[MUSIC] You have now seen that symmetry is all around you. You can find symmetry in common objects, in architecture, and in nature. But our description of symmetry to date really has been qualitative. Now we have to begin to think about a quantitative description of symmetry. So in this last section of part one of the course. We need to talk about the generation of symmetrically related objects. And do this in a way that we can specify their fractional coordinates. For these exercises, we'll look at square and triangular networks. Secondly, we need to bring together all that you have learned about point symmetry. And particularly, that there are ten point symmetry groups. And finally, and this is to help you carry out your first exercise. You need to learn how to superimpose crystallographic symbols on common objects to define their point symmetry. Let's begin by looking at square nets. Previously we talked about chiral objects. And you must remember that chiral objects are always produced by reflection. Also remember that the symbol for reflection is a single solid line. So if you look at the top of this slide. You can see an open circle and a circle with a comma inside. This is how we formally represent the relationship between chiral objects. From the application of a mirror we create the chiral object. Also recall that the reflection is what's known as a symmetry operation. And the mirror line itself is what is known as a Symmetry Element. Now, let's start to use this terminology and these symbols on nets. And we begin with a square net because that's the most straightforward. If you plot any object on a square net, you can give it coordinates x, y. In this case, we have the x axis pointing downwards. And we have the y axis pointing towards the right. Recall from our discussion on tiling, that this is the standard way to represent the x and y directions. If we introduce a horizontal mirror, illustrated by the blue line, then we will generate a second object, minus x y. Because the object is generated through a mirror, we show that it is a chiral object, with respect to the first object. What would happen if instead of placing the object at position x y? We placed the object shown by the green circle, in this case, right on top of the mirror line. Would we generate any more objects? The answer is nothing. We do not generate any more objects and nothing happens to that green object, when it lies exactly on the mirror line. So you can see that for objects to be generated, they have to lie away from the symmetry operator. In total in this particular square net, we have two objects. If the object lies away from symmetry line, when we use the mirror line once we generate one additional spot which has a chiral relationship. We can do the same thing with a vertical mirror. In this case, again we generate two symmetrical objects. For both of those square nets where the mirror is horizontal or vertical we say that the point symmetry is m, because we have only one mirror present. Similarly, with the network on the right, if we place a green object on that vertical mirror, we do not generate any additional objects. When the objects lie away from the symmetry operator, the xy objects we generate the maximum number of objects possible. And this is known as the general position. And we'll keep returning to this concept of general position throughout the course. So now let's look at the square net again. But in this case we'll introduce two file rotation. The two file rotation point will be at the intersection of the x and y axis. If we start with an object xy. Apply the 2-fold rotation, which is a 180 degree rotation, sometimes known as a diad rotation. We generate the object minus x minus y. Notice that that there is no mirror involved. We do not introduce corrality, so both the circles are open. If we decide to put an object directly on the rotation point. So again we place a green circle right at the center. What happens? Absolutely nothing, because objects which lie on the symmetry operator do not change. Now, let's begin to combine symmetry elements. On the right hand side, we're introducing not only 2-fold rotation, but also a vertical and a horizontal mirror. Now, you can see that we generate a total of four objects. Their coordinates are shown, as well. Also recall that it wouldn't matter, if we did the rotation first or the mirrors first. We can apply those symmetry operations in any order. And we will end up with the same result. For the example on the left, we would say that the multiplicity or the number of general positions is two. Xy and minus x and minus y. For the example on the right, we would say, that the general position has a multiplicity of four, because we have generated four objects. Now we go to slightly more complicated examples. And I introduce a little bit of matrix algebra to help explain what is happening. However, this will not be a testable part of the course. The matrix algebra is for those who would appreciate the rigor with which it helps us to find the change in symmetry. If we take again the example on the left hand side, we now have a 4-fold rotation point. In this case, because we're rotating through 90 degrees, which means we introduce objects which are unchanged in aspect through the 90 degree rotation. We have a general position which as a multiplicity of form. One thing that I should emphasize is that, in crystallography and in symmetry nomenclature generally. The rotations always take place in an anti-clockwise direction. So for those of you who are not interested in a little bit of matrix algebra below, which illustrates why that is the case. Just remember that the rotations will always be anti-clockwise. If we now combine the 4-fold rotation with mirrors. Then what we will find is that we generate eight objects inside the network. So now the general position has a multiplicity of eight. You can see that as we put more and more symmetry operations together, the multiplicity of a general position increases. And this again, is generally true. Now let's come to the final examples. This is where we look at a triangular network. And we use triangular and square nets to describe these symmetry operations, because these are the ones which are commonly observed. It's a little bit more tricky, and again, do not be concerned about the small of matrix algebra, if it's not of interest to you or you don't have the Math to cope. But again, the main thing to emphasis is that the rotation must occur in an anti-clockwise direction. Because we're dealing with a triangular net. And we place an object x y in a general position, we generate three objects. So the general position will have a multiplicity of three. And here we have the final examples. And this is where we combine the mirror lines with the 3-fold rotation which is illustrated by the triangle. In this case, it does matter where you place the mirror lines with respect to the rotation point. And you can see the two examples. The point symmetry is either 31m or 3m1. The first symbol as we've already discussed will indicate the rotation point. The second symbol, and the third symbol indicate whether there are mirrors or not. In the first example, 3m1 the positioning of the mirrors will place a combination of chiral and achiral objects as shown. In this case, the reflections align parallel to the x and the y axis. In the right hand case, we also generate six objects. But now the reflection planes are perpendicular to the x and y objects. So let's collect together what we've now learned. We've given many examples of point symmetry. In flowers, in wheels, and so on. In fact it turns out there are ten point symmetry groups. And these are summarized in this slide. Most of these we have already encountered. And it's not necessary for you to remember these it's only necessary to understand what they are telling you. You'll notice in these point symmetry groups that where you have a single symbol indicating just rotation. None of the objects are chiral with respect to each other, in fact all of the circles are shown as open circles, but as soon as we introduce mirror lines. Then we have chiral relationships between the objects. And so you see open circles and circles with a comma inside. There are only 10 point symmetry groups that we need to consider. Now, let's collect together all of the symmetry symbols we have encountered so far. First, there is translation. And translation is always indicated by a line with an arrow showing the direction of the translation. In general, translation would be a distance which is equal to half the edge of the tile or the unit cell. But not always, we'll encounter different instances of this later in the course. Reflection is shown as a continuous, solid line. We know what happens in reflection, we create a mirror image. And in so doing, we create chiral objects. For rotation, we have four different types of rotation that will be used in a formal description of symmetry. There is 2-fold rotation, which is 90 degrees. 3-fold rotation, which is 120 degrees. 4-fold rotation, which is 90 degrees. And 6-fold rotation, which is 60 degree rotation. We apply those rotations repeatedly, until we return to the original position of the objects. Another way to describe these rotation points, is the diad, triad, tetrad, or hexad rotation. Rotation does not create a chiral relationship between the objects. And finally, we had a compound operation known as glide. This is where we put together reflection and translation. Glide lines are illustrated only through a dashed line. So to summarize this part of the course. Objects are generated through the application of symmetry operators. When the object sits away from the symmetry operation we generate the maximum number of objects. This is known as a general position. On the other hand, if the object sits right on the symmetry operator, we don't generate any new objects. And finally, we discovered that there are 10, and only 10 point symmetry groups.