 Hi, I'm Zor. Welcome to Unisor Education. I would like today to discuss some very fundamental geometrical problems and issues, basically starting from the beginning of the geometry. Well, in theory the beginning is associated with the name of Euclid, a Greek mathematician who lived many years ago. I don't remember, like, third century BC or something like this. Basically he was the first one who tried to put geometry on some kind of solid axiomatic foundation. It was not, like, 100% successful from the contemporary rigorous viewpoint, but it was extremely successful, judging from the progress geometry made during his time and all subsequent centuries. So what is basically geometry? According to Euclid, it's basically the science which is studying properties of geometrical objects. What kind of geometrical objects? Well, the most important ones are points, lines, circles, planes, and three-dimensional space, where everything actually is located. Planimetry is something which is studying everything which happens on the plane. So our plane will be this whiteboard, and whatever we will discuss about planimetry will be illustrated with drawings on this board. So let's start from one of the most fundamental concepts of geometry and planimetry, which is a point. Now, point is an object, obviously, and Euclid was trying to define this object in whatever not exactly rigorous way he could. According to him, it's an object without widths, without lengths, and which has no parts in it. Well, I would rather say it's a very good explanation of what the point actually is, not the definition, because it doesn't really identify the object itself, it's not really a rigorous mathematical definition. Similarly, he has defined lines and planes, and then he came up with five postulates, five axioms about interrelationship between lines and points and planes. Like, for instance, there is one and only one line which can be drawn on the plane and contain two given points. So that's basically one of the axioms which he put as the foundation of geometry, and then he built certain number of theorems, he proved a lot of different things. He was definitely a genius at that time, definitely. So let me address this kind of a definition which he came up with. As I was saying, it's not really a rigorous definition, but what is important in this particular case is the fact that points are some objects which satisfy certain axioms. Contemporary viewpoint is that we should not really define an object, we should not really think about what the concrete implementation of the concept which is called point actually is. Is this dot a point? Well, obviously there is an open criticism about whether this particular dot is a point or not. First of all, it has finite dimensions and point in our ideal world has zero dimension. So we will not concentrate on exact object which exemplify the point, we will rather concentrate on the properties. And now here is a very important leap of faith which we have to all make, thinking about points and any other mathematical objects. We do not try to identify the real life objects which resemble our abstract ideal objects like points. We don't. There are many different implementation of an abstract object which is called point in mathematics. What is important is the quality, the properties of the object. So if certain concrete object satisfies all the properties which we attributed to point, we can definitely consider it to be a valid point. And whatever theorems we derived about the points will definitely be true for this particular object regardless of what that particular object is. Now obviously we will use traditional implementation of the point which means basically a dot which I will draw on this whiteboard and the line which is the line basically. But I would like to concentrate on this very, very important shift in logic that point is anything which satisfies certain axioms. And as an example of the object which is completely different from this type of a point, I would like actually to present the certain fundamental objects which satisfy the concept of a point, satisfy the axioms which point actually satisfies. And that's why we can call that particular object a point in exactly the same rights as this dot on the board. And it will be a completely different object. So again, it's an illustration how important abstract thinking in mathematics actually is. It's not a real concrete object which we are looking for in the real life which would resemble our abstract ideas. We are saying that anything which satisfies axioms we put as the foundation of let's say geometry in this particular case. Any object if it satisfies the axioms of the point and the line and the plane then these objects can actually be considered as our new points, new lines and new planes. And whatever theorems were true for traditional points and lines and planes and circles and whatever will be true for any other object which satisfies these axioms even if it completely differently implemented. Okay, so here is my example of something which is completely different from the point in its traditional sense but which still can satisfy the axioms. And that's why we can apply all the geometry, the whole building which we build with theorems, etc. to these new objects. Here is my example. And again, I suppose that everybody understands what the point in the traditional sense is point and line and plane. So here is something which I will try to build which is completely different from the traditional point but which still can be called a point. Imagine a three dimensional space, let's say these are coordinates, let's say it's x, y, z. And imagine a point somewhere here, fixed point. Let's say it has coordinates 0 and x, 0 on y and 1 on z. It doesn't really matter where it is. Now consider this plane, x, y plane, this plane. It's basically sticking out from the board if you wish. Now for every point, traditional point on this plane, I will draw an array through this point from our chosen point. Let's call this point m and this is a. So with a traditional point a, I associate array called m a. Well array is basically half a line with a direction if you wish. Okay, now my most important point is, well it's not that point. My most important issue here would like to discuss actually is about the traditional point a is that this particular array drawn through our fixed point m and traditional point a. In a different world, in a different geometry can actually be called a point. And here is why. So I call this array a point. So that's number one. Number