 Hi, I'm Zor. Welcome to Unisor Education. I would like to introduce today a concept of similarity in three-dimensional space. This is the part of the Unisor.com set of lectures on advanced mathematics for high school students. And basically this is very much similar to similarity on the plane. I mean the same concept of scaling, factor, etc. There are a couple of maybe not very significant additions which I would like actually to introduce for three-dimensional space. And that's why it will not be a long lecture or anything like that. So I would like to remind you certain principles of scaling and similarity. But it would be really great if you go to the corresponding lectures in the plane geometry. There are many different problems which are solved for related to similarity and scaling, etc. So I think it would be really very very beneficial if you will familiarize yourself once more with the similarity on the plane. Okay, now 3D similarity. So first of all we introduce a concept of scaling. Now scaling, sometimes there is another very scientifically sounding name. I think. So scaling is kind of a simpler word and basically what it means is the following. Now we are talking about three-dimensional space. So first of all you have to have a point, let's call X, which is a center of scaling. Then for any other point in the space we do the following. First we choose certain number, real number. It can be positive or negative but non-zero. Let's say the number is 1.5, it doesn't really matter. So for any point in the three-dimensional space you connect it with X and you extend it further. In such a way you choose point M prime in such a way that length of XM multiplied by this factor F is equal to length of XM prime. Now this point M prime should be chosen on the same side from X as point M for positive F, for positive factors. Now if the factor is negative, let's say it's minus 1.5, what do we do in this case? Well, we extend it but to the opposite direction. That's where our M prime will be. So the length would be the same except I probably should add absolute value of F. So point M prime is chosen on the same side from X on this line, XM line, for positive factors on an opposite side from the point X for negative factors. And the length from X to the new point should satisfy this particular equality. So there is nothing new here, it's exactly the same as on the plane. That's the definition. Now that's how we transform the point. So during the process of scaling this is step-by-step instruction of how to transform a point. Now let's say you have a segment. I will use positive F just for simplicity. So let's say you have a segment, M, M. So this is your segment. Now to transform a segment using the same procedure of scaling you basically have to transform each point on that segment. So let's go to the end point. So that probably would be M prime, that probably would be M prime. What's important is, and we have actually proven that thing in one of the lectures on the similarity on the plane. Because if you have a segment, together with this point, these three points define a plane. So everything actually occurs in that plane. Because if you have two points, X and M, which belong to this plane, then the entire line belongs. Same thing here. Now obviously if M and M belong to this plane, then each point on this belongs to this plane. And that's why each point on these lines belong to the same plane. So we have plane geometry basically. So everything is happening within a plane defined by these three points. So in one of the lectures on plane similarity on the plane, we proved that the image of the line is a parallel line. Now as far as the segments are concerned, obviously if these are two segments which have a ratio of their lengths equal to F and these two the same, then from the similarity of these triangles X, M, N and X, M prime, N prime follows that M, N and M prime, N prime also related by the same equality. M, N multiplied by absolute value of the factor equals M prime, N prime lengths. So the length of the segment is proportionally increasing and the transformed segment during this scaling is parallel to this one and that's basically the picture of this transformation of scaling. Now there is nothing new as far as the three-dimensional space yet. So what's the new? Well, the new is transformation of the plane in the three-dimensional space. So let's imagine you have a point and a plane, a center of the scaling and some kind of a plane. So question is if this is a plane and each point on this plane I transform. Now obviously you understand that it's supposed to be a plane. Now how can we prove that the plane is transformed into a plane? Well actually relatively easy because we can always take two lines on this plane. Let's say this line and this line. Now their image would be other two lines as we know. So now what I'm saying is that every other point on this plane would belong to the plane defined by these two lines. Now it's quite obvious because all the lengths is proportionally increasing. So we can always talk about similar triangles. So whenever we have something like this triangle it will be transformed into another triangle with all the lines parallel. And since we know that this point belongs to this plane, now this point belongs to this plane obviously and this one also. So everything in between also belongs. So every line on this particular plane is transformed into a corresponding line on this plane defined by our two initial lines. So basically that's the relatively trivial explanation that the plane goes to the plane. Just choose two lines, reflect them, transform them using the scaling and they will be two lines as we know from the plane geometry. And they define the plane which is an image of this plane. And then for each point we can very easily build some kind of a triangle like this and show that the image will be on that plane. So the plane goes to a