[MUSIC] Some of the most beautiful examples of plane symmetry can be found in Islamic architecture. When you look at this wall at a mosque in Singapore, you cannot be anything but amazed at the beauty and the depth of the symmetry involved. The subtlety in these sorts of designs are amazing. Clearly, when you look at this wall, there are triangles within triangles. There is an overlaying hexagonal symmetry shown by the gold tiles. But we need to explore the origins of why symmetry is so important in Islam. And to do this, we need to speak to our experts at the School of Art, Design, & Media within Nanyang Technologic University. >> Well, Islam, if you notice, when they started off, one of the important things they looked at is that we will not look idle. And we will not be worshiping any form. So that was the first step to saying what we should not do, right? But then the Quran itself says, I'm just paraphrasing, that I have not placed anything out of place. Everything is in order. I created for you perfection. So there's the leading thing, the statement in the Quran, where it says the whole universe has been made to be beautiful. So as the Islam started spreading westward towards [INAUDIBLE] and eastward towards India. They found out that there's already artisans and craftsmen were working and and together with that Islamic poetry started evolving over a few hundred years, and geometry played a big role. They had to build mosques to start with, which had a very specific direction to face to. They wanted to put facades to those mosques. They needed tiles, and each area the craftsman came out with interesting patterns as a form of expression to replace anything that could look at as somebody's worshiping an idol or anything like that. >> When anybody looks at an Islamic design. You instantly recognize what it is. We're all familiar with it. If you looked at patents in Spain verses patents in the Middle East, verses patents in East Asia, could you tell that they're different? >> You can tell. There's influence of the local culture. For example, as you're going to India the intricacies of patterns become much more interspersed in terms of art. The Indian art. The Rahjistani, the influence. And then you can see the gold inlays being used. Silk becoming an important aspect of the whole thing. So you can see that the fundamental patterns and the repetitions are the same, but then the intricacies, the kind of nuance Is different. It also, again, if you go to Indonesia and look at their music versus their architecture versus their Islamic art, there seems to be a kind of correlation to the whole thing. So there's this beautiful way of combining these arts together that made art, architecture, and music, and performance as well. As Islam moved towards the east, more performance came into being. People singing and dancing. >> One of the things of course that the Islamic countries are very famous for was their mathematics. And of course mathematics and geometry are natural bed fellows, they go together. What came first do you think? What drove, what was it, the understanding of math that drove the geometric expression In the mosques, or the did the other aspect, the religious part drive the development of mathematics in Islam? >> Everything starts with a circle, a very beautiful circle, and intersecting sections of circles started giving geometry. I have this feeling that geometry came frist, the attachment to beauty as it evolved and then they started seeing relationships happen and they were also influenced, we have to say, that by the Romans and the Greek, so they started seeing arches. They started putting things together. Now, mathematics traditionally, as I understand, came up from their interest in astronomy and distances and I'm sure that someone along the way who is an engineer, and astronomer, and an artist must've seen the connection together. >> You've now had a very brief introduction to Islamic design and its impact upon architecture and art. But, of course, there's been a great deal written on this topic, and in front of me here, I have a number of books. You can find any number of books in libraries. This slim volume Islamic Design A Genius For Geometry by Daud Sutton, is actually one of my favorite books. When you look inside this particular volume what you find is a very concise description of the way the intersection of circles provides a basis for creating all the wonderful geometric designs that we find in Islam. I highly recommend this book if you just want an introduction to this topic. There are many books on Islamic designs. This one, for example, really gives very beautiful representations of the patterns which are possible and the colors which are used. And then also goes on to the very intricate designs that can be incorporated inside the Islamic tiles. So now, let's begin to look at nets and net works. Nets are a very useful way to begin to understand extended playing symmetry. Networks provide a means of describing symmetry in a very simple way. To do this, you will begin by looking at the most common regular nets. The regular nets we'll look at are square nets, triangular nets, and hexagonal nets. In addition, we'll start to look at semi-regular nets. And in particular, a net known as a Kagome net. The second thing we want to do is to make sure you can identify net symmetry and also define the connectedness. How are the nets put together, and derive their Shlafli symbol, which