[MUSIC] Welcome back. We're finally in a position now to construct a model for our sunflower head. So, let's see how this model will work. We're going to assume that as the sunflower head grows the new florets are going to be created close to the center of the sunflower head. And the florets once they're created will move radially out. With constant velocity as the sunflower head grows, radially out from the center. But we need to try and fill up the sunflower head, so before the floret moves out it gets rotated by a constant angle. So, the floret moves out, but then the next one will be rotated by some angle and then rotated by some angle before it moves out. And we can denote the angle by 2 pi alpha. So 2 pi is the angle of the whole circle, so alpha will be the fraction of the circle that the florets are being rotated. So, let's see how this will work. What happens if alpha is 1/7? So, think about the model. So, we're moving out radially, we're rotating 1/7 of the circle for each new floor wrap that gets created. So if you like, you can pause the video and think what the model is going to look like. So, here we go, we can run the model. And you start seeing these radial lines from the center, right? One, two, three, four, five, six, seven. There are seven lines coming out of the center. So of the first floret was here, and it moves out radially. The second floret is up here. So, we rotated 1/7 a way around the circle, and it moves out radially. The third floret is here. So after one, two, three, four, five, six, seven rotations, we're back to the line that we started with. This doesn't look like a sunflower, the florets are not filling up the sunflower head in any way. What's the problem? The problem is that alpha is a rational number. If you use a rational number for alpha you will always get back to where you started from. So, you'll always end up with lines. What matters is the denominator here. So, seven means you'll end up with seven lines. So, the answers is not to use a rational number. So, you can use that irrational number, so let me give you the first example of an irrational number. We can use the fractional part of pi, so pi minus 3, that's the 0.14, whatever, okay. And we'll use that as our rotation angle. So, we will not now ever get back to the initial line that we started with. So, what should this model of the sunflower head look like? If you want to try and think about it yourself you can pause here. If you want to hint, you can remember that pi is very well approximated by a rational number 22 over 7. So pi minus 3 is very well approximated by 1/7. You can pause again if you want to see what that hint implies or we can, let's continue now, so here we go. Well, once again, we're getting seven curves now instead of seven lines. They're kind of like spirally like, right? Spirally like, but certainly not our sunflower head. Why does it spiral like this? Well, pi minus 3 is slightly less than 1/7, so we don't quite get back to the line that the first floret followed. We're slightly short because pi minus 3 is slightly smaller than 1/7 but as we get closer and closer to the center the curve will spiral in. Because the angle here is steeper for the ones nearer the center and less steep for the ones farther away. But because of the nature of moving closer to the center, you start to see something that looks like a spiral. So, pi minus 3 is irrational, but still doesn't give us anything that looks like a sunflower head, so what should we use? Well, the problem with pi minus 3 is because it has a very good rational approximation. 1/7, which number doesn't have a very good rational approximation. The golden angle, so we'll use the golden angle for alpha. The golden angle is 2 pi times 1 minus little pi. So, that's the angle of rotation, is the golden angle. So alpha then, is just 1 minus little 5. So, what will this one, this angle look like then, in our model of the sunflower head? So the florets are moving out, you don't see lines now. Because there's no good rational approximation to the golden angle but we see spirals. Now are starting to see spirals and were starting to see spirals in both directions just like our sunflower head. So, this is starting to look a lot like our sunflower head and we can look at two of these implementations. The difference between these two is that I changed the velocity at which the florets move out. We can count the number of spirals. Here we have 13 or 21, depending on whether you're counting the clockwise or counter clockwise spirals. 13 and 21 are Fibonacci numbers. Here we have 21 or 34, also two Fibonacci numbers. And these are, in fact, the two Fibonacci numbers we saw in the real sunflower head. So, this picture here then actually looks very close to a real sunflower. The one we saw, why do we get Fibonacci numbers? Because the golden angle, the rational approximations to the golden angle from the continued fraction are just the ratios of Fibonacci numbers. It's actually Fs of N divided by f sub n plus 2. These form rational approximations to the golden angle. So the denominator is a Fibonacci number. If the denominator is a Fibonacci number, then you see that number of spirals. I hope you found this to be a very interesting topic. I've showed you then how the golden ratio and the golden angle can show up in nature. And together with the golden angle here for the case of the sunflower you also get the Fibonacci numbers.