 Hi, I'm Xorv. Welcome to Unisore Education. Today's topic will be one of major component basic elements of geometry called lines. The previous lecture was dedicated to points, and points were one of the fundamental elements which we do not really define as a particular object. However, we do define certain properties of points. Same thing with lines. We consider this to be an undefinable percent geometrical object. However, lines do have certain properties, and any geometric object which satisfies these properties can be called a line. I mentioned before the name Euclid, the Greek mathematician who was one of the first, if not the first actually, who put geometry on a relatively solid axiomatic foundation. Today I will actually talk about postulates or axioms which Euclid came up with. We will consider certain details about them. The points and the lines on the plane, right now we are talking about planimetry, geometry on the plane, do have certain properties, and Euclid characterized these properties in certain set of axioms, five axioms to be exact. Anything which satisfies these axioms, any objects which satisfy these axioms in theory can be called points and lines on the plane. Again, I would not characterize these postulates by Euclid as extremely rigorous from the contemporary standpoint. I will talk about this a little bit later, but right now let's just consider his attempt, his theory, his postulates to put geometry on a relatively solid basis. So, first of all, points and lines on the plane are not definable objects. However, the first axiom of Euclid says that if you have two points, there is a segment which can connect them, and this is basically one and only one segment on the plane which connects these two points. The second axiom says that the segment can be expanded both sides into an infinite straight line, where straight line was actually kind of explained by Euclid. Again, not exactly in a very rigorous way, it's something which has no breadth, basically only the lengths. He was talking about segments, and the lines are something like a segment of an infinite length or something like this. But again, this is not the definition, it's kind of an explanation of what it is. So, here we have two first axioms, two points can be connected with a segment, and the segment can be expanded both ways. Then, he also considered a circle as yet another object, basically specifying that if you have a segment, then one end, which is a point, can be used as a center, and the segment can be used as a radius of a circle around the center. So, that's the third axiom, third posture of Euclid. Then, the next one is related to right angles, basically saying that right angles are always congruent to each other. Any two right angles are congruent to each other. Again, it's questionable whether there is a place for this particular postulate in a body of geometry among the axioms of geometry. However, that's what he actually wrote, and then there is a very interesting and very, I would say, controversial fifth postulate. And we will talk a lot about this today. So, what is the fifth postulate, fifth axiom of Euclid? Here it is. If you have two lines, and the line which crosses them, if some of these two angles is less than two right angles, well, the right angle is 90 degrees, so it will be 180 degrees. If this is true, then the lines on this side of the crossing will eventually cross each other somewhere. Well, the opposite is that if this sum is not less, then they will not cross each other in this particular direction. And one of the very important things is that if the sum is exactly equal to two right angles, which means this angle plus this angle, if they exactly 180 degrees, then the lines are parallel and they are not crossing each other, because if this sum is less than 180, equal to 180, then some of these angles also equal to 180, as it's very easy actually to prove, which means that on neither side the sum is less than 180, which means neither side actually has a crossing point. Well, another way we can say that lines are crossing in infinity. Well, it's just a statement, which means that they're not crossing in any finite point. And in a regular geometry, actually, we don't have anything but finite points, and I will talk a little later about another kind of geometry where there is such a crossing. Well, in any case, I just wanted to present five axioms of Euclide, and they were relatively sufficient for centuries actually to be the foundation of geometry, to be able to prove certain theorems about many different geometrical objects. Enough basically to define certain objects, like for instance, what is a triangle? Well, if you know what the segment is, then a triangle is basically a set of three segments which have common vertices. So definitions and properties of more complex geometrical objects were actually derived from these axioms until people kind of realized that these axioms are not really sufficient to be called a rigorous foundation of geometry. At the end of 19th century, famous German mathematician Hilbert came up with another set of axioms. Actually, I think it's 20-21 axioms which are rigorous from the contemporary standpoint. Hilbert was extremely productive and extremely talented mathematician, and his foundation of geometry is considered right now the right thing to do, basically the right foundation on which other theorems are derived and principles, etc. And the game, he actually decided that there are certain undefineable objects in geometry, like points, lines, planes, three-dimensional space, etc. And then he basically specified certain properties, like a game. If you have two points, there is one and only one line which contains them on the plane. So having all these axioms allows us to build a very rigorous foundation for an entire building of geometry, all these theorems, principles, properties, etc. I still would like to talk about the fifth postulate of Euclid, this