two, I have to define the line. Okay, let's have a traditional line. Here's another point b on the same plane x, y and I put another array through this point b. It's also a point in my new geometry. Okay, now let's consider all these rays which are going through every point on this traditional line, all these rays before and after. Therefore a plane, traditional plane, right? A plane which contains our point m and entire line ae. So I'm calling this plane a line, a new line in my new geometry. So in as much as the point traditional point a corresponds to the new point which is array m a, my line a b, traditional line a b corresponds to an entire plane, traditional plane m a b, traditional plane, which I can call a line. Now, why is it possible? Well, let's consider one of the first axioms of geometry. Namely, if you have two points, there is one and only one line which contains them on this plane. Now, if you have two new points m a, array m a and array m b. Well, obviously there is only one and only one plane which I call a m b which contains these two points, which contains these two new points which are traditional rays. So it looks like the first axioms which I pointed out here that there is only one line which contains two chosen points is basically satisfied in this particular case as well. Because this new line which is actually a traditional plane m a b contains, there is only one new line, a traditional plane which contains two different new points which are traditional rays m a and m b. So this axiom is satisfied and just take it on face very basically that every axiom which we know about from traditional geometry where points are basically something like dots on the plane will be satisfied in this geometry as well. It's a completely different geometry obviously because these objects are rays and planes rather than dots and segments which connect them. However, all the axioms will be satisfied which means the whole building of geometry, all the theorems which we build upon these axioms will be satisfied as well. Now, why do we do this? Well under certain circumstances, certain other implementations might actually be beneficial for certain cases. And here is a very important difference between this geometry and the traditional geometry. And important point is related to the concept of infinity. Well, we might actually talk about the point which is ghost to infinity. For instance, it moves along the line to the right and more to the right etc to the right and finally it just disappears and we're saying well it's an infinitely remote point or something like this. This is quite frankly not a very good mathematical statement because there is no such thing as infinitely remote point from let's say a given point. There is no such point. Infinity does not exist as a point so to speak. Infinity might exist as a process during which this distance is increasing and increasing and increasing and gradually it will become larger and larger than any other segment which we can think about. That's what it means that the point goes to infinity. But there is no real point which is located at the place which is called infinity. There is no such place. But here's what's interesting story about this. Let's forget about the plane. Let's just think about the point right now. So we have only one new point which is actually a ray. Here is our point A and here is our ray. What happens if this point really goes to infinity? What happens with our new point which is that ray connecting M to A? Well it will move more and more horizontally. Let's say just for simplicity purposes we go along Y axis. So our ray will have this position then this position etc. So the ray will be more and more horizontally located on the plane. But what happens when this particular point infinitely goes to the right? Well this ray will actually be closer and closer to something which is parallel to this plane. So if I draw the ray which is parallel to the plane we are talking about this actually is a limit of the positions of all these rays as the traditional point A goes to infinity. So in this particular geometry we have a concrete element, a new point which is a ray horizontally stretched horizontally to the plane which actually does represent the infinity as a real object. So in this geometry not only we have all the properties of traditional geometry and the point and segments and lines and circles etc. We also have a concept of the point which represents an infinity. And since we can actually have more than one ray from the point M parallel to the plane we can do it this way, this way, this way. These are all parallel rays. We have many different infinite points. Every one of them is an infinity but one of the points is an infinity in one direction and another is infinity in another direction. So this type of geometry has even a certain advantage for certain researchers when we are saying that the point which doesn't really exist in real geometry, well traditional geometry, actually the object which represents infinity does exist in this new geometry and this infinitely remote points actually are numerous. Every ray which is parallel to this X or Y plane would represent a point which we can call infinity located in one or another direction. So I just wanted to point out very important again issue that it's not the object itself which is important, it's the properties of this object. And if properties of two different objects are the same, if they satisfy the same maximums, we should not really differentiate them in our research. So mathematics doesn't really differentiate objects if their properties are the same. Let me give you just a real life example. There is a concept of a player. There is a CD player, there is a DVD player, there is a VCR player, there are many different players. But the concept of a player is something which is in our mind associated with certain ability to basically to produce sound, a rheumage or something like this. So that's actually what is important. It's the property to play which is sufficient to call an object a player. We don't really say that okay player is only this particular object because there are many different ones which satisfy exactly the same criteria they can play and every one of them is a player. Same thing is this. This implementation of a concept called point or traditional implementation of the concept, they are all just different implementation of different objects which satisfy the same axioms and that's what's the most important part of it. Alright, thanks. That's what it is for tonight. Thank you.