then point existence tells us that there exists a point. And since we can go from points to lines, we have lines as well. But I'm getting a little ahead of myself. I'm starting to prove things about the incidence axioms. We'll get to that in the next video. The last axiom incidence, number four, I'm going to call this non-colinearity. Remember that points are collinear if they live on the same, if they're incident to the same line. Non-colinearity says that not all points live on the same line. If I take the set of all points, that is a non-colinear set. There is no line that contains all of those points. And this was also often taken as an axiom for many of the Financiams who studied as we went forward. If I remember correctly, if I'm incorrect, you don't have to correct me here. But I think on Young and Fano Geometry this was axiom three of our axioms and then with three point and four point, they might have had something similar to that. I don't recall at the top of my head, I do apologize. But these are the four axioms of incidence established by David Hilbert for which if a if a geometry satisfies these four axioms, we call it an incidence geometry. Some people call it a linear geometry or linear space. I prefer to avoid that term because it starts to sound a lot more like a vector space, like from linear algebra. We don't want to confuse that there because actually the geometry of a vector space has a more structure than this. They are incidence geometries, vector spaces are. I'll give you some clarification on that a moment. But there's also other geometric structure going on there for which an incidence geometry in general doesn't have. So I'm not a big fan of that term, although it is used. So in our lecture series, we will refer to this as an incidence geometry. Some people call it an incidence system or something like that. But that's again starting to sound more combinatorial and less geometric. So we'll call this an incidence geometry in our language here. And so going forward for this video and other videos, you might want to copy down these incidence axioms and put them in a notebook as you're studying geometry here because we're going to refer back to them over and over and over again. But rarely are you going to see them on the screen. So take a moment, pause the video if necessary, and copy these before we go on. So next what I want to do is actually discuss some examples of incidence geometry, for which I've mentioned them already, but let's be more specific here. Three point and four point geometry, which we discussed in lecture three, are examples of incidence geometry. The five point geometry that we discussed in the homework, at least for my students, if you've been following the videos and not doing the homework that you might not know what that is, but it's also an example of five point geometry. Excuse me, five point geometry is an example of incidence geometry. And then Fano and Young Geometries, that we did in lectures four and five, are also examples of incidence geometry. These are examples of finite incidence geometries. What about an infinite incidence geometry? Well, I mentioned vector spaces a moment ago. The vector space R2, this is the Euclidean plane. This is sort of like the archetype of what a geometry is supposed to be, although there are many non-Euclidean geometries, which we'll talk about in other lectures, of course. But R2 is a incidence geometry. Incidence geometry, it does have points, it has lines, it has incidences, these are undefined terms. So to tell you that R2 is an incidence geometry, I need to interpret the undefined terms. What is a point? What is a line? What is incidence? And then with that interpretation, argue why the four axioms are then statements, why they're true statements, I should say. So what is a point in the set R2? Well, R2 is it's the Cartesian product between R and R. Every element of the set R2 is an ordered pair, X and Y, where X and Y are real numbers. It's very common convention that when you talk about a geometry, you denote it as a set, and that set is the set of points. So ordered pairs X comma Y, those are points in the geometry. Again, common convention that when we refer to a geometry, we refer to it by the set of points, and then the set of lines and incidences implicit. It's not specifically stated, although we have to be very, we have to be explicit at this moment so you know what that means. So a point is just an ordered pair. It's an element of the set R2. What is a line? A line in this interpretation is actually a linear equation. Ax plus By equals C, where of course A, B and C are not all 0. Two of them could be 0. One of them could be 0. None of them could be 0. But they can't all be 0. That's not considered a linear equation. I guess I should say A and B can't both be 0. C could be 0. I don't care. But A and B can't both be 0. Which that's the problem. A and B are never both 0. Otherwise, it's not a linear equation. And so linear equations are lines. And this is really, when we get down to this, to the basics here, that's why we call these linear equations. Because linear equations give us lines. We often like to think of like an analytic geometry. The graph of a linear equation is a line. Hence why we call this a linear equation. We're kind of going the other way around here. We're saying a line is this equation. You can call it the graph if you want to. But I didn't want to get into anything like that. But the line is this equation. And because there are equivalent equations that give us the same solution set basically, one has to be a little bit more careful because you could have the multiple lines. So really, we have to place an equivalence relation on this. We're really talking about the graph of the line. That the graph of the linear equation, that's