 All right, we're into part three of lecture 32 for geometry 31-30 here. All right, math 31-30, I should say. And so this one's related, this last portion is related to the previous two parts, but also it's a little bit different, which kind of makes sense to modulate it away from the rest of them. We've been talking in the last couple lectures about the notion of area inside of a geometry. I wanna talk about the notion of a, what distance is. If we treat distance as an undefined term, or sometimes this is referred to as a metric, if we treat distance as an undefined term, how might we, what axioms might we attach to a distance function to capture an intuitive notions of what distance ought to be inside of a geometry. And so treating distance or metric as our undefined term, we have the three metric axioms, which are actually commonly discussed in real analysis or topology, but let's talk about from a geometric perspective here. So our axioms of metric are the following, much like area, we have a positive axiom. That is, if we have two points A and B, they're two distinct points, we want that the distance between the two points is always a positive number. We can say the distance from a point to itself is zero, but there's no such thing as negative distance. It should be a positive real number. That's what we mean by distance there. There should be a symmetric axiom as well, that if we have two points A and B, the distance from A to B is the same as B to A. So if we think of like these two points in our space, in our geometry, we don't allow for things like one-way roads, right? The distance from one should not be like longer or shorter than the distance to the other, right? Something like this would be no, no, we want a symmetric distance operation here. So we have the symmetric one and then we have the additivity axiom, which sometimes this is referred to as the triangle inequality when it comes to the metric axioms. That's the more common term in a topological setting right here. So if we have three points A, B and C, which are non-colonial points here, then the distance between A and C is less than the distance between A and B plus B and C. And so the idea is we have three non-colonial points, we might have something like the following, right? So let's say A is right here, B is right here and C is right here. The distance from A to C would be the length of that segment right there. And if that should be shorter than the distance from A to B plus the distance between B and C. So the idea here is triangle. So really, I think I'm gonna revise it. I'm taking it back. We're gonna call this the triangle inequality because that's all it is, the triangle inequality. We take as our third and final metric axiom. Now, the way that this triangle inequality is phrased, we are specifying that the three points are non-colonial. So we have to make, if we adapt this axiom as it is, we have to have some notion of incidence so we can describe what it means for points and lines to be incident to each other. Therefore, we can talk about collinear sets and non-colonial sets. One could modify this definition where we just say that three points A, B and C are just points. And then we say that the distance from A to C is less than or equal to the distance between A and B plus B and C. And then we put this little caveat that equality happens if and only if B is between A and C. Like so, because if you have a degenerate triangle like this, the distance between A and C is in fact the sum of A to B and then B to C. So we could try to have that modification. That way, if we don't wanna talk about collinearity, we could talk about betweenness, but even still betweenness is a geometric notion, right? So we have to, this triangle quality is somewhat dependent on we have a triangle. How do you, how does one describe a triangle? We can use non-colonial betweenness to do those. And so there are some options there. And so like in a topological setting, right? In a topological setting where topology is basically just the weakest form of geometry for which one can start doing calculus is we could talk about what a limit is. Cause this is based off a notion of open neighborhoods, which is a weaker form of distance, right? Metric spaces are one way as we're trying to just capture this thing. So in the case of a general topology, you might actually just have that the distance from A to C is less than or equal to the distance to A to B plus B to C. And you might have any caveat about when equality happens. So such a thing that happens. Well, in the presence of geometry, we do act that extra specification to the triangle inequality. And if we have an incidence geometry that satisfies the three metric axioms, we call this a metric geometry. And I want to mention that every congruence geometry that we've discussed so far can be given, it's automatically a metric geometry because I mean, there is an issue about how do you define a number to it? There's a bit of an issue right there, but it's automatically symmetric in the triangle inequality we actually proved as a theorem. So as long as there's some way of connecting segments to numbers that is a length, then you have a metric axioms there. So for example, the segment measure theorem that we did in neutral geometry shows that every neutral geometry is a metric geometry. There are of course non-neutral geometries like the rational plane Q2 for which we can give a distance to length. So that's not, that has to be neutral, but the segment measure theorem gives us that every neutral geometry will be a metric geometry. But like I said, Q2 can be done. Basically any Archimedean congruence geometry can be made into a metric geometry. There's ways of defining a measure to a distance, a length to each segment. So that's basically what we're doing right here. Congruence geometry basically already had metric geometry inside of it. But I do want to point out