 Hi, I'm Zor. Welcome to a new Zor education. This is my last introductory lecture to solid geometry, geometry in three-dimensional space. Now, I call all these lectures introductory because I'm just explaining concepts and objects we will be dealing with without getting into any kind of properties, no theorems, no problems, it's just a language, basically, if you wish. So this is the language of solid geometry, where I'm talking about what kind of geometrical objects we will be dealing with. What are the elements of these objects? Now, in this last lecture, I would like to devote to construction problems, not to solve them, but to explain what is a construction problem in a three-dimensional world. You see, if I'm talking about plane geometry, two-dimensional geometry, I can always use my board as basically a plane where everything happens. And I have usually two main instruments for construction as straight ruler and a compass. Now, these are the main elements for three-dimensional geometry as well. However, I understand that there is a problem constructing a plane in the three-dimensional object. I mean, physically, it's difficult to construct. You cannot use just a regular sheet of paper and a pencil, for instance, and a compass and a straight ruler to do the construction. So we have to make certain assumptions. Now, I would like to assume something, and then I can say, OK, if these conditions are met, then let's assume that construction is basically done. All right, so here is what I actually mean. For instance, our purpose is to create, to construct a plane in a three-dimensional world. Well, what the condition I would like to impose is the following. If I have three points in the space where I definitely know our plane is supposed to pass, then I consider that the construction problem has been solved. So if somebody asks, OK, could you construct the plane which satisfies such and such conditions? And I found three points where this plane is supposed to pass. Then I consider my job is finished. My construction problem is solved, as long as I found these three points. Another example of this is if I have only one point and a line, and the point is outside of the line. Well, according to certain axioms, according to certain principles of 3D geometry, there is one and only one plane which passes through this line and the point outside it. Which means that I can say that, OK, fine, my job is finished. As long as I found my line and the point, my plane has been properly and completely defined. So my construction problem is solved. Now what else about the plane? Another is you have two intersecting lines. There is one and only one line which goes through two intersecting lines in space. And the fourth part is if I have two parallel lines in space, then there is one and only one plane which passes through these. So if I know that I have two lines which are intersecting or I have two parallel lines and I'm looking for a plane and I know that the plane actually passes through these lines or these lines, I consider the job is done. Now this is all about the planes. Now how about other geometrical objects? OK, how about cylindrical surface? Now cylindrical surface is defined by, if you remember, a straight line which is called generatrix. And my cylindrical surface is basically when I'm taking every point on this curve which does not necessarily be a flat curve. It can be whatever in space I can think about. And through every point on this curve I drew a line parallel to this one so the set of all these points and all these lines represent a cylindrical surface. So I consider that if I have a directories and I have a generatrix then I have my cylindrical surface, I don't really have to physically construct it. Now if a particular kind, for instance a prism is needed to be constructed well it means that I have certain directories as a flat polygon and I have another line which is a straight line called directories and then I consider everything is done. Now if my prism is supposed to be a right prism of certain height what does it mean? Well it means I have to have my polygon at the base and I have to have the altitude or the height, the prism which is just a segment actually. Now I can build a prism based on these two. Why? Well this polygon lies in certain plane, let's call it base plane, right? Then there is another base plane which is on this distance from this one and that's sufficient to build a prism like this. So all I need to know is again directories and the altitude, the height of the prism of the right prism in this particular case. So this again is an example of sufficiency of knowledge of certain elements to basically consider that the job of construction is done because there is one and only one prism in this case which satisfies these conditions. This is the directories and this is the altitude. What's next? Next is a cylinder for instance. What do I have to know about cylinder? Well again I have to know the directories. The directories of a cylinder is a circle. So I have to know actually what is this circle about which means by the way to know the center. If I have the plane where it belongs to I need a point on that plane which is the center and the radius. So if I know this I consider my circle defined and I have to know if it's a right cylinder I have to know again its height and I can build it this way. I can build it in my imagination not necessarily in three-dimensional space and again the sufficient condition is plane where the base is located, another plane difference the distance between these planes which is the height of the cylinder and location of the base circle which is basically a center and the radius. Now let's go to conical surfaces. What defines conical surface? Well conical surface is defined by a point called apex and a directories, some curve in space. So again if these two things are given if this is my apex and this is my curve then I can consider my job is done because I can always draw all these lines through every point of this and this point. Now a particular kind of conical surface like a pyramid for instance what do I have to know about the pyramid? Again, apex and some kind of a plane where I have a polygon which is a base polygon of the pyramid. Now I can just connect every point etc. So I consider my job is done if I have this polygon on the base plane and the point in space which is in apex. Same thing with cylinder. Cylinder is supposed to be a circle not cylinder, circle. We are talking about conical surfaces. So I need a circle in the base which means again I have to have a plane and a circle in it defined by center and the radius and then I have to have a point which is in apex of this cone. What's left? Sphere. Okay, what do I have to know for sphere? Center and the radius of the sphere, right? So if I know these two it means I can construct it at least virtually in my imagination but nevertheless it's sufficient to consider the problem as solved. So here is for instance here is an example. For instance you want to construct a sphere which is tangential to all the faces of a tetrahedron. So this is your tetrahedron and I would like to have a sphere which is inside and touches its tangential to every face these three and the four. So all four faces are supposed to be tangential to the sphere. I did not define properly what tangential plane to a sphere is but you basically understand it if this is a ball and this is a tangential plane, right? So how can I solve this problem? If my tetrahedron is given in some way then maybe for instance I know all four points where they are located or something like this. Then if I can, based on this, find a center of this sphere and a radius then I consider my job of constructing is done. So that's basically the approach which I would like to take with all these construction problems in 3D. You don't really have to draw it in three-dimensional space because it's kind of difficult but what you do have to do is to establish sufficient number of elements of a three-dimensional figure, three-dimensional object which define this object uniquely. So for a sphere for instance it's a center and a radius for the right circular cylinder it's the circle in the base plane and its height, etc. So that's basically it. That's my approach and I will use this particular approach when solving all the different construction problems in the future. Well, that's it. Thanks very much and good luck.