 Welcome back to Lecture 35, Part 2, The Wrath of Khan, in which case we are actually going to transition to Section 7.3 about duality from the Wallace and West textbook here. And so we've saw in the previous part of this lecture that in elliptic geometry, if we have a line, there exists this point called the pole, for which all lines from the pole to the line are equidistant. And also, if you're given any point, you can construct a line for which the given point is a pole to that line. And so this right here, this idea of the pole on a line in elliptic geometry illustrates a concept in geometry we refer to as duality, where a dual geometry, so given any incidence geometry, a dual geometry, it'll interchange the roles of points and lines. Because we define points and lines as, well, JK, we didn't define them, right? If we go back to the very axiomatic beginning of this course, points and lines are considered undefined terms, and incidence is just a relationship between points and lines. And so the dual geometry is formed given any incidence geometry, and it doesn't even have to be an incidence geometry. All that has to happen is if we have a geometry with a notion of point and line and incidence, we can reverse the roles. So you swap the roles of point and line, and you keep the same incidence relationship. This forms the dual geometry. And so this is a construction that works whenever we have points and lines, such as an incidence geometry, and you have other things as well, and this typically creates a new geometry of some kind. So I wanted to consider the order to affine geometry, the so-called four-point geometry we referred to earlier this semester. Now, in this geometry, we had four points, which in the diagram to the left here, you're going to see that you have this four-point geometry. We have the four points we'll call them P1, P2, P3, P4, as illustrated. And by line determination, there were six lines between these four points. And these were just handshake lines, two points per line. And so you can see the possibilities. There's L1, L2, L3, L4, L5, L6, right? L1 is the line determined by P1, P2. L2 is the line determined by P2, P3. L3 is the line determined by P3, P4. You get the idea right here. And so in the dual geometry, we could construct a so-called four-line geometry, which what we do is every point over here in the four-point geometry becomes a line. And so in this diagram you see right here, I've labeled, so in both diagrams, the points are labeled in green, and the lines are labeled in yellow. And so as we move from one diagram to the other, that color scheme is still preserved, but I kept the same names there. So the four lines you have is this one right here. I need a different color for that. We have P1, which now becomes a line. It's a line that contains the points L1, L5, L4. There's P2, which is a line that contains the points L1, L2, L6. There is the line P3, which contains the points L2, L3, L5. And then finally there's the line P4, which contains the points L3, L4, L6 here. So we have four lines in this geometry, and there are six points. In the original geometry, there was four points and there were six lines. So we have this duality going on here. We swap those, and let's discuss a little bit how this worked out. So the six lines in the original geometry L1, L2, L3, L whatever, those become the six points in this geometry over here, right? And then the incidence relationships are preserved. So if we think of, okay, what lines are associated to the point L1? Well, that's going to be the lines P1 and P2. So if we come over here and find L1, there you are. It has two lines incidence to it, okay? And then if we take the point L2, what are the lines incidence to it? Well, that's going to be P3 and P2. So if we come over here and find L3 right here, it'll be incidence to two lines. There's the line P3 and the line P2, right? And so you can carry this through with the other points and lines right there. So this is what we like duality. We switch the roles of points with lines and lines with points. We have the exact same incidence relationships, but we have a slightly different configuration of these two drawings right here. Now this one over here was an example. The original one was an example of an affine geometry. This is an affine geometry of order two. Not commonly used in the literature, but sometimes if you take the dual of an affine geometry, it's referred to as an if fine geometry. And not commonly used. You don't see that a whole lot. On the other hand, the first geometry satisfies the Euclidean parallel postulate. So some people might say this is Euclidean. Again, Euclidean as an adjective typically refers to as the geometry R2, but it does satisfy the Euclidean parallel postulate. A little bit more commonly used in this term if fine. Sometimes people would use describing this dual of the affine here. They call it Euclidean dual, sort of like just an amalgamation of Euclidean and dual right here. And so sometimes people use the term Euclidean dual to refer to as the dual of the Euclidean plane, but sometimes it's used to refer to as the dual of any affine geometry. So Euclidean dual is probably used more commonly. So this is the four line Euclidean dual to the four point affine geometry right here. I also want to point your attention that the reason I drew the four line Euclidean dual geometry the way I did is what happens if we were to overlay these two geometries? You place these two geometries on top of each other. You would get a picture that looks something like the following. I would draw it again. You get something that looks like the cloak of invisibility. We would get something that looks like the resurrection stone. And we would get three elder ones. Oh my goodness, this is affine geometry. If you completely overlay those pictures. Now there's a slight abuse there because we're identifying points with lines and such like that. But again, the configuration right here, you can see that just from a configuration point of view this overlay does give you the phantom plane. And this is actually not a coincidence that if you