to you that there do exist metric spaces that are very different from the type of geometries we're talking about right now. So like for example, there's what's called the discrete metric. This is kind of a weird counter example by mentioning it as it is. You take any set X and you're gonna define and so then you take points, we'll say little a, little b, which are inside of X. Then you define the distance formula between a and b to always be one, right? You just say the distance between 82 point is always one. And so this kind of has some weird consequences to your geometry, right? If you have three points a, b and c, you can then define all of these distances will be one, right? This does satisfy the triangle inequality that is the distance, the distance from a to c is strictly less than the distance from a to b and b to c, right? So it satisfies that it's symmetric. It is positive because everything's equal to one. So this kind of in a very trivial way satisfies the metric axioms. And so even a finite geometry could be given this discrete metric. So like things like, I mean, you see in front of us essentially the three point geometry we talked about earlier in this series that could be made into a metric space. And so the idea of a metric function, a distance function could be done very weakly. If one wanted a strong notion of metric you might need some more notions attached to it. So like the idea of a distance inside neutral geometry is very, very strong. But there are still even benefits in this discrete setting about having a metric of some kind. So I wanna take the next few minutes and talk about the metric associated to hyperbolic geometry. It's kind of interesting here. So if let's take the half plane model, I'm sorry, no, not the half plane model, the other one, let's take the disk model right here. So everything inside the unit disk is considered our geometry here, those are the points. So we have two points, say a and b, two points like that. And so there exists either a diameter or a semicircle, a semicircle that forms the hyperbolic line between the two things. And remember, the semicircle should be orthogonal, perpendicular to the boundary circle, which is not belonging to the space there. But because hyperbolic geometry is an incident geometry, it has, there are lines determined by any two set of points. Take these two points at infinity, right? This, even though the boundary circle is not part of the geometry, it is part of the Euclidean geometry, it sits inside. And so take the boundary, the points at infinity here, call them p and q. And I wanna mention that this model of hyperbolic geometry, it sits inside of Euclidean geometry. So we can describe these things as Euclidean points, but we wanna give the hyperbolic distance between them. So what we're gonna do is we take these points at infinity, q and p, and we're gonna draw the Euclidean segments that go between them. Here's the Euclidean segment, p, b. Here's the Euclidean segment, ap. Here's the Euclid. So let me actually do these. So you get these, and then also you get these ones right here. So we get ap, qb, aq and bp. So we get these four Euclidean distances. So each of these numbers right here, the aq, that is a Euclidean distance. The bp, the ap, the bq, those are all Euclidean distances. And so what we're gonna do is we're gonna calculate this product of four Euclidean numbers. We take the natural log and then take the absolute value. That's what we define to be the hyperbolic distance between these points, all right? And so this kind of gives a, this is sort of an interesting little function. And in the homework questions that you can find, of course, in the script that's linked to this lecture right here. In the homework questions, we can exactly try this out with specific numbers, specific points, and I'm gonna have you do that. It can get a little messy, but please be patient and work through it. But this formula right here turns out, it actually does give us, it satisfies all the axioms of a metric for a geometry. It's positive, that's pretty clear because you take absolute values. It is symmetric if you switch the roles of a and b, that'll switch the roles of p and q. And this formula right here, it'll be symmetric in that regard. The hard part is proving that it satisfies the triangle inequality, which one can do. I'm not gonna make anyone do that here, but you can do that. It really comes down to, I mean, the reason why we have the natural log right here is that the natural log, where if you have a product or quotient inside of the natural log, this turns into a sum or difference of natural logs. And so it's that homomorphic property of the natural log, which actually will cause the triangle quality to be preserved. Because the natural log, I said, it turns multiplicative expressions into additive expressions. And this also sits inside of absolute values. Well, the absolute value function satisfies the triangle inequality. So you put those things together, something that's homomorphic with something which is a triangle inequality type function, it's a norm. When you put those together, the resulting object will also satisfy the triangle inequality. And it turns out that kind of like the area we talked about earlier, this is the right function to define to give you the notion of distance when you're describing a hyperbolic setting. And so we'll see some more of this in the future, but that's where I'm gonna cut it off for today. If you have any questions, feel free to post them in the comments below or feel free to email me directly. As always, there's a script, the link to the script you can find in the description. So you can find a complete script of these lecture series here. Feel free to subscribe if you wanna see more videos or find more information about these geometric issues or other cool, interesting mathematical objects and topics. And I'll see you next time.