So, today I’m gonna tell you about how the Fibonacci sequence appears in the Mandelbrot set. Hopefully, your mind will be blown by the end of that sentence. So the Fibonacci sequence, the rule is that you take the previous two numbers, and you add them together to get the next. Right? So we started with 1 1. Their sum is 2.
The next number will be the sum of 1 & 2, which is 3. The sum of 2 & 3 is 5. The sum of 3 & 5 is 8. I’ll do one more. And then, you can continue on. Brady: “Easy.”Easy, right? And this occurs everywhere that has interesting connections to things in nature, and all of that, but I just want to show you where it appears on the Mandelbrot set. So, slightly less easy. So, the Mandelbrot set is a special object inside of the complex plane.
So the plane of complex numbers. And the way you cook this thing up is by considering a certain type of dynamical system. So if you give me a complex number c, so here’s a picture of the complex plane, so these are the real numbers and these are the real numbers times i, which is that square root of minus one, if you give me a complex number c, it’s in the Mandelbrot set if, when you take the function z squared plus c and you look at what happens to zero when you plug it into this function repeatedly, if that number doesn’t get large, then c is in the Mandelbrot set.
So I know that sounds sort of complicated, but just let me do an example, right? So if you look at c equals minus one, right? Then you look at the function z squared minus one, you plug in the number zero, zero, if you plug it into this function, gives you minus one. Minus one, if you plug it into this function, gives you one minus one, which is zero again. And so you’re just gonna repeat this pattern.
And so this number doesn’t get large, no matter how far out we go. And so this number is in the Mandelbrot set It’s about right here. So for each complex number, you make this decision based on what happens to zero under iteration. So it looks something like this. So there’s this big piece in the middle, all this interior is included, by the way, and a little disc around minus one, and there’s some more pieces coming off of here, and some kind of funny tendril-y stuff goes on out here, and a few more pieces this way, another heart-shaped piece. Mathematicians love this thing. Even non-mathematicians love this thing, but maybe for different reasons. All right, so where is the Fibonacci sequence? I mean first of all, this thing has nothing to do with whole numbers and addition, and arithmetic, and the kinds of things that you think about with Fibonacci, which is why it’s weird that you can see it in here. But so, let me show you how to find it.
I didn’t draw too many of them, but there’s a bunch of these little components coming off of this main piece. Brady: “What are they called?”They’re called the hyperbolic components, but let’s not get into that. To these components I’m going to assign a number, and that number is going to be the number of branches that come off the sort of tendril-y bit, which is called an antenna. So like here, this component, there’s this part where we have these tendrils, and there’s three different directions you can go in the antenna. And so this component will have number three. And similarly, over here, it actually turns out there’s only two directions you can go in that antenna, so this is component number two. Now if we look for the next biggest one, the largest component between the two and the three component, I’ve drawn it here. I haven’t drawn the tendrils, but I’ll try. It turns out that there’s exactly five directions that you can go from that antenna. And the next biggest between these two, if you draw the antenna, I’m not sure I can fit it in here, since I think you already know what the answer will be, is eight. So if you go through the Mandelbrot set, and you start with these two components, the two and the three components, and you look between them for the next biggest component, the next biggest one will be the next Fibonacci number. So I want to explain why. Brady: “You’d better.”All right. I’ll explain at least part of why. How about that? I called this big piece, the main component, it’s called the main cardioid. Brady: “What’s its number?”Its number is one, actually. That’s a really good question. But it’s not so obvious to see from antennas, so.
So the main cardioid here, number one, it turns out that there’s a very natural way to stretch this thing back into a disk, which is something we understand really well, right? So this is always useful in sort of geometry or that kind of study, if you can change something just a little bit and get back to something you understand really well. So there’s a natural way to view this thing, just by some stretch, so this disk maps under this stretch to a cardioid. And I want to look at what happens to, first of all, my center point, it turns out it goes to zero, this ray will map to this ray. This ray is, I guess, zero of the way around the circle, right? If we go halfway around the circle, it maps to this line inside the main cardioid. If we go, say, a third of the way around the circle, it maps to a kind of a funny curve inside the main cardioid. And same for two thirds, and so on.
