[MUSIC] So of course, despite your best efforts, Victor, some tiles really have been lost. But you can do some research and find out what the patterns really were from all those years ago. >> Well, a lot of design in the old days you can't. It's already lost. So we are lucky that we have a book, a catalogue. This catalogue is from Italy, Ing Ghilardi, okay, and it's 1929. Okay, all the design here inside, all the design. This is a very old book. I got it from a antique shops. All the design is here. So even although they are gone, but with this catalogue we can reproduce it. This is one that I just reproduced. It's very beautiful. Look at the tiles are so perfect. And this kind of tile has got glaze and non-glaze over here. There is two types. One is matte, one is glossy. So this design, this motif, we also can change it to the different design. You just flip it up. It is so unique, you know, they are so smart in the old days. See, with this design, you flip it over, you see it can become another new design. Don't you think that it's nice? >> Oh, that's beautiful, yes. >> So this is the beauty of the geometric design. >> You can put them together in different ways to create something. >> Yes, that's correct, yes. >> So when people use these tiles, do they use them primarily for walls or as large sections? I mean, I notice these particular tiles also have a border, a frieze. >> Basically this is all for floor tiles. They are floor tiles. They are not glazed. They are unglazed because they can walk, okay, for heavy traffic. So a lot of these is on shop houses, the five foot way, inside the temple, in the shop houses, all from front through the courtyard, all with this design. >> So I can see you've got many designs here. So what I think would be very nice is if we just look at each design, and you can show us how you can rearrange to create different patterns [CROSSTALK] for each one. >> Oh, sure, okay. [MUSIC] >> Now that we've seen some of the more complicated tilings that are found in of the shop houses, we should examine how we can analyze these in a formal way. In short, we have to understand the basis of their tessellation. The main two objectives of this short lecture are to teach you some more about counting objects in a symmetric tile, also known as a unit cell. And also to locate the symmetry operations in these tessellations. This perhaps is a more complicated activity for you. Let's look at this arrangement of roses. We stick with the rose theme because the Peranakan tiles use roses so often. You can see now that we have roses in a couple of orientations. So that means that we must have a larger unit cell. If you study that pattern carefully, you would find a unit cell which is of this size. Again, I begin by showing a unit cell where the origin is at the bottom of the stem. We could also find alternate unit cells, as you know, with different origins. If we take that unit cell and apply the same descriptors as we've now become familiar with, in terms of two directions, x and y, and two distances for those edges, a and b, the tile might look like this. So what symmetry element can you find inside that tile? But I hope that many of you will see straight away that there is a vertical mirror line. Are there any other aspects of this tile that we should look at? The most obvious one is how many roses are inside the tile. The unit cell contains two roses. So what must be the asymmetric unit? Let's recall what definition of the asymmetric unit is. It is the smallest component of the pattern upon which we use symmetry operators to generate the whole pattern. Therefore, in this case the asymmetric unit must contain a single rose. It doesn't matter whether you choose the rose on the left or the right of the tile. If we apply the vertical mirror to that asymmetric unit, we can create the entire tile. Evidently, the asymmetric unit contains one rose. The full tile contains two roses. Here we only have a vertical mirror, so the point symmetry descriptor must be m. Let's look at another pattern. Take a moment to examine this, and try to find the repeating tile. Again, if you're not sure, pause the video for a few moments, look at your printed handout and see if you can find the tile. In this case, I've drawn the tile with the origin between the roses. We can describe that tile formally. You should be getting very familiar with this now. x and y for the directions, a and b for the edges of the tile. Now we come to the part of the exercise which is a little more complicated. What are the symmetry elements inside this pattern? I'll tell you straight away, there are three. The first and perhaps most obvious would be the vertical mirror. In addition, there is a horizontal mirror, for two mirror lines inside this pattern. In addition, and this is perhaps a part that you have to think about for a moment, the point at which the vertical and the horizontal mirrors intersect generates a two-fold rotation point. Two-fold rotation is where you rotate the object through 180 degrees and the pattern remains unchanged. So the intersection of the mirrors provides you with the two-fold rotation point and this is indicated by a lens shape. This is the formal crystallographic representation of two-fold rotation. So we have mirror planes in x and in y, and we have two-fold rotation axes. We need to count the number of flowers in the tile, and you can see immediately there are four flowers in this tile. So you already get a sense that as we increase the number of symmetry operators which are working, the number of objects inside the tiles also increase. What would be the point symmetry of this pattern? Well, we've identified what the symmetry operators are. We know there's a vertical and a horizontal mirror, a two-fold rotation. So following the terminology which we're now familiar with, the point symmetry must be 2mm. Let's look at this in a little more detail, working with the same pattern. The asymmetric unit evidently contains one rose. I think we would all agree about that. We know that we have three symmetry operators working, the mirrors and the rotation point. So how do we create the complete pattern? Is there a particular way that we should do it? Well, the first thing that we could do is think about applying the vertical mirror. So if we look at the left-hand example, we apply the vertical mirror and we get the second rose. We could then apply a horizontal mirror to get the combination of four roses which create the unit cell. Do we have to apply the symmetry operators always in that sequence? The answer is no, it's not necessary. We could begin by applying two-fold rotation, then applying the horizontal mirror and we arrive at the same tile with four roses. Or we could begin with the horizontal mirror, the vertical mirror, and arrive at the same result. This characteristic of symmetry operators, to be applied in different orders but arriving at the same final tile size, the same final number of objects inside the tile, is known as being a commutative set. Let's now look at one final example. Here we have a more dispersed pattern of the roses. What would be the repeating tile? If you need some time to think, just pause the video and do that. Here is one way that you can draw that repeating tile. We can define that tile again in terms of directions x, y, distances a and b. What are the symmetry elements inside this tile? This is a slightly tricky example. And I'll tell you the answer and then you can look for where the element is. It is actually a glide plane. Recall that a glide is a compound operator. It involves a mirror and a translation. It's here, it's a horizontal glide. Recall that the formal representation of plane glide is a dashed line as shown. And if you take one of those roses and you reflect across the glide, translate through half the length of the unit cell, and this is also a critical definition, is through half the length of the unit cell, then you'll get to the next rose. Then if you apply that operation repeatedly, then you will generate the entire pattern. So we have glide planes in x. How many flowers are inside the tile? I think this is now a trivial question for you. There are two. How many flowers in the asymmetric unit? It must be one. What must be the point symmetry of this particular pattern? Well, the answer is m. We don't use a g. When we talk about point symmetry we don't use a g because, remember, g's don't belong to point symmetry nomenclature. They belong to plane symmetry. But the glide contains the mirror. So the point symmetry of this pattern is m. As we've worked through these examples, we've used more and more symmetry operations. We talk about high symmetry and low symmetry. High symmetry implies we are using many symmetry operators. Low symmetry implies we are using a few symmetry operators. In this case the final pattern that we looked at, with the roses containing four flowers, would be said to have high symmetry, the first pattern said to have low symmetry. So to conclude this section, we've learnt that symmetry transformations can be applied in any order. And in this regard they are said to be commutative. Also, the tilings or tessellations have a symmetry hierarchy. The greater the number of operations in the tile, then the higher the symmetry.