plane. Alright, what else is important as far as scaling is transformed? Now I can actually do a little bit more. I can actually talk about similarity as a concept. So what is the similarity in three-dimensional space? Well, I can say that two different objects in three-dimensional space are similar. If there exists some kind of a scaling of one object which means there is a center of scaling and the factor which will transform it maybe not to another object but to an object which is congruent to another object. Congrance in this case means that maybe we have to turn it a little bit after. So let's just consider for instance a square. So this is a cube. Let's have a three-dimensional case. So this is the cube. And this is a cube. Now how can I transform this cube into this cube? Well, let's just measure the edges. Divide this edge into this to get some kind of a factor. Now using actually any point in space as a center and this as a factor, I will reflect each point of this cube and I will get something like this. Now since I have chosen the scaling factor which is a ratio between this and this, between the edges, this resulting cube will be of exactly the same size as this maybe positioned differently in the space. But again, that's what congruence actually means. So I can always take this cube if it has exactly the same edges and obviously angles. I can always somehow turn it, rotate it and convert it into this cube. So basically to the degree of congruence if one object can be scaled to another congruent to the one which we are looking for, then the objects are similar. Now in this case congruence means that we can shift the resulting object. We can rotate it and actually we can also asymmetrically reflect it because symmetry also is considered to be a congruent transformation. So that's how we can convert one into another. So if such a scaling exists which transforms object Xe to some object Xe prime and the scaling requires center and a factor. And if this object is congruent to our object Eta then C and Eta are considered to be similar. So that's the definition of similarity. Similarity is based on existence of the transformation of scaling which means existence of the center and the factor which will transform one object into another which is congruent to the one which we need. Now here is an interesting point. Now let's say object C is transformed into C prime using Xf center and factor which is congruent to Eta. So that's what Xe is similar to Eta means. Now if I will take Eta, maybe transform it a little bit using some congruent transformation which means I will rotate it, shift it, etc. into C prime. And then I will use a transformation which has exactly the same center but 1 over F as a factor. What will I have? Well obviously I will have C. So if I convert this object into this, if I convert this into this with a factor F I can convert this into this. So this is my C, this is my C prime. I can convert this into this with the factor 1 over F. If this is double, this is 1 half. So which means that if object C is similar to object Eta object Eta is similar to object C which means we have a symmetrical relationship. So the similarity is symmetrical. If one object is similar to another then that other is similar to the first one. So that's called symmetry of this relationship. Another interesting point is obviously any object is similar to itself. Why? Because I can always choose any point as a center and the factor of 1. The factor of 1 will convert every point of this object into itself. If you have an object and I have a center then this point is converted into itself because the factor is 1. So the length is multiplied by 1. So that actually is called reflexivity. The similarity is reflexive which means any object is similar to itself. So it's reflexive and it's symmetrical. Okay, fine. What else? Angles. All right, angles. Now since we preserve, well since we proportionally increase the sides of the segments then any angle which can be included into some triangle with any point of center of scaling will be transformed with every factor. So this triangle will be transformed into this triangle. Let's say factor is 1.5 approximately in this case. Now obviously these triangles are similar because everything is proportional which means that the angles are preserved. Now this is a plane geometry but dihedral angles which is a three-dimensional angle between the planes they're also preserved during the transformation of scaling. How can that be done? Well, again that's kind of easy because if you have two planes and angle between them, okay, so these are two planes. Now what is dihedral angle? Angle. Dihedral angle is measured by the angle between two perpendiculars to the intersection, one perpendicular within one plane, another perpendicular in another plane. So the angle between these two perpendiculars is actually a measurement of the dihedral angle, right? But if I will scale it to another, so the line of intersection will be parallel obviously, the plane will be parallel, it will be just further, let's say. That would be our plane. But now this point will be translated into this one. Now this line would be parallel and this line would be parallel. This belongs to this plane and this belongs to this plane and the same thing here. So the angle will be exactly the same and these lines are parallel. Scaling transforms a line into a parallel line. So that's why the dihedral angles are also preserved. Okay, next two topics are very important and not very difficult to control. Now let's talk about the area. So for instance we can have a prison and we have area of all the faces of this prison, right? So how the area is transformed in the three-dimensional space? Well, that's kind of easy. Let's take a small square with a side equal to one and let's take a scaling with a factor f. So for instance this is my, so I'm scaling this twice. That's how it will be. So this is my result. Well, obviously