one. So if the sum of these two angles, this one and this one, is less than 180 degrees, then the lines will cross somewhere. It's a very interesting axiom and many famous mathematicians were trying to basically derive this axiom from other axioms, making it a theorem rather than an axiom, and they couldn't. Until at one particular time it was actually proven that fifth postulate is independent from the rest, which means we can just take it as is and it will be one geometry, or we can take some other axiom, which is not exactly this. For instance, even if the sum of these two angles is equal to 180, then the lines will cross, for instance. Or maybe if the sum is less than 180, the lines do not cross. So if we take that particular postulate, or axiom, as a foundation of our geometry, we will have a completely different geometry. What's interesting is that there are actually two very famous places, I should say, where geometry is not Euclidean. I will talk about this in a little while. What's interesting is that this particular axiom is not really dependent on anything else, and whenever we take it as Euclidean did it, we have our Euclidean geometry on the plane. Now back to other two cases. There was a case of one famous mathematician called Riemann, and he basically came up with the idea of geometry on a sphere. So if you have a sphere, like our planet Earth, for instance, which is almost like a sphere, the geometry on this particular surface is completely different from the geometry on the plane. Well, what are the points? Well, you can probably say the points are more or less close to what we think about the points on the plane. But what are the lines? Well, lines are very, very interesting things. Lines on a sphere are, well, something like meridians, for instance, these meridians are lines. Equator is also a line, and basically if you take a plane and cross this sphere in such a way that the center of the sphere is contained in that plane, whatever the circle this plane cuts off the sphere is basically a line. Now, how about this particular axiom that for two different points on the surface, we have one and only one line? Well, very easy. We have two different points here and here on the sphere. Now, we can have a plane which is crossing these two points and the center. Now, three points always define the plane in a three-dimensional space. So, there is only one plane and that's why there is only one segment which connects, which is part of the big circle. Big circle is basically a terminology of a circle which is crossing between the plane going through the center and the sphere itself. So, there are certain properties of this geometry which are exactly the same as in the Euclidean geometry. However, the fifth postulate about the parallel lines is not working here. There are basically no parallel lines at all in this geometry because no matter what kind of two big circles we will take, there is always some kind of crossing. So, there are no parallel lines. No matter what the angles between this particular thing is, now all of these lines are actually some kind of big circles. No matter what kind of angles they make among themselves, they will always cross. So, this postulate of Euclidean geometry is not actually working here. It should be replaced with another one that two lines are always crossing. And the crossing gas actually is in two points, one at this point, this side and one on this side. Why? Because two big circles are always crossing each other on one and on the opposite side of the sphere. So, that's one of the examples of geometry which is different from Euclidean and the difference is only in this fifth axiom about parallel lines. Now, let me give you another example, kind of an opposite. So, here we have an example of geometry where there are no parallel lines. Now, here is another example. Consider you have something like a funnel basically, this type of surface. Now, I don't want to go into too many details, but something like this would be a line and well, something like this would be the line too. Now, my point is that in this particular case there are lines which do cross each other, but there are lines which do not cross each other no matter what kind of angle they make among themselves. So, these two lines, let's say this one and this one, these lines definitely cannot be called parallel in a kind of a tourist extent viewpoint, but they don't cross each other on this surface. So, the person who came up with this particular model is called Lovachevsky, it's a Russian mathematician. So, this surface gives you the geometry where two lines might not actually have the angle between them equal to 180 degrees, but still do not cross each other. Okay, these are just examples of different geometries and the purpose of this is that you really have to understand that lines are not only something which is really like drawn with a ruler and a pencil and a piece of paper. Lines are abstract objects in geometry which satisfy certain set of axioms and if we change the axiom then we can actually have a completely different geometrical object. And these examples, this one and the previous on the sphere, show that they do have certain practical implementation. So, thinking about geometry only from the Euclidean position is not actually correct and if you go even further down the line, if you will take the progress of physics which was made in the 20th century, especially general theory of relativity in Steinkeinapolis, it actually talks about the curvature of our space. So, our space is not really Euclidean. Masses of gravitation change the geometry in such a way that the lines which seem to be parallel are not parallel anymore. Alright, being as it may, I would like to return to an example of a different, again yet another, different geometry which I touched during my previous lecture about the points. If you remember, I was talking about the same geometry as ours, well you can call it conditionally