 Hi, I'm Zor. Welcome to Unizor Education. I continue introductory lectures into solid geometry, basically specifying certain objects which we are dealing with. Today I will explain another object which is called the prison. Now this lecture is presented on Unizor.com. That's where the whole course of advanced mathematics for teenagers is presented, and I do suggest you to watch this lecture from this website, because there are some explanatory notes very important, so you can just read the notes as well as a textbook, basically, in parallel to listening to the lecture. Alright, so let me just talk about prisons a little bit. So far we have introduced points, lines, planes, and angles between the planes. Then we have introduced a cylindrical surface. Now I am going to use the concept of a cylindrical surface in this particular lecture, which was just a prior lecture to this one, and if for whatever reason you forgot the material which was presented in that lecture, I do suggest you to watch this lecture again, or read the notes on Unizor.com website about what is a cylindrical surface, because it will be extensively used in prisons and cylinders, obviously. Okay, so, prisons. Alright, first of all, let's imagine that we have a plane. I will try to position it somehow. It looks like a horizontal plane, right? Let's call it alpha, and I will call it a base plane. Then we will have another base plane, which I call beta, and let's imagine that these two planes are parallel. Now, I did not explain exactly, I did not define the parallelism, but obviously you intuitively understand that the parallel planes are just like that, and they never intersect. This is, among all other introductory lectures, the lecture without any kind of exact definition or proofs, etc. I am just explaining the concepts which I will be dealing in the future in more detail. So, this is the concept of a prison, and I would just like you to understand how I make this particular object. So, first of all, we have two parallel planes, which I will call the bases, alpha and beta. To differentiate, I can call this one the bottom base, and this one the top base. Fine, no problem. Now, next what I also have is, I have a line, which is not parallel to these, so basically it intersects both of them. Let's call it D. Then, on this particular, let's call it the bottom plane, the alpha plane, I have certain polygon, something like this. A, B, C, D, E, F. Well, in my case, the polygon with six sides doesn't really matter how many sides. It can be three, it can be 25, some kind of a polygon. Now, what I'm going to use next is, I'm going to use this polygon as a directrix of a cylindrical surface, and this line as a generatrix, which means basically that from each point on this polygon I will draw a line parallel to my generatrix, and I will actually make this line long enough. Let me just make this a little bit differently. Let's make it this way. It will be better. So, I will make these lines long enough so they will intersect this particular plane as well. These are parallel planes, right? So, from each point, I will go to the corresponding point here. So, it will be something like this, something like this, something like this, and I will call the corresponding points whatever goes through the point A intersects at A prime, B prime, C prime, D prime, E prime, and F prime. So, now I'm considering this cylindrical surface, which is formed by all the lines which are going through every point on this polygon and they intersect the opposite base, the top base beta. Alternatively, you can consider that I have one line from A, let's say, which is parallel to D, and I'm just moving this line, and that's how I'm forming a surface. So, actually, my lines, this one should be solid because it's visible, and this one should be solid, the E line should be visible, all right? And the D line is visible. Now, these are not visible. I think that would be, and A also is visible. So, I think that would be a better representation of my figure. So, the visible lines are solid and invisible lines are dotted. Now, what is a prism? Okay, the prism is the object which consists of part of the cylindrical surface, which is in between these two planes, and these two polygons on the planes. So, this part is called a prism. Well, that's it basically. I mean, that's the definition of what is a prism. Now, as far as the terminology is concerned, points A, B, C, D, E, F, A prime, B prime, C prime, D prime, E prime and F prime. And again, it doesn't really matter. This is a hexagonal or triangle or square, whatever it is, any kind of a polygon. So, all these points, the end points of every segment are called vertices, exactly the same way as with polygons. Now, the segments from A to A prime, from B to B prime, so all these vertical segments, as well as segments within each base polygon here and here, like A prime, B prime, B prime, C prime, etc. All these are called edges, and all planes also have their own name. One plane is the polygon which is at the bottom as a base. Another polygon is on the top, and we have many different sides here. So, all these pieces of planes, pieces of corresponding planes are called either faces, so all of them can be called faces, or you can alternatively call these faces