it will be a square because all angles are preserved and the lengths of the segments are multiplied by a factor of f, right? Well actually absolute value of f. So what does it mean? It will be a square. Now what will be the side of this square? Well if this side of this is one, then this one will be one times f, which is f. Well actually absolute value of f. Just in case I have chosen a negative f. And what will be the area? Well the area will be f square. So what happens with our square of a unit length? The area of the square of a unit length is multiplied by f square. So that's the ratio. So every line, every edge of the square, every side of the square is longer by absolute value of f. But the area is f square. Okay, what if I will choose another square? Let's say square instead of n, instead of one unit length I will have one over n. Now what will be my side length of the resulting? Well absolute value of f times one n. That would be my length, right? So what would be my area? Obviously the area would be f square divided by n square, right? If f over n, absolute value of f over n is a side, then this is the area. Now the area of the initial square is obviously one n. So the ratio is again f square. So any square of one over n actually is transformed in such a way that the area is multiplied by f square. Okay, now this is the square. How about any flat polygon? For instance you have a hexagonal, and using the center you transform it into another hexagonal, which is kind of bigger, something like this. This is the transformation. And again the factor is f. What happens with the area? Well let's do it this way. Let's consider just the concept of the area. The concept of the area was defined using squares actually. First we put like unit squares everywhere, wherever we can. Then we have something which is not really filled up. We can take a smaller, let's choose n equals to whatever, one-tenths, and put smaller squares around. And gradually we will fill up the whole area with squares of smaller and smaller and smaller sides. Now I'm not pretending this is a rigorous proof, but in any case you definitely should feel that this process can be continued to any degree of precision. I mean choosing significantly smaller and smaller squares to fill up whatever remains unfilled, we can actually approximate the area of the hexagonal with certain number of squares of different size. And since every square is transformed by this rule, multiplied, the area is multiplied by f square, then the sum of these obviously is also multiplied by f square. And the sum of them is with any degree of precision approximates the area of the, in this case, hexagonal. Which means that area of any polygon can be considered in this transformation as multiplied by f square. So this is basically the rule in as much as every segment is transformed by absolute value of f as far as its length is concerned, every flat polygon or not even polygon, it can be a circle or ellipse or anything, any flat thing which has certain area is transformed in such a way that the area is multiplied by f square. So that's my point. Area is multiplied by f square by the scaling by the factor f. Now, volume, that's the next thing, right? It's also the same kind of concept. First we will start with cube. We start with a cube of a unit length, okay? Now, whenever we transform it, if this is the one, the new one, the transformed cube, well, it will be cube, right? Because everything is retained and proportional. So it will be our size absolute value of f. The volume of a cube, by definition we know it's multiplied, it's a prism from the concept of prism and its volume, we know that it's the result of multiplication of three dimensions, one, two and three, and each one of them is f, so it's absolute value of f cube. Now, that's the unit. How about if it's one over n, the size of the h? Well, obviously it would be absolute value of f over n here, and the volume of this thing would be, again, the multiplication of absolute value of f over n three times, which is this. And this guy has the volume of one n cube there. So again, the factor is f absolute value of f cube. So that's the factor when we are talking about cubes. And the same exactly consideration. If you have any kind of an object, a pyramid, a prism, a cylinder, whatever else, we approximate the volume of this figure with unit cubes, then whatever is not filled up, we reduce the size of the cube, let's say one tenth of something, some unit centimeter inch, whatever. And smaller and smaller, and that's why we approximate to any degree of precision, we can approximate the volume of this three-dimensional figure with our cubes, in elementary cubes. And since every cube is transformed by this scaling in such a way that the volume is multiplied by absolute value of f to the third power, then the same thing is applicable to any three-dimensional object. Its volume during the scaling is multiplied by a factor absolute value of f to the third degree. And I would like actually to pay attention to this interesting detail. A segment, length of the segment is multiplied by absolute value of f, area is multiplied by f square, and volume is multiplied by absolute value of f cube. And that's related to dimensionality of the set. The segment is a one-dimensional object. Some kind of a flat polygon is a two-dimensional object, and something like a pyramid is a three-dimensional object. So dimensionality is related to this power, one-dimensional multiplied by absolute value of f, one-dimensional f square, and three-dimensional f cube. Isn't that interesting? So dimensionality and this power is very much related. And whenever something more advanced like geometry in n-dimensional space is studied, we can see that there is a concept of a volume, obviously, in n-dimensional space. And in the scaling procedure, this would be exactly n-dimensionality. Well, that's it. Thank you very much and good luck.