[MUSIC] Welcome back. I'm standing in front of the figure of the spiraling squares where I've drawn the golden spiral. You see, this golden spiral seems to spiral into a point. That point is the accumulation point of all of the spiraling squares. How can we determine the location of that point? Well, you can look at this figure, and this figure has drawn two diagonals, a diagonal of the large golden rectangle and an opposite diagonal of the second largest golden rectangle. And they seem to intersect directly at the accumulation point of all these spiral and squares, which will then be the center point of the golden spiral. How can you prove that? Well, you look at this middle rectangle here. This is also a golden rectangle, and in fact, it's an exact copy of the outer golden rectangle, just reduced in size. So if you can show that this diagonal crosses the corners of this rectangle, these two corners, and this blue diagonal line crosses the two corners of this slightly smaller rectangle here, then those diagonal lines have to go all the way down into the accumulation point of of our spiraling squares. And their intersection point, then, is the center of our golden spiral. I'll leave that to you as an exercise. If you'll also look at this figure, you see that the center is in the right-hand side of the figure and in the bottom half of the figure. Because this is a rectangle, and the center of the spiral is not in the middle of the rectangle, then there must be other ways of drawing this figure so that the center can be up here, say, or it can be over here, or it can be down here, right? So let's look at the four possible ways we can draw this figure. Here they are. The two figures on the left are drawn such that the first square is on the right side of the figure. And the difference here is that the second square here is drawn on the bottom, while on the lower figure, the second square is drawn on the top, and we still spiral the squares. So the spiral here goes clockwise, and the spiral on the lower curve goes counter clockwise. So those are two possibilities. The other possibility is that the first square can be on the left of the figure. Here, they are on the left side, and then we can have the second square on the bottom or the top, and then as we spiral, the top one here will be counter clockwise, the bottom one here will be clockwise. So we have four possible centers, four possible starting points for these golden spirals, four possible accumulations points of the spiralling squares. So, we can mark them. Here's how we mark them. We draw all these diagonals, so the intersection between the red diagonal and the blue diagonal will mark the center of the starting point of the golden spirals. So we have one, two, three, four of them. Because of the symmetry of the rectangle, we can connect these four and get another rectangle. So if we connect those four to get another rectangle, here we go. It's rather amazing, I think. We actually get another golden rectangle. So the golden rectangle that we get comes from the corners being at the center of the four possible spirals. And that golden rectangle is just, as I mentioned, that is reduced by square root of five. So, the length of the rectangle is square root of five, shorter than the length of the initial golden rectangle that we start with. I think this is really cool, and I'll leave it to you to show that this is, in fact, true. I'll see you next time.
