 Hi, I'm Zor. Welcome to Unisor Education. I would like to talk today about similarity of geometrical objects. First, before going into mathematically rigorous or almost rigorous definition of similarity, let's just think about what kind of geometrical objects we may call similar. How about two points? Are they similar? Intuitively, yes. I actually would like to refer to intuition right now, before we go into any kind of rigorous definition. Okay, fine. How about two segments? Are they similar? Well, I understand they're not congruent, not equal in lengths, but are they similar? Well, again, intuitively, similarity is referred to shaped more or less the same way, and they are shaped the same way, more or less. So, I would say these are two similar objects. How about a little bit more complex, something like this and this. Are they similar? Well, probably not. Again, we are talking about only intuitive kind of understanding of similarity. Now, why are they not similar? Well, because this is kind of angular, it's a triangle and this is circle, it doesn't have any angles. All right, fine. Let's just do it slightly differently. Let's just have two angular figures. Let's say this one and this one. Are they similar? Again, probably not, because there are three angles here and four angles there. All right, how about two triangles? This one and this one. Are they similar? Again, although they do look alike, however, it doesn't look like this is the same kind of a shape, if you wish. Yes, they both have three angles, but angles are different. So it looks like an angularity by itself is not sufficient for being similar. We need something which is something like this. These two, although not congruent and not equal in size, et cetera, do look similar. And why do we feel that they are similar? Well, primarily because their angles are the same and obviously the number of angles and number of sides, et cetera. So how can we define two different geometric figures to be similar to each other in a little bit more mathematical fashion? Well, we can always talk about equality of the angles and proportionality of the sides, but I don't want this to be defined this way because how about two circles? This one and this one. Are they similar? Well, seems to be, right? But we can't really define any kind of angles here or sides. There is no such thing. So I would like to approach similarity from slightly different angle and it would be more universal and applicable not only to triangles, quadrangles, et cetera, but basically to all geometrical figures. For instance, this one and this one. They also look similar, right? Okay, so let's go and try to define similarity in a little bit more rigorous fashion. First of all, let me recall what transformations we already know, which we usually call congruent transformations. Now, if you remember, we can do something like a parallel shift, which means a figure like this will be transformed into a figure like this. Now, the parallel shift sometimes is called translation. We also know that there is a rotation, which means something like this can be rotated into something like this. Just turn it and shift it. And we also know that there is another congruent transformation which would transform something like this into something like this, which is symmetry relative to the axis. Now, having basically recalled these transformations I would like to add one more transformation and using this transformation, I would define the similarity. So what transformation I need right now? Well, it's called scaling. Scaling basically means the following. You have one particular point on the plane which is called a center of scaling. And then each point wherever it is would be transformed into another point using the following group. So the center of scaling, let's call it Q, is fixed and also fix something which we called a factor of scaling. Now, this factor of scaling can be any real number. So let's say factor of scaling is one half. What does it mean? It means that any point wherever it is would be transformed along this particular line in such a way that the distance, whatever it is right now, will be multiplied by a factor. So if it's one half, this point will be shifted to half of this segment. So if that was A, and this is B, so A would be transformed into B for factor equals one half. Now, if the factor is equal to, let's say two, then this segment will be multiplied by two in lengths and it will be here. I meant transformation. Now, what about the negative factors? Well, negatives is basically the same thing except we go to this direction. So we do two things. Number one, we basically reflect A to A prime. So if factor is equal, let's say minus one half. First, we reflect it to A prime relative to this center from A to A prime. It's a symmetry relative to the point. And then we basically multiply the lines of this by absolute value of the factor and you will have B prime. And finally, if A is transformed by the factor of minus two, it will be, again, first, we symmetrically reflect relative to the point and then we multiply these lines by the absolute value of the factor and we will have C prime. So it's quite easy to understand what actually is the scaling. Now, any other point, let's say, let's call it this point M, would be transformed according to exactly the same rule, which means I will connect it to the center of the scaling and then multiply this particular length by whatever the factor is positive. It will be this way, negative will be this way. Now, what's important to understand is that every point on the plane would be transformed in exactly the same way, regardless of what direction we are scaling. So this direction would be the same factor as that direction or this direction or this direction from the center. This property is called isotropic. Isotropic is basically equal in all directions. So that's why I put this word there. So I would actually refer it to certain words so we will know where we are. So we have basically decided that scaling is supposed to be isotropic, which means in every direction we are scaling by the same factor and we are basically either stretching or shrinking the distance from the center of scaling of any point along the line which connects the center with this point. And in case the factor is negative, we just go to another side of this line, but in any case it's exactly the same thing. So this is a transformation of scaling. Now, why did I define it? Well, let's look again at this picture of something which I consider, well, it's kind of similar, right? So my purpose was to preserve the shape but to change the size. And the factorization, the scaling, that's what actually is changing the size. So I presume that I have a transformation of scaling which will transform one into another. So this point would be transformed into this point, which means they're supposed to be on the same line from some center of scaling. And this point would be transformed, this point would