[MUSIC] Welcome back. We've talked about the golden spiral, now I want to talk about the Fibonacci spiral. To understand how to get the Fibonacci spiral, we can go back to an identity that we derived, that the sum of the first n squares of the Fibonacci numbers, sum from i = 1 to n of Fi squared = the nth Fibonacci number times the n+1 Fibonacci number. So we can consider the geometric interpretation of this equation, of this identity. The left hand side is a sum of squares. We can interpret squares as the area of a square. The square of a number is the area of a square with the side given by the number. So we can interpret the left-hand side as n squares. Each square has a length given by the Fibonacci number, and an area given by the Fibonacci number squared. We can interpret the right hand side of this equation as an area of a rectangle. The rectangle has side length Fn and side length Fn+1. So let's see how we can interpret this equation as an equation about areas to construct the Fibonacci spiral. So let's start with the first two Fibonacci numbers F1 equals 1 and F2 equals 1. We put two squares next to each other. Each have a side length of 1 and an area of 1. And that identity tells us that 1 squared + 1 squared = 1 x 2. So this is the first Fibonacci number 1, the second Fibonacci number 1 = the second Fibonacci number times the third Fibonacci number. So this 1 x 2 is the area of this rectangle. This rectangle has a width 1 and a length 2. To this two Fibonacci squares, we can now add another square with a length of the Fibonacci number F sub 3 which is 2, F sub 3 equals 2. So, here we go, we add one more square. So we can put it on the bottom here. And then that identity says the area of this one, which is 1 squared + the area of the square of the second Fibonacci number is 1 squared, + the area of the square with the side of the third Fibonacci number, which is 2 squared = the area of the rectangle, which is 2 x 3 rectangle. 2 width, 3 length. We can continue, we can add the fourth Fibonacci number. And here we go, the fourth Fibonacci number is 3. And we do this in such a way, that we do this so it will look like a spiral. So 1, 1, and then bottom, and then right. And then the next one will be top, and the next one will be left, and the next one will be bottom again. So we can add it like a spiral. This one is simply 1 squared + 1 squared + 2 squared + 3 squared adds up to 15 and that's equal to 3 x 5. This is a 3 x 5 rectangle, and we keep adding. So here, we've added up to Fibonacci number of 8. One, two, three, four five, six, the first six Fibonacci numbers. We went 1, 1, 2, 3, 5, 8. So we add as a spiral. We can draw the spiral. Here, the easiest way to draw the spiral is just to draw quarter archs of circles connecting the inside of each of these squares. Connecting the two opposite sides of the square. So we can draw these spiral, and here it is. This is called the Fibonacci spiral. I think it's a rather pretty spiral as spirals go. I think it's a rather pretty one. And then I colored this figure to make it the icon of this course. So here is the icon of the course. I think this is one of the prettiest sort of figures that you can see in mathematics. Illustrating both the Fibonacci numbers, illustrating an equation, which is the sum of the squares of the Fibonacci number, is equal to the area of the rectangle. And also illustrating a spiral called the Fibonacci spiral. In fact, the Fibonacci spiral is very closely related to the golden spiral, I can show you both of them. The upper figure is the Fibonacci spiral, the lower figure is the golden spiral. You can see visually that these spirals are very similar. You can view the Fibonacci spiral as starting from the middle and adding squares and spiraling out. You can view the golden spiral as starting on the outside and breaking up the golden rectangle into squares, and spiraling in. The main difference between these two spirals is the golden spiral spirals into a point, while the Fibonacci spiral spirals out but not from the origin of the spiral. But as you go out because the ratio of the Fibonacci numbers approaches the golden ratio, the Fibonacci spiral, as you spiral out, approaches the golden spiral. So these two spirals become the same spiral as you move out from the center, and that's why they look very similar. So now we've learned about two spirals, we've learned about the golden spiral and the Fibonacci spiral, two spirals considered among the most beautiful spirals in mathematics. I'll see you next time.