 Hi, I'm Zor. Welcome to a new Zor education. This lecture is yet another introductory lecture into solid geometry where I will just explain a particular kind of objects which we will be dealing with as far as theorems are concerned or problem solving etc. This lecture is part of the advanced course of mathematics for teenagers presented on Unizor.com. I'd certainly encourage you to take the whole course. Let's just listen to this particular lecture. Also, it's important to view this lecture from from the website from the Unizor because there are some side notes which are very very useful just to refresh your understanding after you have already listened to the lecture. So today's lecture is about polyhedrons. It's just another concept. I mean we were dealing with planes, points, lines, spheres, pyramids, etc. So this is just another type of solid geometry object which we can deal with. Okay, so what is polyhedron? Well let me just give you an example of polyhedron first. Something like Egyptian pyramid. Well, I hope it looks like Egyptian pyramid. It's a polyhedron. Basically, a hydron means a face of this particular object and poly means there are many faces. So basically anything cube is also a polyhedron and there are many other examples. So what is characteristic property of polyhedron? Well, all faces faces are attached to each other along the edges. Now edges are also attached to each other on the vertices. So basically that's as much as I can say about the definition of the polyhedron. I understand it's not precise. However, I think you're understanding with the really strengthened by these two examples. So what's important? All the faces are flat and faces are attached to each other to basically form a solid figure. Now, what's the simplest polyhedron we can just think about? Well, I think the simplest one is the one which has the least number of elements, the least number of faces, the least number of vertices, the least number of edges, etc. Well, I think it's very easy to make a claim that something like a triangular pyramid is a simplest polyhedron. Now it has one, two, three, and four, four faces. It has one, two, three, four, five, six edges and one, two, three, four vertices. Something like an Egyptian pyramid which has a square as far as its base. It has four, five, five faces. Ages one, two, three, four, and four, eight edges and one and four, five vertices. Something like cube has one, two, and four. It has six faces. Ages four and four and four, 12 edges. And how many vertices? Four and four, eight vertices. Just examples, basically. So I think based on these three examples, you understand that what basically polyhedron is. Now, if you want some nice pictures, you can just go to internet, enter polyhedron into a search and you will have lots of very nicely presented pictures of hexahedron and tetrahedron and all other hedrons, basically with all kinds of number of faces. Okay. Now, obviously all the pyramids are polyhedrons. All the prisms are polyhedrons because they contain flat faces and faces are attached at edges and edges are connected at vertices, etc. So basically what I can say in this case, there is no face whose edges are not adjacent to other faces. Something like this might actually work. And there is no edge which on both ends does not connect to some other edges. So that's just another characteristic of polyhedrons. Okay. Now, most likely we will be dealing with convex polyhedrons. What's convex? Convex always means basically the same thing. If you take two points on the surface of polyhedron, for instance this and this. So this one is on the side and that one is on the back side. And if you will connect them with a segment, well, I put the dotted line because it's inside the pyramid, right? Now, all points of this segment lies inside. So basically we are talking about polyhedron to be a such a such an object which divides the whole three-dimensional space in inside and outside parts. So there is always something which is inside and there is always something which is outside. Now, it means that one part which is called inside cannot be connected to another part to one point which lies inside cannot be connected to a point which lies outside without intersecting the surface. That's what it means. So first of all, the polyhedron divides this entire three-dimensional space into two parts, inside and outside. And secondly, it means that if you have two points which are lying inside, then the connection between them also lies inside of the polyhedron. So then this particular polyhedron is called convex. Now, cube is a convex. Such a pyramid is also convex, but you can always imagine something which is not a convex. Let's say you have this cube and I will build certain pyramids on each face, something like this would be a pyramid and on this I will build another pyramid and on this I will build another pyramid, etc. Now, it will also be polyhedron. However, if you want to connect for instance this point to this point, it might actually go outside of the of the whole polyhedron. It will be outside of the whole thing. But without these pyramids, the cube itself is a convex polyhedron. All right, so that's the difference between convex and not convex. And now what's a very interesting thing is convex polyhedrons have a very important equation which connects the numbers of edges, faces and vertices. This is called an Euler's formula. Euler was a very famous mathematician. He lived all over the Europe and in Russia in the 18th century. As far as the Russian spirit, he actually was one of the people who created the mathematical school of Russia in the 18th century from which lots of good things actually grew from all the different subjects and mathematics. So anyway, this formula actually belongs to him and the formula is the following. Number of vertices, minus number of edges plus number of faces is equal to 2. For a convex polyhedron, let's just check it for these three cases. Vertices is 4, edges 6, 4 minus 6, minus 2, plus faces we get 2. Minus 2 and 4, it's 2. Now here faces is 5, vertices is 5, that's 5 and 5, 10, minus edges 2. 10 minus 8. Now here you have 8 vertices and 6 faces, that's 14, minus edges, minus 12, 2. You see how interesting? Well, the proof of that actually belongs to Euler We will talk about this at some other later point, but I just wanted to tell you that this is actually quite a remarkable equation and whenever you deal with something, you might feel that this something has certain property. Personally, I never felt that this is a property of any polyhedron, any convex polyhedron. Well, Euler has proved that this is the case and it's a very simple formula and and actually there is a pretty involved logic when you want to prove it. And the last but not least I wanted to mention was something which is called a regular polyhedron. Now you remember what regular polygon is, right? Regular polygon is something which is something like this. All edges are the same and all angles are the same. Well, the same means concurrent, basically. Now the regular polyhedron is a 3D equivalent of this. So what can we ask about this polygon to call it regular? Well, obviously all faces of this polyhedron must be regular polygons. So in this particular case, every face is supposed to be equilateral triangle and they're all supposed to be the same. So if they are all the same, that's basically a base for being called a regular polygon. And obviously as a consequence of this all edges are the same because, you know, if the polygon, this polygon is congruent to this polygon, edges obviously are also congruent. And what's also important, all angles between the planes, like this plane and the bottom plane or this plane and the bottom plane or this plane and this plane. So all these angles between the planes and I did actually explain what angle between the planes is. So they're also supposed to be congruent to each other. So again, we have all faces the same. As a consequence, all edges are the same. And all 3-dimensional angles between the planes must be the same. That actually is sufficient to call this polyhedron regular. Now the next regular is a cube, obviously. This is also a regular polygon, a polyhedron, because all faces are squares congruent to each other and all angles between different planes are actually 90 degrees in this case. They're all perpendicular to each other. Alright, so basically that's all I wanted to explain about polyhedrons. Again, this is introduction. What basically this object is. So we will be able to to deal with its properties and the problems related to this when regular lectures will actually start. So this is still an introductory lecture. I might actually have one more introductory lecture, after which I will go in details about each particular kind of object in solid geometry. So that's it for today. Thank you very much and good luck.