be transformed into this line, which means these two lines are on the same scale. Now we are dealing with this center of scaling. And if the proportionality of these two segments, let's say it looks like it's about two, right? This is about half the size of this. So if proportionality between this and this is the same as between this and this, then we are talking about scaling relative to this center and the factor equal to ratio between this length or this length, this length, and this length, et cetera. So these points are corresponding to, not the probability, call it A prime, B and B prime, C and C prime. So these points are scaling one into another, one into another. And thereby, we are transforming this particular geometrical figure into this. So it looks like scaling and similarity are related to each other concepts. I think we are ready right now to define what actually similarity is. Now, one more thing before that. Now, let's consider these two are similar because we can scale one into another. How about these two? So I turned this particular figure this way. Well, because I'm turning, which means I'm transforming congruently, transforming one into another, similarity should not really change. I mean, this is as similar to this as this, which means that if I would like to use transformation to be able to transform one figure into another, I should use not only scaling, but also congruent transformation of parallel shift, and which is translation and the entrantation and symmetry relative to the axis. So all my congruent transformations also should participate. So now the definition. Two figures which can be transformed one into another using these transformations of rotation, translation, reflection, and scaling using some center and some factor. Then these two figures can be called similar. This is a definition. So existence of congruent and scaling transformations is basically a definition of similar geometrical figures. So I don't have to really talk about angles or lengths of segments, et cetera, because they might not actually exist. You can have a form as non-angular like this one. This is a more universal definition, and I think it's always more preferable if you're dealing with something as universal as similarity. You don't want to be restricted only to triangles or quadrangles, et cetera. Now, all these properties of triangles and quadrangles with preserved angles and proportionality of the sides, I will address all these, but they will be not a definition of the similarity. They will be the consequence from this particular definition which is based on scalability. Okay, now let's talk again a little bit about scaling. If I have scaled one figure into another using this center and let's say a factor of two, like in this particular case. What does it mean actually? Well, it's an operation on geometric objects, right? Transformation, any transformation is some kind of an operation. Now, if you remember, when we were talking about operations, we were talking about certain sets of operations which are kind of complete to each other. Now, what does it mean in this particular case? Number one, is there a scaling which is a unit operation? Which means it does not change the geometric object. Well, the answer is yes. Have any scaling with a factor equal to one and you will have basically transformation which doesn't really move any point. Any point will just remain at its own position after this transformation. Now, next, is there a reverse transformation? So if I have a factor two for this particular scaling, can I transform back from here to here? Well, answer is obviously yes. We just use the inverse factor. If this is two, use one over two. If this is minus two, use minus one over two. If this is minus one, use another minus one. One over minus one, which is minus one. So whenever you're using one over this factor, you will have a reverse transformation which means all the transformations of scaling are well complete for anyone who is a reverse one. There is a unit transformation which doesn't change anything. Now, how about commutative and associative laws? Well, yes, they are preserved as well because if you think about, it's actually very easily, it can be very easily approved from the properties of multiplication because what are we doing? We are multiplying this length by the factor. Now, if you are multiplying it once by a factor of two and then another time by the factor of three, it actually means you're multiplying the lengths by two and by three. But it's the same thing as if you multiply it by three and by two, which is six anyway because the multiplication of the numbers is commutative. So the scaling of different factors using the same center of scaling is commutative. And for obvious reason, it's associative as well because of associativity between the numbers. All right, what are the elements? I think I have covered practically everything. All right, just one more obvious remark. If you have two congruent objects, are they similar? Well, obviously they are similar or object is similar to itself with the factor of scaling of equal to one as I have already mentioned. And other than that, I think we have covered this completely. Now, the purpose of this lecture was to introduce the definition of similarity based only on transformations of scaling rather than resorting to definitions for triangles separately from quadrangles and then we don't know what to do with circles, et cetera, et cetera. So similarity based on scaling is much better definition because it's much more universal. And there is one more word, which I wanted to talk about. This scaling is in mathematics, sometimes is called homogeneity. It's just the same word. I mean, the same meaning, different word, the same meaning. It sounds more scientific. That's why probably people like to use it, homogeneity. Anyway, I'll probably use scaling anyway. So again, the purpose of this lecture was just to introduce you to the definition of similarity. Now, next lectures will be devoted to how similarity is related to angles or to segments or to triangles or to circles and segments together, whatever it is. And then some problems will be introduced. This is just an introductory lecture to concepts of similarity. Don't forget that everything is on unisor.com. Gradually, this website is getting populated with lectures. There are about 100 lectures right now for different kinds of aspects of mathematics. I do recommend to go to this website and just take the whole course, whatever is available right now. And I do recommend parents to participate in this educational process of their students as well. Or maybe teachers who would like to introduce this way of learning. So people would learn from the lectures and then the teacher of the group can actually answer some more difficult questions or put some light on certain difficult parts of this. Okay, so again, it's unisor.com and it's free. So basically use it as you want. Thank you very much.