[MUSIC] Let's recap what we've so far learned about plane symmetry. We began by looking at the five Bravais lattices. We then applied these to corranigan tiles. And most recently we've just looked at the use of the Bravais Lattice as a basis for Escha creating his wallpaper patents. But what we need to do now is look at these ideas mathematically. We have to understand them in a formal way. And in particular we begin by looking at the difference between the so-called general and special positions. General and special positions are created depending on where an object is with respect to the symmetry apparatus inside the tile of the unit cell. By having general and special positions, we can replicate objects in different proportions. We also need to learn how to read the plane group tables. By reading the plane group tables, we can understand the diagrams for the general position, and also the symmetry operator positions. And this is a great short hand for figuring out where objects will be located through the application of plane symmetry. Now let's discover the difference between general and special positions. For this exercise, we'll use a rectangular Bravais lattice. The repeating tile, or unit cell, is now superimposed on that lattice and illustrated by a blue outline. On top of that rectangle, we want to insert symmetry operations. And for this example, we'll use two-fold or diag rotation points and mirror lines. You'll notice that the mirror lines, which are both vertical and horizontal, intersect at the rotation points. Inside that repeating tile, we place one object, the red square, in the top left hand corner. The question now is through the application of the rotation and the mirrors, how many square objects do we produce per unit cell? It turns out that we produce them as shown in here. In the uppermost and lowermost corners we have a total of four red squares. When you look at the diagram in this way it might not be so obvious where those four red squares came from. But if you look at the distribution around the origin it becomes more obvious. By applying mirrors, both vertically and horizontally we will generate those total of four red squares. What would happen if we placed a blue square on the horizontal mirror as shown now on the diagram? How many additional blue objects would we create? In this case, there would only be one additional object per unit cell. So the total would be two blue squares per unit cell. And this is because only the vertical mirror plane or the central 2-fold rotation point is operating. The mirror lines which are uppermost and lowermost in the unit cell do not create any additional blue squares. If we then extend across all of the tiles, we see the total pattern of blue squares in this example. Let's do the same thing with the vertical mirror. In this case, we use a green square. As you have probably guessed, you can produce two of these per unit cell and we can extend that throughout the extent of the tiling. Finally, let's place a yellow square at the central rotation point. Through the application of the mirrors, both horizontal and vertical and the rotation, how many additional yellow objects to be produced? In fact, we don't produce anymore. There are only one of these per unit cell. And again, we can extend through space to show all of these. So what you can see is that the difference between a general position, where the object lies away from any of the symmetry operators, and the special positions, where the object lies on top of a symmetry operation, is the number of objects created. The general position always creates the maximum number of objects possible. In this case, four per unit cell. But as you place objects on positions of higher and higher symmetry, you produce fewer and fewer objects. If we look inside the International Tables of Crystallography, you'll find this section about plane groups. It shows that there are 17 plane groups and only 17 of these. Every page of the plane group tables contains specific information. At the top left hand side you will always see the symmetry operation diagram. On the righthand side is a general position diagram. And then underneath here, we see where the general positions will lie with respect to their fractional coordinates x and y. Every plane group table looks like this. And we provided you on the internet the resources where you can find this information. So let's return to this arrangement as shown before where we generated the special and general positions for the different colored squares. How do we use the information in the space group tables to look at this more formally. In fact, that arrangement of dyad rotations and vertical and horizontal mirrors conforms to the plane group p2mm which is actually number 6 in that series of 17 plane groups. The p implies that the cell is primitive. The two shows that there are two fold rotation points and these are located at the origin of the unit cell and also in the center and the edges. Wherever we have a two fold rotation point, we have two intersecting mirrors, hence the symmetry symbol p2mm. Formally the origin of the unit cell is shown in the top left hand corner. The a direction is vertical and the b direction is horizontal. So the fractional coordinates x and y of course are also vertical and horizontal. If we place an open circle in the top left hand corner and use the p2mm symmetry to operate upon it, we create chiral objects because there is mirror involved. So we create chiral objects as shown using the vertical mirror but also the horizontal mirror. And if we apply either of those mirrors again, we create another object in the bottom right-hand corner which is no longer chiral with respect to the original object. Again, just as we already demonstrated with the squares, we can generate a group of four objects around every origin, but now we are carrying one additional piece of information, which is that it's two pairs of chiral objects. We can then look at the fractional coordinates for the positions of those objects inside the unit cell. The first object which we began with, might be given the nomenclature, xy. We can then label all of the other objects inside the unit cell as shown. So we have x-y, -x-y, and -xy. In crystallography, the minus sign is normally placed above the x or the y. This then defines using fractional coordinates all the general positions. And we list them in number, one, two, three, four, with the actual general position written next to it. So how is this information presented in the plane group tables? We show here the two diagrams involved. On the left, we have the symmetry position diagram. On the right, we have the general position diagram. I'm showing, as well, precisely where the mirrors and the diads lie. But we also have some additional nomenclature that we have to learn. Multiplicity, which we've already mentioned in our discussion of the Escher tile, but also the Wyckoff position, site symmetry, and the positions or coordinates of every object. An object which does not lie on a symmetry operation is the general position and it has multiplicity four. We have a total of four objects inside the unit cell, two open circles and two circles with the comma. The Site symmetry is 1 because the object is not sitting on an operator. If we place an object on one of the symmetry operations, such as one of the mirrors, we produce a special position. And the multiplicity must be less than a general position. In this case we can find four special positions laying on mirrors as shown here. We can also place objects directly on top of the two fold rotation points. But here at the two fold rotation points two mirrors intersect so the side symmetry is 2mm. You'll also notice that the multiplicity must only be 1, and we have four of these sites which have that type of multiplicity. Finally, to complete the information in the plain group tables, we need to assign Wyckoff symbols, or Wyckoff letters. This is very straightforward. All we do is label from bottom to top in sequence of the alphabet. So, A, B, C, D, and so forth, starting at the bottom. Now, let's take a moment to look at how the site symmetry symbols work. For the general position, the site symmetry is one. Indicating that the object is not lying on top of any symmetry operation. In the case of the sites, which are produced by placing an object on the mirror lines, we use the m symbol but the position of that m symbol is important. Generally we use three positions to describe site symmetry. These could be indicated by three dots. The first dot is always used to represent the rotation axis. The later dots indicate mirror lines or glide lines. The middle dot is used to indicate mirror, which is at right angles to the x direction. The final dot is used to indicate the mirror at right angles to the y direction. For the diad rotation points, the site symmetry is 2mm. And that's because the mirrors intersect there. We can also discover from the general position diagram what the asymmetric unit must be. Remember the asymmetric unit will only contain one object upon which the action of the symmetry produce all of the other general positions. So in this case the asymmetric size of this particular symmetry, p2mm, is the top left hand quadrant of the cell. These lectures have been particularly information rich. We have spent a lot of time talking about general positions and special positions in the plane groups. For a general position, the object does not lie on top of any symmetry operator. In the case of a special position, the object superimposes one or more of the symmetry operators. When the object sits directly on top of a symmetry operator, it has no effect, in other words, it does not generate any new positions. Consequently, the multiplicity of the general position, always exceeds the multiplicity of the special positions. Because we can create special positions that are different, but have the same multiplicity, we need to be able to differentiate these. And to do that, we use Wyckoff symbols. The Wyckoff symbols are assigned from the highest symmetry point to the lowest symmetry point.