[MUSIC] In the 16th and 17th century there was a great influx of Chinese migrants to the Malay Peninsula. Over time the cultures of the Malays and the Chinese combined and interacted and created something very new. It was known as Peranakan Culture. The Peranakan Culture had it's own style, it had it's own cuisine, it had it's own fashion and it had it's own architecture. And one manifestation of that architecture were the shop houses, which were built in Malaya and subsequently Singapore. And these are examples of those shop houses. An important characteristic of these shop houses were the beautiful tiles which were used to decorate their facades. And as you can see here, those facades often contained flowers, but the range of tiles that were actually available was tremendous. Unfortunately, many of these tiles have been lost. And we are so fortunate in Singapore to have individuals like Victor Lim, who have devoted their life to collecting these tiles, restoring them, and then using them in conservation of old buildings. This is Victor's showroom, where there are so many tiles, it is simply bewildering. When we look at these various tiles, I can see patterns with flowers and plants. Other patterns with animals. Purely geometrical patterns. The vibrancy of the colors is also absolutely remarkable. Now these tiles which were once used on walls or floors are pieces of art. How did you become interested in preserving these tiles, and how long have you been doing this? >> Oh it's back to 1979 when I was going to NS. Before I went to NS. >> This is National Service. >> National Service, yes, correct, okay. >> When I saw a lot of buildings they demolished. I look at a house, it's so beautiful. And it's a waste the way they damaged it. So I tried to collect it. >> Over the years, how many tiles do you think you may have saved? >> Over years of proximity about 15,000 tiles I have right now. >> 15,000, wow. And when you collect a tile, how do you discover the history of these tiles? Cuz I imagine it's not so well documented. >> The tile itself is pretty in the old days. Then came to 15 years ago, then people keep on asking me, you collect so many tiles. You know where they're from? How they from? I don't know. We don't know. Start to collect some books. Started to get some books. Okay. When we read through the books, oh, very interesting. So the books tell us where the tiles was manufactured, the year, even the copyright. >> We noticed when we were looking around the shop houses that roses often feature in the tiling. Can you tell us why that was? Why were the, the roses so popular? >> It all depends on the buildings, okay? When this building was built. At that time in 1920s Japan had not produced titles yet, only England. Okay, so only England tiles was in Singapore. And this is one of the good example of this tiles. And you can found this design in Blair Road as well as Ann Siang Hill. A lot of this. And this flower is, okay, this flower, this motif is just standalone flowers. But when you join all together, I show you the sample, okay, here's joined together, it can come out with a new motif in the center. It's so clean and it's so nice. These tiles, okay, approximately about 100 years too. And this is all Victorian design. >> And this is a very apt tile to finish talking to you because when we go back to the studio we're going to be teaching the students how to look at tiling in a mathematical way and also how to look at symmetry in a very quantitive way. And for those exercises, we're also going to use roses. So thank you very much. >> Thank you very much. Thank you. >> So now you've seen the beautiful tiling and the beautiful architecture associated with the shop houses in Singapore. The Peranakan tiling is really something to behold. I'm sure you will agree. Now we want to consider that tiling more formally. Again, how do we describe these tile patterns mathematically? So in this section, what we would like you to learn and understand is how to describe and explain the concept of tiling in a general sense. We would also like you to be able to identify the unit cell in these repeating tiles. And remember, the unit cell need not be equivalent to the asymmetric unit. The asymmetric unit is the most simple, basic aspect of the tiling pattern upon which the symmetry operators work to create the whole pattern. We need to use the asymmetric unit, the same asymmetric unit, it is possible through the application of different symmetry operators to create different overall patterns. You saw already that flowers were a favorite amongst the Peranakan tiles. And roses were particularly popular. Shown in this slide is an arrangement of roses. If we want to find what is the unit cell or what is the basic repeating unit in that pattern, we have to focus on some feature in the roses, and we use that feature as the origin of the unit cell. If we choose the bottom of the stem as the origin of the unit cell, then we can draw a unit cell that looks like this. The origin of the unit cell, at least in terms of these patterns, is not specific. Multiple