[MUSIC] Welcome back. In this lecture, I want to show you how to do a classical construction of a golden rectangle. What's a golden rectangle? A golden rectangle is a rectangle that has sides in the ratio of the golden ratio. So the length, the longest side of the rectangle, divided by the width, the shortest side of the rectangle Is equal to capital Pi one plus the square root of five over two. So classical constructions of these planar geometrical figures goes back to the ancient Greeks. The idea of a classical construction is to construct this figure using just a straight edge which is like a ruler without the markings and a compass. A compass is that device that has a point that you can draw a circle off of this point. Okay, fix the radius and you can draw an arch or a circle. So a lot of these common figures can be drawn using classical construction. And the idea of classical construction was also very popular during the renaissance and after. Not so popular these days because we have computers and we can just program the computer to draw anything we want. But nevertheless, it's kind of a fun thing to do particularly for students secondary school students, even university students. Okay, so how do we construct a golden rectangle? We start by constructing a square. So by classical construction using a straight edge of a compass, you can construct a square. So I won't go into that here because our goal is really the golden rectangle. So you start by constructing a square. We just called the size of unit left so we put a 1 on the sides. Then the next step is to draw this red line here. This is from the midpoint. So first, you have to find the midpoint of this bottom side of the square. There's a classical construction using the compass that allows you to find the midpoint of any line segment. I won't talk about that here. So we assume that you can find he midpoint. And then you can use your straight edge to draw the line to the opposite corner here, okay. Now we have Pythagorean's theorem, so we have a right triangle with a side of 1 and 1/2. So one squared plus a half squared is one plus a quarter is five quarters, and we take the square root to get the hypotenuse, so we have a square root of five over two. So you see the reason of drawing this line segment is to introduce the square root of five. The golden ratio is the square root of five plus one over two, okay? So what do we do next? Well, we can put our compass point here. At the midpoint, at the bottom side of the square. We can make this line segment the radius, okay? Make that the radius and then draw the arc. So we can use the compass to draw this circle, arc of the circle. Okay, and then we can take our square side and use our straight edge to extend the side, all the way to the arc, okay all the way to the arc. And now what do we get? Well, this means here the square root of five over two, and this red line at the bottom here is also a square root of five over two because it's the radius, right? It's the radius. And then this total length, one-half plus root five over two is one plus root five over two, which is the golden ratio, right? So now we've drawn a bottom line that has the length of the golden ratio. All we have to do now is complete the square. We need to draw a perpendicular line to this bottom line, either perpendicular to this line, which you can do in classical construction. Or parallel to this line what you can also do in classical construction, so we draw the line here up. And then we draw the top line over using a straight edge. And we have a rectangle. And this rectangle then has a length one plus root five over two and a width one, this is a golden rectangle. So this is the classical construction method of the golden rectangle. Kind of fun, okay? Just for fun. Okay, in the next lecture then we'll all use this golden rectangle to show, I will use this golden rectangle to show you some very interesting figure, okay? I'll see you next time.