as base faces. These faces on the sides, you can call lateral faces, or just sides, sometimes it's called sides. So, that's basically all the elements of the prism. So, we have vertices, we have edges in between them, and we have faces which are different. It can be side face, it can be lateral, or lateral face, it can be base face, or another base face, top base face, bottom base face, etc. So, that's the terminology. Now, it's very easy to prove that every lateral face, every side is a parallelogram. Now, why it's because these two lines are supposed to be parallel to these ones, right? And these two lines will also be parallel to each other because the planes are parallel. It should be proven absolutely, and I'm going to do it when I will discuss the prisms in more details. Right now I'm just introducing, and I would like you to have a feel what exactly the prism is. So, these are parallelograms. Obviously, two bases, two base polygons are congruent to each other. Again, it's all the consequence of every side being a parallelogram, for instance, and some other properties of the angles. So, then I would like to introduce another terminology. If this line D, the generatrix, is perpendicular to bases, then the prism is called the right prism. Well, the same way as a triangle, if you have a perpendicular one line, a side perpendicular to another, it's called the right triangle, right? So, the prism is called the right prism if every edge on the side of the prism, which is parallel to this one, is perpendicular to bases, then the prism actually stands right. Well, that's why it's called right. Now, otherwise, if it's not perpendicular, it's called oblique. And it's actually tilted like the Pisa Tower, the Tower of Pisa, remember? Well, it's actually round, so it's kind of a cylindrical, but cylinders also can be oblique, and we will talk about this. But anyway, if it was a prism, it would be an oblique prism. Now, what else? Another classification of prisms is based on how many sides have the polygon at the base. Now, if the polygon is a triangle, the prism is called triangular prism, obviously. If you have a rectangle at the base, well, the prism would be called, well, guess what? Rectangular. Something like this, something like this. That's a rectangular prism. Now, rectangular prisms are usually considered only those prisms which are right rectangular prisms. So, not only these are rectangles, but also every side is also a rectangle. So, it's a right prism where every edge on the side of the prism is perpendicular to the base. Okay, what else? If you have a square here, and obviously there, and it's a right prism, and this edge is equal to this edge, so all edges are the same, you will be dealing with a cube. So, a cube is also a special prism. It's a right prism. It's a rectangular prism with all edges equal to each other. So, the rectangle in the base is basically a square, and every parallel pipette, not parallel pipette, the parallelogram, parallelogram which is a lateral face is actually a square as well. What else? I just mentioned parallel pipette. So, if you have a parallelogram at the base, then the whole prism is called parallel pipette, because every face of this prism is actually parallelogram. The base is a parallelogram by definition. Now, all the sides, all the lateral faces are always parallelograms in the prism, so every side is parallelogram. In this case, one, two, three, four, five, six. Six different sides, two bases, and four side faces. All of them are parallelograms, so that's the parallel pipette. What else? Well, again, if you have something like a six angled polygon, hexagonal, if you have a hexagon at the base, then the whole prism is called, obviously, hexagonal. Now, there is another special kind of prisms. If you have a regular N polygon with N vertices, but it's regular, like regular polygon with five vertices, with 25 vertices, doesn't matter. And it's a right prism, which means all these edges are perpendicular to the base. Then we usually can talk about N prism. So N prism is the regular N vertices and an angular polygon, the regular polygon with N sides and N vertices. And it's a right prism, then we can talk about N prism. Well, that's all terminology, and again, the purpose of this lecture was just to introduce all the different concepts related to prism, all the different kinds of prisms, et cetera. So in the future, if I will, for instance, discuss some kind of a problem, I will say, okay, let's consider we have a parallel pipette with such and such dimensions or angles or whatever it is. You will know what I'm talking about. The parallel pipette is a prism with every side being, every face being a parallelogram, et cetera. So this is just terminology, no concrete properties or theorems or anything like that. That will be in the future. And again, the purpose of this lecture is just to introduce you to a concept of a prism. I do suggest you to read again this lecture on unison.com. It's basically like a textbook denotes to this lecture, and you will read basically the same information which I am right now talking about, but it would be better for you just to remember all the terminology because it's written and you will read it. That's it for today. Thank you very much and good luck.