[MUSIC] Welcome back. In this lecture, I want to tell you about logarithmic spirals, and a type of logarithmic spiral, which is called the golden spiral. So the logarithmic spiral has this relationship that the radius is equal to a constant a times the exponential function, e to the b theta where theta is the angle. So, the relationship is written in polar coordinates with the radius r and the angle theta. This is what a logarithmic spiral looks like. So, we start at some center here, and we're spiralling out here. We're spiraling counterclockwise. You see in this first spiral here, we've gone out one unit. Each circle in this figure is one unit spacing apart. So we've gone out one unit, and then when we spiral again, we've gone out three units from one, two to three. And then we spiral again, and we go out four, five, six, seven, eight, nine units. So we go from one to three to nine. So every time we do a spiral, we are increasing the radius by a factor of 3. That's how a logarithmic spiral works. Because of that, if we look at a picture of a logarithmic spiral, we can change the magnification, so we can zoom in. [BLANK AUDIO] And it looks the same. On all magnifications it looks the same. When we zoom in, and zoom out it looks the same. That's the key property of logarithmic spiral. Because of that property, the logarithmic spiral shows up in nature. This is a picture of the nautilus shell taken from Wikipedia, the previous diagram is also from Wikipedia. Here we see that the nautilus shell grows in such a way that it traces out a logarithmic spiral. A very beautiful picture. So what is a golden spiral? A golden spiral is a logarithmic spiral, but the radius of the spiral increases by a factor of the golden ratio, 1 + square root of 5 over 2, each one quarter turn of the spiral okay? So one quarter turn is one quarter of a circle. A circle is 2 pi, so one quarter is each pi over 2 angle. So, if we then use that to determine the exponent. Remember it was R equals A times E to the B theta. We can use this condition to determine B, so we get the radius is equal to a constant. Here should be an A. Constant A times the golden ratio phi raised to the 2 theta over pi. So when theta increases by pi over 2, the radius then increases by a factor of phi. Here's our golden rectangle divided into spiraling squares. There's a square here, a square, a square, a square, a square. And see we a spiral here, because each of these squares is being reduced by a factor of the golden ratio. Each time we go one quarter over the way around the circle, we can fit into this golden rectangle and golden spiral. Here's what it looks like. So, this is a beautiful golden spiral fit into this figure of the spiraling squares. Now, the famous mathematician Jacob Bernoulli was very fond of the logarithmic spiral. Next to me here is his tombstone. Jacob Bernoulli was one of the Bernoulli family members, bunch of famous mathematicians. He was of the early proponent of the calculus, and one of the founders, together with his brother Johan of the calculus of variations. Most notably, I think Jacob Bernoulli was a major contributor to probability theory. He has a few things named after him in mathematics. He has the Bernoulli differential equation, the Bernoulli numbers, and the Bernoulli distribution. Now Jacob Bernoulli apparently had a somewhat mystical fondness for the logarithmic spiral, and he wrote, in the English translation, that the logarithmic spiral may be used as a symbol, either of fortitude and consistency and adversity, or of the human body, which after all these changes, even after death, will be restored to its exact and perfect self. We can look at a translation of the inscription on his tombstone. It's interesting to see what his wife wrote here. It says, Jacob Bernoulli, the incomparable Mathematician, Professor at the University of Basel for more than eighteen years, Member of the Royal Academies of Paris and Berlin, famous for his writings had a chronic illness of sound mind in the end. Succumbed in the year of Grace 1705, the 16th of August, at the age of 50 years and 7 months. Awaiting the resurrection, Judith Stupanus, his wife, for 20 years and his two children have erected a monument to the husband and father they miss so much. Now if you look at the bottom of this tombstone, you can see a spiral. We can blow up the picture of that. Is here. Jacob Bernoulli had requested that a logarithmic spiral be engraved on his tombstone. Together with the inscription eadem mutata resurgo, which translates as although changed, I arise the same. Referring to both the logarithmic spiral and the way that it's invariant under different magnifications as well as the hope for the resurrection of the dead at the end of the world. Unfortunately [INAUDIBLE] the stone masons carved an Archimedean spiral at the bottom of his tomb stone and not a logarithmic spiral. You can see how the spacing here is constant, while we know that for a logarithmic spiral, the spacing should increase by some common factor. Also there's a bit of error at the end here where the spiral seems to come back together. I guess the dead man can't complain. I'll see you in the next video.