take an affine geometry and construct it Euclidean dual you overlay the two geometries. If you do this in a strategic way, you can actually construct a projective geometry. The phantom geometry is being an example of this. And the idea is actually the reason I mentioned this is that a projective geometry is actually an example of a self-dual geometry. Self-dual meaning that if you take the dual of the geometry, you get something isomorphic to the original geometry. That if you interchange the roles of points and lines it makes no stinking difference to the geometry whatsoever. So to try to make that more explicit, let's think about the following. Whenever you have a geometry, you can switch it to the dual statement. The dual statement meaning that you switch the roles of lines and points. So if we were to say something like, okay, in an incidence geometry we might say that there exists a point off a line. So we would write the statement in the following way. So for all lines, there exists a point off of... So when I say off of let's be more specific, there exists a point not incident to the line. So this is just a regular statement. We'll call it statement A for a moment. This is a regular statement in a geometry. And this is actually just the non-colonial axiom for incidence geometry. But this can be interpreted as a statement in a geometry. Well, we can switch to its dual statement, which we'll call A' which would say that for all points, there exists a line not incident to the given point. And so this right here is sort of a statement about non-concurrents. That given any points, there's not one line that is concurrent to all those points, right? And so this is what we call a dual statement. These statements are dual to each other, a dual statement. And dual statements, you just switch the roles of lines and points and like this. So whenever you have a geometry, which has a statement, whether it's true or false, you can rewrite that statement as the dual statement. That becomes a statement for the dual geometry. Now if you have a theorem, let's say this thing was like a theorem of the geometry, then the dual statement will become a theorem for the dual geometry with the exact same proof. And when I say the exact same proof, I mean the dual proof, you switch all the points to lines and the lines to points. And so one nice thing about duality is that whenever you prove something about geometry, the dual proof gives us the dual theorem for the dual geometry. You don't have to provide a new proof. You can just use the same proof, but you change by the principle of duality. Well, the reason this is relevant for projective geometry is that projective geometry is self-dual, which is to say that it's isomorphic to its dual. So whenever we make a statement in projective geometry, its dual statement also applies to projective geometry. That is, whenever we prove something in projective geometry, automatically its dual theorem would be true as well by the exact same proof. And this comes from the fact that the axioms of projective geometry are dualistic here, dualistic. So let's actually take a look at them real quickly. So if we take the five axioms of projective geometry as we have them before, let's take their dual statements, take their dual statements. So if we take line determination, which line determination, if you take dual, it would become point determination, right? So line determination says that for each two distinct points, there exists a unique line incident to both. The principle of duality here, the dual statement will switch the word points to lines and the word lined up point. So point determination tells us that for each two distinct lines, there exists a unique point incident to both of them. Now, this idea of point determination, it has a different name that we've used to before. We've talked about this statement. For each two distinct lines, there exists a unique point. Oh, this is just saying that all lines intersect and they do so uniquely. This point determination is none other than just the elliptic parallel postulate that we've seen in the past. Elliptic parallel postulate. It has a slightly different name, but the dual of line determination axiom is the elliptic parallel postulate, which is sort of an interesting fact to notice here. So if you take the dual axiom of line determination, you get the elliptic parallel postulate. Notice projective geometry already does that. All right, let's take the dual of tricancy. Well, tricancy says that for any elliptic or for any projective line, there exists at least three points on the line. The dual statement would say that for every point, there exists at least three lines concurrent to it. Because what does it mean for a point to be on a line? It means that the point is incident to the line. And so if we have a point and there are three lines on it, that's the incidence relationship right there. So while tricancy for projective geometry requires that we have three points on the line, the dual of tricancy just says that there are at least three lines incident to a point. Now, be aware that this type of property, dual tricancy, happens in projective geometry. Projective geometry does have this dual tricancy property. All points have at least three lines incident to it. So I want you to notice so far that point determination is a theorem of projective geometry because it's just the elliptic parallel postulate. Dual tricancy is also a theorem. That one's not immediate from the axioms, but one can prove and we've seen this before that it holds. What about line existence, right? Projective geometry has the axiom of point existence because that's just an axiom of incidence. Line existence would be its dual, that there exist at least three distinct lines. We switch the word point to line there. And you can see that a projective geometry, the smallest of all projective geometries of course is the phantom plane, right? Which has seven lines, but and every projective geometry that exists will contain an isomorphic copy of the phantom plane. It's the smallest example. So point existence, although the axiom line existence