Okay, now it's time to discuss mathematical sciences: mathematics and technology involved in this structure. Please remember it was 750 AD when this was constructed. So that's already 1,300 years ago, approximately speaking. We stopped at the previous lecture at this picture, and I mentioned what these bumps could contain. Actually they contain some secret. And then I mentioned this cracked stone part also, the centerpiece, which is very very heavy. And what's the secret of holding the whole structure from falling in? And the secret is here. This is the model which is actually a replica of the real thing in Gyeongju. And there, they actually show this particular model. And those bumps were actually the nailheads of these pegs. That's interesting. And so these pegs actually, which are covered under the mound, balance the weight. And I'm going to talk about that. One of the older structures, this is a 2000-year-old structure. The great world heritage, the Pantheon of Rome, Italy. That has a fantastic dome ceiling. And this is a much larger ceiling. But then, it does not have pegs. It does not have the centerpiece. So this hole is just a hole. A hole is there. And there are controversies, but it's generally understood that the centerpiece was not blocked because they couldn't actually cover it. And so you can see the comparison between the two. One does not have pegs but is smooth outside, and the smooth outside has a hole at the center because they couldn't cover it. But the pegs somehow hold the whole thing in, and this grotto is covered. It's the same dome structure, spherical dome structure. So how do pegs balance the weight? The dome structure from the side approximately looks like this. After they place one particular stone, then they put a peg over here. This is really large, very very heavy, a large peg. Some of them weigh as much as 2 metric tons. Quite heavy. And of course, this might retract. So to prevent a peg from being dislocated, it has to have a nailhead so that it holds in. And that was the bump inside the grotto. So you can actually see the nailhead inside here which is tightly attached to the ceiling. So that was one of the brilliant features, brilliant features of this dome construction. And so the centerpiece is very heavy, but it's balanced by this particular peg. Of course these pegs add weight to the whole structure, so the forces that come out to the dome structure eventually becomes flat. But still you need to know how to put what kind of pegs precisely. On the other hand, there was some interesting newspaper article which said that Kim Daesung, the original builder in 751, was the the first faulty architect. But this piece, which was a firestone piece, two metric ton heavy, was not very easy to replace. And so they decided to use it correctly, as you saw. And it had been there without falling in with no problems for more than 1,270 years. And so I don't think that was unfaithful construction. It's just brilliant construction using pegs. So you see those nailheads inside. Now you understand what those are. And this was broken, but it doesn't matter because pegs hold them from falling in. Then you see the halo disk behind. This is a usual halo disk which shows the holiness of this Buddha, showing the brightness, the bright light, shining light from his teaching in his ideas. Then this is the front view. And from the view, the halo disc is round, or actually shows something like an ellipse. Interesting. But then, but then it looks round to us. But the view is an interesting thing because if you look at it from far away, you have a podium here. Here's the podium of Seokguram. And then the statue is here. The Buddha statue is here. And the actual measurement shows that this is 1.6 meters. So that's more than... It's over 5 feet. And then this statue is as tall as 3.5 meters. So it's already over 5 meters from down to here. It's a tall structure. So when a visitor comes, his eyes are usually about on the level of this podium. And as I said, from the visitor to the podium is already 10 meters. It's pretty far. And so you have to look up. And the halo disc is on the wall, almost vertical here. The halo disc looks round. But if you look from the bottom, then the round disk does not look round. It becomes oblong with the side stretching a bit longer than the vertical distance. To balance the view, the halo disk had to be built differently. So this is what you see from the visitors' view. The halo disk doesn't show very well in this camera's eye, but it actually shows the whole picture. And it looks very round. From almost 8 to 10 meters approximately, that's 25 to 33 feet away, the visitors look at the statue sitting on the podium, and the halo disk looks very circular, but it's actually an ellipse. The actual measurement shows that the distance, the diameter from top to bottom (mathematically speaking it's the major axis) is about 228 centimeters. And the minor axis, from one side to the other, the halo disk shows a diameter of 224 centimeters. And so actually it was built intentionally as an ellipse so that it shows round. Then you see two hands here. This particular position has a meaning in Buddhism. The pose is like this: the right hand points downward, and the left hand is horizontal. But there's a distance difference between a right hand position and the left hand position because the statue is large. And because of projective perspective, a far-away thing looks smaller than a close-by matter. So close-by objects look bigger. So if you build the two hands the same size, then they wouldn't look balanced. And so indeed, the actual measurement shows that the left hand was built larger than the right hand. But now from the viewers' perspective, they look perfectly balanced. And also the statue shows a larger head. The actual measurement shows that to the neck, this distance, I'm sorry, and then this distance to that, and those are equal. And so it's very unnatural. If you measure an actual person, it's not so. Actually it should be longer down there slightly. But it's balanced in this way. And so the close-by view might not be very natural, but the far-away view looks completely natural. Okay, all these are from projective geometry. That's impressive that they actually understood projective geometry when they constructed this. Well, maybe not the theory, but they knew how to do it. And if you look at the history of projective geometry of Europe, as I said, the primitive form was already... only in... The theory was developed only at the end of the 18th century. And then from the pictures as early as... It goes as early as 1600 something, or maybe if you insist, it was done in 1500 AD. So probably you might feel that I say that the Silla people knew projective geometry way ahead of Europeans. I'm not saying that, but... But for the constructional purposes from an engineering point of view, Silla people knew this technology. And also this is not the world first. I want to point out that Karnak Temple of Egypt which is almost at least 6,000 years old, or maybe even 7,000 years old. Karnak Temple of Egypt has statues which are taller and larger than this particular statue. And those show this type of construction, a larger head and smaller feet, so that from the ground level, they look more natural in this way. So somehow ancient people knew this technology, but Silla people knew this thing clearly. We see that from 751. And this cannot be done by just trial and error. As I said, the structure is made of firestone. Firestone is one of the harder stones. It's very difficult to carve. So if you apply the trial and error method, then you'll take forever. Instead, this whole grotto was constructed very shortly, very quickly And so I think there is an evidence that people knew this precisely. Then for the future studies, one can actually ask many questions because it's a grotto, a very unique grotto. And so many people actually asked about this grotto. What about the sound effect, acoustics? What about the lighting? What about the moisture control? Why does it have to be a grotto? And why does it have to have such a location? And many more questions can be asked. I challenge students to ask more questions and find out about the hidden treasures of Silla culture, technology and science.