 In the previous video, we introduced the important idea of an affine geometry that is an incidence geometry that satisfies the Euclidean parallel posh at all parallel lines are unique. And then as we ended that video, we proved probably the most important result of affine geometry is that all affine lines contain the exact same number of points. It doesn't matter which line you look at, they have the same number of points. And so since all of the lines have the same number of points, we can give that number a name. And this is referred to as the order of the geometry. So just so you're aware, the four point geometry that we introduced previously, so it's this diagram right here, you'll notice that every line contains exactly two points. And so four point geometry is really the order to affine geometry. So going into the future, we're actually going to refer to this geometry as order to affine geometry. This is of course to be, we don't want to confuse it with another order, excuse me, four point geometry. You have of course four point fan geometry, this we call F4. We don't want to confuse with that. This of course is the four point affine geometry are better yet the order to affine geometry. Young's geometry, which we've talked about before is order three affine geometry. But are there, is there an order four or five order six affine geometry? We'll say some more about those in a little bit. We really don't have, I want to develop the theory of affine geometry before we start talking about new affine geometries, which we haven't really considered yet. So it turns out, because all lines have the same number of points, this order, which we're going to typically call it the number N right here. So in an affine geometry of order N, we can say other things, right? So all lines have N points. But the next theorem of affine geometry says that if your order is N, then each point has N plus one many incidence relationships. That is, every point is on N plus one lines. So it's the number of points per line plus one. All right. So how are we going to accomplish this? Well, we're going to choose an arbitrary point because it's for each point. So we're going to start with some point called P. And I want to consider some line, which is not on P. This is a theorem of affine geometry, excuse me, of incidence geometry. But what's the basic argument there? Well, if all lines were on P, this actually violated one of our theorems of affine, excuse me, of incidence geometry. There's a non-concurrency theorem, right? Because we took non-colinearity as an axiom. Not all points are on the same line. But we also proved a form that not all lines are on the same point. So there is some line L, let's call it that. That's not on P. We can take that for granted because we proved it for incidence geometry. And affine geometry is incidence geometry, but with the Euclidean parallel postulate. All right. So then by assumption, because this is an affine geometry, we've assumed the order is N, right? We know there isn't a order we just given a label. And I want you to be aware that in this proof, this one and all the other proofs in this video, I'm not assuming that N is a finite number. This could be an infinite cardinal as well. It won't make any bit of a difference, although my diagrams, I'm going to draw it as if it's finite. So L is going to have N points on the line, because all lines in affine geometry have N points. So then from here, using line determination, we can construct a unique line from P to the first point on L, to the second point on L, to the next point on L, right? And I'm kind of assuming that I can do this in an inductive manner. I don't need to worry about that. I can construct a line between any point on L with P. Line determination gives me that. Even if they're done cannibly many points, I can construct all of these. And how many lines have I formed? At this point, I've now constructed N lines in my geometry. Why are they different lines? Right? How do I know they're not the same line? Well, if one of these lines intersected L at two different places, then we would have two lines that contain these two points. There's L and then the blue line we just constructed. So that would be a violation of line determination. So in fact, these all have to be different lines. And you'll notice the diagram we have right here is basically just fan geometry. It's like, oh, you have a sub-geometry inside this affine geometry. That's fan geometry. This will give you N of the lines on P. Okay? Where is, why N plus 1? Where does the other one come from? Well, this comes from the Euclidean parallel postulate. If you take the line L and you take the point P, which is not on L, then you're going to form a unique line parallel to L that passes through P. Right? So that's going to give you another line. And that's going to give you the N plus 1 lines that we're talking about. So if you counted for N plus 1 lines, why can't we have any more than that? Right? So suppose, you know, suppose for the sake of contradiction, there's another line that contains P. We'll call this line M just so we can talk about it for a moment. M contains the point P. I claim that it's one of these lines you've already counted for. Because first of all, M cannot be parallel to L. Well, why not? Well, we already have a line parallel to L passing through P. By the Euclidean parallel postulate, that line is unique. So this line M must, it can't be parallel to L. So it has to intersect L somewhere. So it's like, okay, we have a point of intersection like this between L and M. But wait a second. M contains P. M also contains a point on L. We'll call it Q. But we've already considered the line that contains P and Q. So by line determination, this line is unique. There's only one of them. And therefore, if M was a different line, then we'd get a violation of line determination. So M can't intersect that point Q. But honestly, if we considered any point on L, we get the exact same argument. And so therefore, the line M isn't possible. An additional line would always lead to a contradiction. Therefore, we've then proven that there's exactly N plus one many lines on each point. So every point is incident to N plus lines. So we've proven that every line has N points. So there's N points on a line. We've now proven that there's N plus one lines on a point. And with those observations, we're now ready to classify how many points are in the geometry. In an affine geometry of order N, there will always be N squared many points. Think about four point geometry, for example. Four point geometry that is order two affine geometry has two square, which is four. So we call it four point geometry. And young geometry, right? We have three points per line. It's order three. You square and you get nine points, which is exactly the stuff we've seen before. All right. What