is a theorem of projective geometry. So this dual axiom is satisfied. Non-concurrents, right? We have the non-colinearity axiom that says that there is no line that contains all points. It's dual, which be the non-concurrents axiom. Non-concurrency here would tell us that not all lines are concurrent to the same point. That is there's no point that holds all the lines. And so there's always got to be a line that's not on a point. So you always have something like this happening, which kind of looks like the same type of picture we had before. But the point is the starting point with the non-colinearity axiom. We start with the line and we're guaranteed a point off of the line. With non-concurrents, we start with a point and then guarantee there's a line that's not on the point. And this is in fact a theorem of projective geometry so far so good. And therefore, the last one we're going to look at is the dual of elliptic parallel postulate. The elliptic parallel postulate says that all lines intersect, which if we rewrite that phrase in a little bit, we're saying that given any pair of lines, there's a common point incident to both. Well, the dual statement of that would then tell us that given all pairs of points, there's a common line incident to both. But wait a second, this right here, any two points, there's a line that's on both of them. That actually is just a restatement of line determination. Given any two lines, given two points, excuse me, there's a unique line between them. Now, the uniqueness statement was hidden up here in this point determination here. But because the elliptic parallel postulate generally doesn't take a uniqueness statement there because line determination gives it. I want you to be aware that this dual of the elliptic parallel postulate is holding here. So one thing to mention is that line determination and the elliptic parallel postulate are dual statements of each other. And so elliptic geometry gets these things. So these five dual projective geometric axioms are all theorems of projective geometry. And therefore any any so projective geometry is a dual projective geometry. But by similar reasoning, the projective axioms are actually also theorems of the dual geometry there. And so that's what we get that projective geometry is this. It's this self dual. The dual projective axioms are theorems of the projective geometry. Thus, any theorem that can be proven from the dual axioms can also be proven from the original axioms. This implies that for every incidence theorem and projective geometry, its dual is also a theorem of projective geometry automatically. And of course, you get the other way around as well. And so this actually helps simplify proofs in a projective geometry because we can use duality. Once you've proven one thing, you get the other one automatically. So like we want to prove, for example, there exists a set of four points, no three of which are collinear. The dual statement, there exists a set of four lines, no three of which are concurrent is automatically true in projective geometry if we can prove the original one. And I want to mention that this is a duality notion between points and lines. So points and lines are these dual notions. This is heavily studied in the literature. But if we go to elliptic geometry, which is a special type of projective geometry, there are other notions going on there as well. So like for example, the notion of duality can be extended to betweenness and congruence. And you should try to convince yourself that this idea of point betweenness, point betweenness if we go to like say neutral geometry or just an order geometry is dual with angle betweenness. We saw a lot of relationships between point betweenness and angle between, not my angle between this, I'm sorry, ray betweenness. Let me fix that ray betweenness. And there are some similarities going on there. And also in neutral geometry, we have this idea of segment congruence and its dual notion would be angle betweenness, not angle betweenness, sorry, angle congruence right here. Now, again, there's this duality going on here. So as we approach like the Euclidean plane, and then we look at the Euclidid dual plane, you'll see there's this duality. You swap not just points with lines, but you switch betweenness of points with betweenness of rays and congruence of segments with congruence of angles. And you actually get this very nice geometric structure going on here. And in elliptic geometry, these ideas of duality are also still self-dual. When it comes to, for example, a line in elliptic geometry, what is the dual of the line? It should be a point, but that's none other than just the pole of the line. This idea of polarity in elliptic geometry is just a tangible way of measuring this self-duality that happens in all projective geometries. We can specifically form a duality function that interchange every line with its pole and vice versa. This gives us a one-to-one correspondence like we talked about previously. So therefore, we can show that this correspondence and incidence and continuum we can do with betweenness and congruence as well is preserved. And we get this self-duality, a very strong sense of self-duality in elliptic geometry. Something that's absent from Euclidean and hyperbolic geometry. This idea of self-duality does not happen there whatsoever. Lines and points act very differently in hyperbolic and Euclidean geometry. But in elliptic geometry, lines and points act the exact same way. And there's really no way of distinguishing between the two except that we just so happen to call one a point and so happen to call one a line. And that brings us to the end of lecture 35. I appreciate everyone from watching this. If you have any questions, please feel free to post them in the comments below. And I'll be glad to reply to them sometime in the future. If you like this video, please click the like button, share with other people who would love to learn more about modern geometry as well. And feel free to subscribe so you can get updates about other cool videos all in the future. I'll see you next time, everyone. Bye.