[MUSIC] Welcome back. To understand the model of the sunflower, it's useful to introduce one more concept which is called the golden angle. We take a circle and we draw this angle in the circle, splitting the circle into two arc lengths, x and y. If we draw this angle so that x and y are in the golden ratio then we say, that g is the golden angle. So what does that say, if x and y are in the golden ratio? It means that x divided by y, x is the larger arc length, y is the smaller arc length x divided by y is equal to capital phi, the golden ratio. The reciprocal of the golden ratio is the golden ratio conjugate, little phi. So y over x is little phi, and little phi is just the fractional part of big phi, so 1 + little phi = big phi. So that gives us the relationships between x and y. What about the golden angle g? It's angle g divided by the full angle of the circle 2 pi should be equal to the arc length y divided by the full arc length of the circle which is x + y. So g over 2 pi = y over x + y. Now we can use this formula to figure out what g over 2 pi is. So to do that we can divide the numerator and denominator by x. So y over x is phi, and x over x is 1 + y over x is little phi. 1 + little phi is big phi, so this is little phi divided by big phi. But big phi is just 1 over little phi. So this becomes phi squared, phi squared is all right, that's a correct result. But if we want to make it a linear result, we can take this equation and multiply by phi. So we have phi + phi squared equals, big phi times little phi is 1 because they're the reciprocal of each other. So little phi squared is just 1 minus little phi, that puts it in a linear form. Putting this together we get the golden angle then multiplied by 2pi. It's 2 pi times 1 minus the golden ratio conjugate 1-5 This is in radians, if we put in 0.618 for phi, but we can also put this in degrees. If we put this in degrees, this is approximately 137.5 degrees, and you will see that in your literature. They call it the golden angle, 137.5 degree. The golden angle will then become an important angle in our modelling of the sunflower. And since we've already talked about continued fraction, it's important for us to know what is the continued fraction of the golden angle. So let's try and figure that out. So here's our golden angle. We know the continued fraction for the golden ratio, for phi, so it's better if we then convert this formula into a formula for the golden ratio. The golden ratio is x over y, so if we divide numerator and denominator here by y, we get 1, y over y = y over y from the second term which is 1 + x over y, x over y is our golden ratio capital phi. So this gives us g over 2 pi in terms of the golden ratio, 1 over 1 + 5. And if we want to know what is the continued fraction for g over 2 pi, then we can just use the continue fraction for the golden ratio. So this is 1 over 1 plus the continued fraction for the golden ratio is 1 + 1 over 1 + 1 over 1 + all the way. All 1s here, so what is this? These two terms combine to get 2. So it's 1 over 2 + 1 over 1 + 1 over 1. So we can write that in our short hand form, a 0 here is 0, because g over 2 pi is a number between 0 and 1. A 1 is 2, this first value here is 2, and then a2, a3, a4, they're all 1s. So we got a 1 here, and it just repeats forever. So the golden angle divided by 2pi normalized by 2pi, has a continued fraction with all 1s at the end here, all 1s. So it is also a very difficult number to approximate by a rational number. It is also the most irrational of the irrational numbers. The same way that pi itself is all ending ones in the continued fraction expansion. So together with our knowledge of continued fractions and our knowledge of the golden angle, we're ready to construct a model for the sunflower. I'll see you next time.