 Hi, I'm Zor. Welcome to Unizor Education. This lecture is a beginning of the whole course of solid geometry, geometry in three-dimensional space, basically. So this is an introductory lecture and there will be other introductory lectures, which will introduce certain concepts of solid geometry certain figures, properties, etc., and after I will finish all these introductory comments, only then I will start talking about each individual figure which is occurring in solid geometry, about its properties. I will prove some theorems. We will solve problems, etc. So solid geometry is dealing with different three-dimensional objects. Now, among these three-dimensional objects, obviously there are those which we already know from regular two-dimensional geometry on the plane, which are points, lines, straight lines, certain curves, like circle, for instance, certain geometrical figures in two-dimensional plane, like triangles, polygons, etc. So I assume that we know all these, all the flat geometry, the plane geometry, the geometry on the plane, we already know. And what's new in the solid geometry, obviously, is the third dimension. Okay. So all these elements of geometry, geometry in two-dimensional or three-dimensional space, all these different figures, objects, concepts, they're all abstractions, as we have already spoken about many times. So there is no such thing as a point in the real life, point is a mathematical abstraction. Same is the straight line and same is the plane. Now, these are very elementary objects, which means it's not easy to define them, but what is a point, what is a plane? You cleaved a couple of thousand years ago, actually put the foundation of geometry and it was a brilliant achievement. At the same time, he was trying to do something which probably he should not, and primarily I'm talking about his attempt to define certain things. Well, I will actually present you his definitions of certain things. I just wanted to warn you that this is not really the definitions. These are more of explanations, but not the definitions in the rigorous sense of this word. Okay, so what's Euclid's definition of point? Point is what has no parts. Well, I mean intuitively, we probably agree with this. Mathematically, we should not consider it as a definition, but as an explanation. Now, line is the length without the breads. So line has the lengths, but it does not have the breads. Now, what's next? The straight line. The straight line is even relative to the points on it, so it doesn't really deviate as far as its position relative to any points. It's exactly the same relative to this point as relative to that point. Again, it's an explanation. Don't criticize it for not being very precise. Next is a surface. Now, a surface, and now I'm having the furthest difficulty because the surface is a three-dimensional thing, and I'm using the two-dimensional boards to draw it, but I will try something like this. I don't know. If it looks like a surface, if I will add these lines, maybe it looks better. So what is the surface according to Euclid? It's something which has the breads and the lengths, but does not have the thickness, if you wish. Now, the plane surface, the plane surface, which I can probably draw something like this, it's supposed to be plain, looking from here. Now, the plane surface, or just simply a plane, is something which is, well, similar to this definition, which is even to any straight line on its surface. So these are definitions of Euclid, and as I was saying, you should not really consider them as definitions, just an explanation. Now, what's interesting is that these concepts, a point, a line, a straight line, a plane, these are primary concepts, and the wise thing to do is just don't define them at all, because to define them, you have to define it using some other concepts. Like, for instance, Euclid was using the concepts of the lengths, of the breads, concept of a part, like point does not have any parts, etc. Now, what are lengths, breads, breads at parts? Obviously, it's not defined. Otherwise, you would probably ask them to define it further using something else. So the best thing to do is do not define the primary concepts in geometry. But however, what you do know about these things, they do have certain properties. So for instance, like if you have a line, for instance, the line has at least one point. It's kind of a property, right? So these properties we really should pay attention to. And then we can just say, whatever objects satisfy these properties, we can call them correspondingly a point, a line, a plane, etc. Now, what's important is that it was necessary to to delineate all the properties, which these concepts, a point, a plane, a line, etc. satisfy. It was important to delineate all these properties, to be able to build the whole building of geometry on the top of it. So we can use these properties as axioms and build all the theorems and prove the properties, etc., etc. Now, David Hilbert in the beginning of the 20th century attempted to do this successfully, by the way. So he came up with 20-something axioms which I present in the notes for this lecture on unizor.com. Now, I'm not going through all these axioms. However, what's important is I would like to present certain axioms which are related to our solid geometry, related to three-dimensional space. I have three different axioms and these will be used in addition to whatever we know about the geometry on the plane, the two-dimensional geometry, to prove all the different theorems. So let me just talk about these three important properties, which we would like to accept as axioms. Axiom number one. Now, before I start, let me just say that I will try as much as possible to use capital letters for points, lowercase for lines and greek for planes alphabet. So, using this principle I will say the following. The axiom number one, if two points a and b belong, I'm using this symbol as the symbol of belongingness, belong to a straight line to belong to a plane alpha. Two points belong to a plane alpha. Therefore, the whole line which passes through these points also belongs to the same plane alpha. Well, imagine this board is my plane, so what it says is if two points a and b belong to this plane, they are on the board, then the entire line which connects them, which passes through them belongs to the same plane. It's kind of obvious, but we have to state it as an axiom. That's number one. Number two, if two planes have a common point, the point a belongs to both planes alpha and beta, then intersection of these two planes is a line, a