about the number of lines? Well, I claim that an affine geometry of order N, the number of lines is going to be N times N plus one. Let's consider our geometries again as well. Order two geometry means that you're going to take two times two plus one, which is going to give you two times three, which is six. And in order two affine geometry, as we've seen before, it did in fact have six lines. What about young geometry, which is order three? Order three means three times three plus one. So we're going to get three times four, which in this case is going to give us 12 lines. That was exactly what we saw in young geometry. We had four families of parallel lines, the horizontal, the vertical, the slants and the other slants. And they came in groups of three, three times four was 12. And that classification, I just mentioned there, like, oh, it came in these parallel families. That's actually where these numbers are going to come from. So these results are satisfied by the affine geometries we've studied so far. And right now we want to prove that this generalizes to any affine geometry of order N. All right. So what we're going to do is we're going to take two lines that intersect each other. So why are there necessarily two lines that intersect each other? Well, this is an argument of incidence geometry. Incidence geometry guarantees there are three points. And in fact, we have a theorem that we can come up with three non-colinear points, in which case fix one of the points, call it P. We can take a line that intersects. That is a line determined by those two points, a line determined by the other two points. So we get two lines that intersect each other. That's a consequence there. I'm going to remove the points and then redraw the lines here. So we have one line, we have another line, like so. Let's call this line L. Let's call this line M. And let's call the point of intersection between the two. Let's see, am I going to give it a name? We're going to give it, nope, I don't really care right now. Let's call it, no, let's not worry about it. What I want to then look at here is L and M intersect each other. Take a point P that is on M right here. By the Euclidean parallel postulate, there is a unique line parallel to L that passes through P. We can call this line LP. It's parallel to L contains the point P. We get this, like I said, by the Euclidean parallel postulate. Now, for each line LP, it contains exactly end points, right? So it's going to have end points. On it. Of course, I should mention that L itself also has end points on it. And particularly, I didn't give this point a name, but whatever this point was, call it, say, Q for a moment. L is, of course, just L, Q in that situation. So what have you? L is one of these lines that are running transversing M right here. And we can, so each of these lines LP contains end points. Well, how many LPs do we get? Well, because M here itself has end points on it, for each of the points on M, we construct one of these LPs. So we have a line like this, a line like this, and these are all parallel lines so they don't intersect each other. We have a line like this, another parallel line, another parallel line. We get all of these. But each of these lines that are parallel to L, they themselves contain end points. Just filling out this diagram. Each of these lines contains exactly end points. So I want us to consider how many things have we considered so far. Which, of course, all of these lines are parallel to each other. This is transitivity of parallelism. All of these lines are parallel to L. They're parallel to each other. So in fact, none of these lines intersect each other. So none of the points on one line are on any other line. So the set of points for each of these parallel lines are distinct from each other. How many do we have? Well, you're going to have end points each. That is, each line has end points. And you have N lines total. Now, I'm not counting all of the lines in the geometry. I'm just counting how many lines are there parallel to L right here. And so you end up with N times N many points. And so this is how we've accounted for N squared points in this geometry. I'm not ready yet to say that these are all of the points in the geometry. But what I can say is that when you look at this parallel family, so we've constructed N many lines parallel to L, including L itself. We have this family of N parallel lines. That accounts for N squared many points. Why can't there be another point in this geometry? Well, we'll consider that one in just a second. Suppose there is some new point, some N squared plus one point, some new point X. That's not in the geometry. Well, by the Euclidean parallel postulate, all right, since X is not on L, because we already accounted for the N points on L, X has to be some new point. By the Euclidean parallel postulate, there's some line parallel to L that passes through X. But then consider the line L, M, and this new parallel line to L. So this would be LX that we talked about just a moment ago here. Well, L intersects M. L is parallel to X. So by Proclus lima, this line, excuse me, L is parallel to this parallel line that contains X, call it LX, right? And then so by Proclus lima, LX must intersect M somewhere, okay? But that point of intersection has already, we already constructed an LP that went through that point, okay? So if this is a new point X, this must be a new line because that is a point X that wasn't on that line. But then we would have two lines parallel to L that contain this, that go through the same point right here. That contradicts the Euclidean parallel postulate. So we don't have this point X. There's not a point beyond the N squared. And so that then establishes that we have exactly N squared many points. That gives us the first part of the theorem, all right? So now that we've established that we have N squared many points, I want to recall the previous theorem that we proved that said that each point has N plus one many lines on it. So if I take the N squared points and you times that by N plus one, so again each point has N plus one many lines on it. This point over here has N plus one many points on it, lines on it. This point has N plus one many lines on it. This counts the number of incidence relationships in the geometry, right? How many incidence relationships do you have? So if you think of the incidence relation as a set of ordered pairs, there's N squared times N plus one ordered pairs in that set, the incidence relationship. We got that by considering how many points are there with the number of lines on that. But then I want to flip the directions around. We know how many points are on a line. So we proved previously in the previous video in fact that each line has N many points on there. So if we take the total number of incidence relationships, N squared