Euclidean although, again, it's much more proper to call it Hilbertian because Hilbert actually put the real origins and rigorous foundation. But in any case, I'm talking about Euclidean-style geometry but not containing dots as points and these lines as lines, but something else. Here is what I was trying to explain during the previous lecture and I will very briefly repeat right now. So, let's consider, this is our three-dimensional space and there is a point, for instance, with cohesion that's 001, which we call M. This is the fixed point. Now, consider this plane X or Y, which sticks out from the Y-board, and consider relatively known traditional points on this plane. Now, for each traditional point, I suggest that we will use ray from the point M to this traditional point A or B or C. So, these rays in my new geometry I will call new points, so to speak. Now, the lines, again, new lines will be the planes which contain two different rays, two different new points, if you wish. So, A, M, B basically prolonged in all directions to make a plane would be my new line. Now, I was talking about these particular objects, rays and planes as new points and new lines. And I was talking that all the euclidean axioms and axioms by Hilbert actually completely satisfied with these objects. And I was talking about this just to emphasize how unimportant physical implementation of points and lines actually is, what is important properties which they satisfy. And these objects, rays and planes which all contain the point M, they do satisfy exactly the same axioms. What's also important, if you remember from the previous lecture about the points, that there is a very interesting advantage of this geometry over the classical geometry on the plane. Traditional geometry on the plane does not have a concept of infinity. I mean, we can talk about the process of a point going, let's say along a certain line, going and going and being further and further from some initial position and we can talk about this point going to infinity. But there is no such thing as infinity. Here, on the other hand, there is a real point which we can say, okay, this point's location is infinity, so to speak. If you take the ray which goes from the M parallel in this direction or in this direction or in any other direction, parallel to plane X or Y, it will never cross the plane because it's parallel. And this particular ray, actually any of these particular rays in any direction represent all the different points which are located at infinity. Now, let's talk about Euclid's fifth postulate. So again, if you remember, parallel lines do not cross. At the same time, somebody might say, okay, these don't cross in any real point on the plane. They cross in infinity. Well, there is such an expression that parallel lines are crossing in infinity. Well, how does it look in this particular geometry? It's very interesting. Let's start again from two parallel lines on the plane, which we have here, one and another. Now, these are traditional lines and what we're talking about was that corresponding to these traditional lines, new lines are planes which contain line and this particular point. So this type of plane is one. If you can imagine it, it goes this way and then another which is going this way. So it's like two planes which go from the same point and cross the X or Y plane. One plane is going at one angle and another at another angle and the crossing between these two planes and the X or Y are these traditional lines. I hope it's a clear description now. These lines do not cross anywhere on the traditional lines on the X or Y plane because they're parallel to each other. At the same time, if we go to my new geometry, the planes, this plane which is characterized by this triangle and this plane, they do cross each other and the crossing is a line and that line is parallel to X or Y plane as it's very easy to prove. What's important is that this line is actually two different rays which go to infinity. They do not cross the X or Y plane and that's why in my new geometry these are both infinitely located points but these are real rays. This is one ray and this is another ray. Each ray is basically a new point. So what I'm saying is that traditional two lines do not cross however or again we can conditionally say they cross in infinity but it doesn't really mean much but in my new geometry my new lines which means this plane and this plane have a real crossing two different rays going into two different directions and are parallel which means both are representing in infinity. So we have two different points, one point on one side and another point on another side. Two different infinite points or two different infinities if you wish in two different directions where parallel new lines do have the crossing. So these traditional lines do not cross new lines do cross and the crossing are two points on two different infinities left and right whatever you want to call it. Well I hope the lecture was interesting because I was trying to touch a little bit more than just traditional lines. We were talking about axioms, especially the fifth axiom of Euclid. We talked about different geometries and different surfaces which do not have that fixed postulate as a real postulate. On the sphere it looks like there are no parallel lines basically all lines are crossing each other and on this final type of surface there are different lines which are positioned at certain angle not actually parallel to each other but still do not have a crossing at all. And finally we were talking about this new geometry where instead of points we have rays and instead of lines we have planes which are crossing one and only one fixed location fixed point and in the three dimensional space and we even showed that the parallel lines in the new geometry do have crossing they do cross each other but in the real infinite point on both sides. That's it for today. Thank you very much.