straight line, which contains the point a. Now here it's a little bit more difficult for me to to draw, but let's do it this way. For instance, one plane, let's say alpha is this board. Another plane is this is another plane. So, let's say this plane intersects with that plane and there is a point of intersection. Well, what this particular axiom says that these two planes, the board itself and this plane intersect across the line, and this will be the line actually, line of intersection, which passes through this point a. Again, it's kind of an obvious, but we have to assume it as as an axiom. It will be necessary to have something like this in the theorems. And the third one, kind of obvious, again, if you have three points a, b, and c in space, which do not, not on one line, straight line. When I'm saying line, I assume straight line. Otherwise I would say curve. Okay, so a, b, c do not belong to the same line, like this. Then there is always a plane and one only plane, which passes through them. In this particular case, this is the plane of this white board. So this is the third axiom. Now, using these axioms, I would like to, as a demonstration, prove a couple of very, very simple theorems, which again are as obvious as these axioms, but I can prove them, really, logically, prove based on these three axioms. And then the whole solid geometry gradually will be built using these three axioms and additional theorems, which I'm supposed to prove based on them. Now, obviously, I'm also based on everything which we know about the two-dimensional geometry. Okay, the theorem number one. If you have a line, straight line, obviously, and a point outside it, there is one and only one plane, which passes through the entire line and this point. How can it be proven? Well, there are actually two statements here. One is existence of this plane that plane does exist, which goes through this line and end the point. And another is that there is only one. There are no two different planes which have this property. Okay, now let's consider, I take two points on the line A. Now, these three points, A, B, and C, do not lie in the same line. Therefore, using the third axiom, the third axiom was through three points, not lying in the same line. We can always put the plane and only one. So, there is a plane which actually goes through these points. Now, the first axiom was saying that if two points belong to a plane, then the entire line belongs to the plane, the line which passes through these points. So, that was the first axiom. And using this axiom, now I can see that the plane which I have built based on these three points actually contains an entire line. It's a logical proof. So, I have proven that the plane does exist which goes through the line and the point. Now, is it the only plane? We have to prove the uniqueness. Well, let's consider we have another point. So, maybe we took some other points and draw another plane which also goes through this line and this point. Well, then A, B, and C would be three points which belong to two different planes and that's impossible according to the third axiom, which says only one and only one plane can pass through three points not lying in the same plane. So, we have proven the existence of the plane and uniqueness. That's it. Very easy theory. Number two. Number two. If you have two lines which are intersecting, there is one and only one plane which passes through these both lines. Again, proof is really trivial. Let's take the point of intersection and a couple of points one on one line and one on another line which are not an intersection. Now, we have three points and we can actually draw the plane and only one plane. So, now we can use the first axiom since these two points A and B belong to this plane, then the entire line belongs. Since A and C belong to this plane, the entire line belongs. So, this plane actually contains both lines. Is it unique? Obviously, yes, because if there are two different planes, then I would have three points which have more than one plane passing through them, which is impossible according to the third axiom. Next. Third theory. If you have two parallel lines in three-dimensional space, then there is one and only one plane which contains them. Well, here I am using the term parallel. What is parallel in three-dimensional space? Well, the parallel lines in three-dimensional space by definition are two planes which are number one, lying in the same plane and number two, do not intersect. So, if I'm saying that there are two parallel lines, I'm assuming already that the plane does exist just because of the definition of the parallel lines. That's the plane where they belong to. Is it unique? That's a completely different statement and it does have to be proven. Well, obviously it is unique because if I can draw two different planes through these two parallel lines, then obviously I will have, let's say, these three points and I will have two planes, different planes, which pass through these three points, not lying in the same line, which is impossible according to the third theory, third axiom. And the fourth one is if you have a line in three-dimensional space, there are infinite number of planes which are passing through this line. Now, I would like actually to draw it. Here is how I will draw it. Let's say this is my line. Now, if this is the plane, then I can choose and basically draw the plane which passes through this point and this line. Now, what if I have a line outside of this plane? Let's say somewhere here. Well, again, I can use this point and this line and draw another plane which might actually look something like this. So, it's like a book, open book. This is one plane and this is another plane. And then I can choose any other point all around this line in three-dimensional space and for each position of the point I can have a plane. So, I will have infinite number of planes. You can actually consider these planes as a rotation of one plane around this particular line as around the axis. Well, basically these are very primitive concepts which are in the foundation of the solid geometry. So, all I wanted to present you is just these concepts and show how very, very simple theorems can be proven based on certain very, very fundamental properties which we can assume as axioms. So, this is an introduction which basically introduces the planes, the lines, straight lines and points and a little bit about their properties. And then I will continue introducing other figures which are used in solid geometry. That's it for today. Thank you very much and good luck.