times N plus one incidence relationships, since every line has exactly N points on it, each line accounts for N of those incidence relationships. N squared times N plus one divided by N is equal to N times N plus one. And so this gives us the number of lines that are going to be on, the number of lines that are in the geometry. And that then finishes this proof right here. And so I want to make mention that this theorem, the previous theorem, and all these theorems related to the order of affine geometry, this is extremely important when it comes to studying finite affine geometries because it turns out that up to, up to isomorphism, a finite geometry is determined by its order, right? There's only one order to affine geometry up to isomorphism. There's only one order three affine geometry up to isomorphism. That's young geometry. Now for infinite cardinals, it can get a little bit more complicated because maybe, you know, because you could have a cardinality the same size as larger. Does that give a different geometry? I don't want to jump into all of that right now, but I do want to be aware that the theorems we've proved, these combinatorial theorems do extend to the infinite case as well that when it comes to adding and multiplying and dividing, we can make all sense of, we can make sense of this even in the infinite setting, right? Addition means you're considering some type of disjoint union. Multiplication means you're considering some type of Cartesian product. Division here means you're considering some type of partition. And so we can make this infinite arithmetic make sense for infinite cardinals as well. So these formulas are also valid by the same proof. We don't have to worry about that whatsoever. Now I mentioned these examples earlier, but we're going to close the video by talking about them one more time, right? The four point geometry, the order two geometry we talked about before, two squared gives you four points, two times three gives you six points, and then there are three lines on each point, right? You also have that parallel families come in size two, the order of geometry. Young geometry, its order is three, you have nine points, three squared, you have 12 lines, three times four. Parallel families come in sizes of four and how many lines are on a point, you're going to end up with, excuse me, parallel families come in sizes of three, there are four parallel families, that's what I meant to say earlier. And then you're going to have four points on each line. And so these in retrospect, when you look at young geometry, the theorems we've proven in this video are actually theorems of young geometry. In many cases, it's exact same proof. In other cases, it might give you a different way of approaching it because really it's just young geometry. The theorems here, if you plug in n equals three becomes young geometry. So as I close this video, I want to approach one last question. So even though we know how many points, lines, et cetera, will follow from a given order, what orders are possible? So we've considered order two, we've considered order three, we know Euclidean geometry is some type of uncannable order, but let's focus on finite orders for a moment. What orders are possible? Are some not allowed? Well, for my students in the class, as you've been doing homework from the textbook, you've looked at some assignments that gave coordinate systems to order two affine geometry. We can, as a coordinate system, it looks like z two squared. You also had a homework question associated to young geometry, which shows that you can give a coordinate system to young geometry, it's just z three squared, like so. So we can give it a coordinate using a finite field. Z two is the unique finite field of order two, up to, up to isomorphism, of course. Z three is the unique finite field of order three. This is actually why we call it the order of the geometry because of the coordinates for the associated to the finite field right there, the order determines it. What other finite geometries can you get? Excuse me, finite affine geometries. Well, in abstract algebra, there is a unique finite field, which remember a field has the four operations of distance, attraction, multiplication, division that satisfy the associative, commutative, distributive properties, and you have identities and inverses. There's a unique finite field of order p to the k, where p is a prime number. We'll call it fp for short, excuse me, fp to the k for short. So you do have a unique finite field for every power of a prime. And so then if you look at the Cartesian product of that, so you look at fpk squared, same thing like we did with this geometry and this geometry. This will give you a coordinate system for an affine geometry of order p to the k. So it is well established that if you square a field, you can form an affine geometry. It is conjectured that this is basically the only way to construct an affine geometry. That is, every affine geometry has a coordinate system and it comes from a field. In particular, this is of great importance to finite affine geometries. At least at the time of the writing of this video, it is unknown whether there is a finite affine geometry whose order is not a prime power. In particular, look at order 6. Is there a finite affine geometry of order 6? The answer, of course, is no. That is actually provable. That's not too far away from what we could prove right now, but we're not going to prove it in this lecture series. And so you could ask things about what about 10 and the answer is no. What about 12? The answer is no. Geometers and combinatorics have studied this problem for a while. But the fact that we haven't yet found a non-prime power order that forms an affine geometry, but that doesn't mean it doesn't exist. But I would say that the vast majority of the mathematical community believes this conjecture to be true, that every finite affine geometry must have an order of power over prime. And essentially, the thought is it comes from a finite field. Because we know the finite fields will always work, but is there anything else? It's a very interesting question, but it's completely open. If honestly anyone watching this video has an answer or suggestion to that, please send me an email. I'd love to get involved in that question. It's a pretty important question in combinatorics here. And that's going to end our with that open question, though. I'm going to end this video here in this lecture. I should say on affine geometry. If you learned anything about affine geometry, please give these videos a like. If you want to see more videos like this in the future, of course, subscribe to the channel. And if you have any questions or answers to important mathematical conjectures, feel free to post those in the comments below and I will reply as soon as I can. Bye, everyone.