[MUSIC] Welcome back. Next to me is a golden rectangle with length capital Phi, 1 plus square root of 5 over 2 and with one. I want to show you in this lecture an interesting thing we can do with the golden rectangle, okay? Remember that the golden rectangle was constructed by starting with a square and adding a rectangle to the side of the square. So, let's put the square back. Okay, so here's our square. We have the unit length. So we have a square here on the left, right? The left side of this rectangle. And then, we have another rectangle here. So what is the dimension of this rectangle? If we call it the longest side, the length, then this is 1, right? And then the shorter side is the Golden Ratio conjugate. Because remember the Golden Ratio conjugate is the fractional part of the Golden Ratio. So, one plus little phi equals big phi. Okay? So, here we have a rectangle which is one by little phi. But remember the relationship. So the main total of the width is one over little phi, but the reciprocal of the golden ratio conjugate is the golden ratio. So the length over width for this rectangle on the side here, is still the golden ratio. That means this is a golden rectangle. So by cutting out a square from a golden rectangle, we end up with another rectangle which is also a golden rectangle. But reduced in dimensions, okay? So we can do that again. We can continue. So here we go. Here, the numbers in the middle I show you the reduction in dimensions. So this square is one by one. This square is little phi by little phi. And then this bottom piece here is another golden rectangle. Right? Its dimension, length over width, is going to be little phi. Now the length is horizontal. And the width is reduced by another factor of little phi. So little phi over little phi squared. And again, we get the golden ratio. So we can continue this process. So let's do it in a systematic way. So we go left, top, right, bottom. So here we have squares on the left, then the top. And now we're going to put it on the right and then we're going to put it on the bottom. So here it's on the right, and then we put it on the bottom. And then we repeat. So we always go left, top, right, bottom, left, top, right, bottom and so on and now we get this very interesting figure of squares. So I call this the spiraling squares, okay? And we're going to use this nice figure to discuss a logarithmic spiral. And in particular we're going to talk about the golden spiral which will fit very nicely Into this figure of spiraling squares. Okay? I'll see you next time.