what a line is. But I'm not trying to get too much into the details here. Incidents is then defined to be a point is incident to a line if it's a solution to that line. That is if you plug in x and y and it satisfies the equation. That's incidence. And that would then we can then show that this is an incidence geometry. What are the what are the axioms again? Let's go back up. So any two distinct points determine a unique line. And this is a practice you do in like college algebra, intermediate algebra. You know, previous algebra classes, you do you do such a thing. It's like, oh, I have a line x1, y1, and I have a second point x2, y2. If these two points live on the same vertical line, it's pretty easy to determine what that point is. But if not, if they're not on the same vertical line, then you can compute their slope. You can use point slip form, put in slip intercept form, yada yada. You can construct the line that contains both of them. And that will be the only line that contains them. Do lines contain, do lines take at least two points? In this geometry, all lines have infinitely many points, uncannably many points, frankly. Do points exist? Yes, there's uncannably many points, but that's at least three. And then not all points lie on the same line. Any line you construct, give me any equation, I can find you something that doesn't solve it. So give you give the equation ax plus by equal c. Can I construct something that's not a solution? Well, if c is not zero, then notice the point zero zero is not on a line. That's pretty simple. Now if c is zero, then by assumption a and b, they can't both be zero. So without the loss of generality, you know, we could assume we could assume that a is not zero in that situation, for which then use the point. Then we use the point, something like one zero, something like that. Because then when you evaluate, you're just going to get a, a which is not zero, which is not equal to c, so it doesn't work. I'm not trying to die too much of the details here, but there's a point, every line has a point none on it, you can construct it. And so the four axes of instance are satisfied. So r2 is in fact an instance geometry. But I want to make mention that this argument about r2 wasn't really specific to r2. What if I moved on to r3, r4, r5, r6, these higher dimensional vector spaces? This idea of incidence makes sense there as well, for which a point is just an element of the vector space. So a point is just a vector. That's something you typically see in linear algebra. Points are vectors. It's just this interpretation right here. Now in that situation, what's a linear equation? Well, there's actually a lot of ways you can interpret it. You could say like a linear, excuse me, you know what a linear equation what's a line? A line usually in the usually of course in the linear algebra setting a line is then going to be the set spans by a single vector. You know you get something like a line is going to be y is equal to you know some tx plus b or something like that, where this is some fixed vector and this one right here is allowed to vary. And so anything of this form forms a line. You can think of that way and that usually and that's that's the way we think of it in Euclidean geometry and higher dimensions but there's something that stops you from taking a hyperplane and be like oh ax plus by plus cz whatever d. You could call this linear equation a line for which incidents would make the same would make the same sense. The four axims will be satisfied. Now again I use the word hyperplane like this is a three-dimensional example something of a plane. Those aren't lines right but we haven't really defined what plane is, what a hyperplane is, what's dimension. At the moment we're really only talking about points, lines, and incidents. Okay and so linear equations in these higher dimensions could be interpreted as lines because they behave the same way that the incidence axioms seem to be telling us they have at least two points. There's at least three points in the geometry. There's not one linear equation which has everything satisfied with the exception of the zero equation where everything is zero you know something like that. That would satisfy but we're not talking about that one we said that oh these coefficients can't all be zero but that's the thing is we don't want that one and so in some essence in these higher dimensional Euclidean geometries if we just stop at the four axims of incidents I could interpret a plane as a line. I could interpret a hyperplane as a line. We don't want that in higher dimensions so we have to specify what's the difference between a line or a plane or whatever. Now in this lecture series I want you to be aware that we really aren't going to distinguish well I should say we really aren't going to leave two dimensional geometry. We're not going to dwell upon things like planes and higher dimensions. We're just going to stick ourselves just to planer geometry so the geometries we talk about are a single plane and so we really aren't going to talk about these higher ones but they are interpreted as incident geometries and if we're not careful there's multiple interpretations. We can put on the same set that is we take the same points when we interpret lines differently. We get different incidence geometries. We want to watch out for such things and so while R2 the interpretation we have here is exactly the one we want. In higher dimensions it can get a little bit weird and so we'll be focusing just on planer geometry here. In the next video I want to start proving theory about incidence geometry in which these are going to be theorems that are true for every incidence geometries. Every incidence geometry including the ones we've already discussed in this lecture series.