is a formal representation of connectivity on a network. If we look at a square net and quite obviously this is a square net, it's constructed by putting square together. One way to represent this square net is to have vertices which in this case we are representing by circles. I'm also foreshadowing how we will use networks later in the course. Because these nets and their vertices can also represent crystal structures and the vertices may represent atoms. The distances between the vertices can be edges if we're simply talking about geometric representation. Or if we're talking about crystal structure, it is a bond length between two atoms. The way that we formally represent these nets is to look at what other end-ons or the polygons surround each vertex. In this particular case, every vertex is surrounded by four squares, or a. So we would say that each vertex can be described as 4.4.4.4. This is a symbol. A shortcut for writing that is as 4 to the 4th. In other words, every vertex is surrounded by four squares. When we look at a net like this, we can also apply the principles we learnt before when looking at the roses. We can define a unit cell. Without any other information, you can put the unit cell anywhere on top of that net. So for example, you can start the unit cell on a vertex, or you could place it between the vertices, right in the middle. Both are equally correct given the information available to you. The other thing that you can do is count the number of vertices inside the unit cell. Again an extension of the ideas when we were looking at the roses. For the square net every unit cell contains one vertex. What would be the symmetry elements inside this network, clearly there is four fold rotation, and there must be a horizontal and a vertical mirror. And as we've already seen, the full representative of the net is defined by the number of squares around the vertex, so we have arrived at the sharply symbol as 4,4. Let's move on to a triangular net. Again we can go through the same exercise we can find the unit cell the unit cell could be located directly on the vertex, or it could be located between the vertices either representation is fine there is one vertex per unit cell. The symmetry elements are six fold rotation and therefor there is also six mirrors inside this network. In this case, every vertex is surrounded by six triangles or a three. So when we specify the net in terms of its shaft symbol we can write it in the long form, 3.3.3.3.3.3, or the shortcut 3 to the 6th. We can also define a hexagonal network. It looks like honeycomb. In this case, we define the unit cell, and you can see that the unit cell contains in this case two vertices. For this hexagonal network, there is sixfold rotation and six mirrors. To specify the net again we count up the endgons surrounding each vertex. We find there are three hexagons around each vertex. Therefore it's a 6.6.6, Or 6 to the third. SchlĂ¤fli symbol for this network. These three examples are regular nets. Because they contain only one type of. We can also define semi regular networks. Perhaps the most famous of these is the kagome net. You can see that the network contains hexagons and triangles. So 6-gons and 3-gons. The name kagome is actually Japanese, and you can see a kagome net which is used for fishing in Japan. The first part of the name comes from the fact that it's made out of bamboo. The second part says that it's a woven pattern. Hence, kagome network. Again we can define a unit cell. Now, in this case you have to be very careful how you count the number of vertices inside the unit cell. Because what you'll notice is that we have a vertex right in the middle of the unit cell. So that's very easy to count. We have at least one. And then we have additional vertices in the edge of the unit itself. And we shouldn't count these twice. So in this case when we begin to count the vertices we count them once if they appear in an edge, and of course we don't count them twice. So we'd find there are one, two, three. Complete vertices within both of those unit cells. So three vertices per unit cell. So we've gone from the square and triangular net where you had one vertices per unit cell to the hexagonal net, two vertices per unit till, this example where we have three vertices per unit cell. We can, of course, locate the symmetry elements which are based on six-fold rotation. You will also be able to show that around every vertex, you have two triangles and you also have two hexagons. To write this properly, you need to record the sequence of those end-gones. So it goes triangle, hexagon triangle, hexagon. So write this down it's a 3.6.3.6 net. Not 3.3.6.6. That would be incorrect because it would give you the wrong sequence of the end gones. So in this very brief introduction to nets. What we can see is that you can always describe the networks as combinations of vertices or atoms at the corners of polygons. And when you describe these networks in a formal sense, you will count the number of polygons and type of polygon around each vertex. Regular networks contain only one type of n-gon, in this cases squares or triangles. Semi-regular nets contain two or more types of n-gons. Later in this course, we will look at more complex examples of networks because these provide a very simplified way of looking at patenting, in for example Islamic tiling and also a useful means of simplifying the description of crystal structures.