origins are possible providing the size and shape of the unit cell is unchanged. So a fairly basic terminology is needed here. We describe the two edges of the unit cell with respect to x and y coordinates. It is always x and y, and there will be an angle between them. The angle might be 90 degrees or in this case it's non 90 degrees. The length of those edges, with respect to x and y are always a with respect to x and b with respect to y. That is a standard terminology which we use throughout this course and which is used in crystallography. If you look at that tile and search for a mirror or rotation point, you can't find one. So for this particular pattern, the asymmetric unit is also the repeating tile. So just to recap, when we look at this pattern, the tile has a fixed size, but has no fixed origins. Look in some more detail at this same pattern. And I've illustrated three different unit cells. One unit cell is where we have taken the origin as the bottom of the stem. The second unit cell is where we've taken the origin as on the rose. And the third unit cell is where we've taken the origin between all the roses. If we look at the first example, the tile labelled 1, the question is how many roses are inside that unit cell? Now you might think there are four, because the unit cell touches four roses. But that's not correct. In fact there is only one complete rose. In a similar way, for unit cell two, there is one complete rose inside the unit cell. Of course for the third example it is very obvious. So that's something which is perhaps a little tricky when you start out looking at tiles. Even though you can move the origin all over the place, the total number of objects inside the unit cell must always be the same. If we look at the whole pattern, how many mirror lines or glide lines can be found in this pattern? Again, it may take a little time to look carefully, but if you do that you will find that there are no mirrors and there are no glides. Therefore, the pattern is asymmetric, and the tiles are asymmetric in their own right as we mentioned before. So what is the point symmetry of the pattern? It's designated as 1, and we say the point symmetry of this pattern is 1. What about the basic properties of tiles? The first thing that we have to say is that tiles must pack together in a way that is congruent. In other words, there must be no spaces between the tiles when we complete the entire pattern. So congruency is a key way in which we define tiling. Evidently, the tile is a polygon and in a general sense, any polygon could be used. It might be four sided, six sided, eight sided, and so on. But for the purposes of a formal description in crystallography, and a formal use of symmetry, we will only be talking about polygons with four sides. And a tessellation is formally described as a filling of 2-dimensional, or plane space with a collection of figures with no overlap and no gaps. So let's look at one final example. Now this pattern looks quite simple and I guess for many of you it looks the same as the previous pattern but in fact it is not. You need to study this pattern very carefully. And in particular, look at the orientation of the roses. You'll notice the roses on one side have large leaves and on the other side have small leaves. Again if we take the bottom of the stem as a starting point outlined is the size of that unit cell. I am sure at this point some of you are scratching your heads and saying why is that the correct tile? So you might like to perhaps pause the video and study it very carefully but that is in fact the minimum size of the repeating unit. And it is because there are two orientations of the roses. Formally we can describe this tile using our vectors x and y. And our cell edges a and b. We need to count the number of roses inside that unit cell and this is going to be the number of complete roses. And if you do that you will find that there are four. Recall that in the first pattern there was only one rose. What would be the point symmetry of this pattern? Again it's related to the presence or the absence of rotation points and mirror planes. And in this case, you will not be able to find either of those symmetry operations. So again, the point symmetry is one. So now let's summarize what we've learned. We now know that repeating patterns in two dimensions can be replicated by placing tiles together. Those repeating patterns are formed when the tiles are said to be congruent. In other words they repeat by placing the tiles without overlap, or without gaps. When the point symmetry of the tile is one, in other words, the tile contains no mirrors or rotation points, then that tile is also the asymmetric unit. In the next section, we're going to learn about tiling patterns where the asymmetric unit and the tile are not equivalent. The asymmetric unit will be smaller than the tile because the tile, itself, contains mirrors or rotation points. Before we do that, let's return to the shop houses, and look at some more complex tiling patterns that can't be described by what we've covered so far.