So you can track what happens to these rays under this stretch. And here’s the thing. Is that the place where the rays end up in the main cardioid are exactly the places where it connects to these components. So here is the connection to the two component, and here is the connection to that top three component, and that bottom three component. We have a five component up here in the Mandelbrot set, so there must be some five ray which lands there, and in fact there is. It turns out that that’s the two fifths of the way around the circle ray. And the point is that, under this stretching map, the number of antennae, the number we assigned to each of these components, is the denominator of the fraction that tells you how far around the circle you went. This question of, what’s the largest component between any two other components, turns into this totally separate question of, what’s the fraction with the smallest denominator between these two fractions? Brady: “Why is the five component at two fifths, and not one fifth or three fifths? Two seems arbitrary.”So the one that I drew is between one-third and one-half, and so that’s the two fifths. But there are five components at 1/5 3/5 and 4/5. You’re totally right, so yep, it works every time with the denominators.
But the point is that you can read off these numbers in two different ways, right? The antennas or the denominators of the fraction for the way around the circle. So we’re closing in on Fibonacci. So I said that, okay, we changed this question completely to a question of, what’s the fraction with the smallest denominator between two fractions? One third, which is less than some fraction, which is less than one half. And the question is, I want a smallest denominator fraction here, how small can it be? Well, it can’t be 4, right? Because 2 over 4 is the same as 1/2, and 1/4 is too small. But it can be 5, because 2/5 really is between these two numbers. Okay, so 2/5 is the answer here.
Now, what if we were to do the next one? The next one was asking, between the three and the five components, right? So we want a number that’s between 1/3 and 2/5 that has the smallest possible denominator. And again you can check, six and seven aren’t gonna do it. It turns out that the answer is 3/8. And in general, the crazy thing is that if your fractions are close enough together, the way to find the one between them that has the smallest denominator, is just by taking the mediant, or the fairy sum, or, as some people refer to it, the freshmen sum, because you get it by adding together the numerators, and adding together the denominators. Brady: “Why is that a freshman sum?”Well, because, I think it’s mean, actually. I mean, maybe a kindergarten sum? Is that better? Brady: “Because why? Because people who don’t know maths would think that was a legitimate…”That’s right, that’s right. Brady: “Well, it is a legitimate thing.”It is a legitimate thing.
I can tell you why it’s bad, right? So why is it a bad way to add fractions? Because it matters how you represent your fractions. So if you try and add 1/3 to 1/2 this way, you’ll get something different than if you add 1/3 to 2/4, right? So, it’s not so good. But. Brady: “It works for this.”It works for this. Right? So 1/2, here’s the symbol usually used for mediant. The mediant of these two fractions is just 2/5. The mediant of these two fractions is 3/8, and so on. So where is Fibonacci? Fibonacci is because, look at the fractions I started with. I started with the first two elements of the Fibonacci sequence, and the third and fourth elements of the Fibonacci sequence. And the way I get the next thing is by adding the other two together. And so it’s exactly the rule which defines the Fibonacci sequence coming up in these fractions, coming up in their mediants, and so coming up in the Mandelbrot set. If you’d like to better understand the Mandelbrot set, and I mean really understand it, then why not check out this? Look, it’s a Mandelbrot set quiz, and a Julia set quiz, as well. We’ve covered the Julia set before.
They’re from Brilliant, a problem solving website that lets you go further into the world of math and science by not just watching stuff, but doing stuff. There’s a lot to like about these curated sequences. But what I like is that they guide you through step by step, help you understand. But it’s not all about scoring your work, or making you feel silly. I mean, you can look up the solutions, you can look up hints.
They help you along the way. And they also help you understand just how beautiful mathematics can be. It really made an impression on me. I really feel like these people know what they’re doing. They’ve made a good thing. To check it out, go to brilliant dot org slash numberphile. I’ll put that down in the description. You can sign up for free, but the first 233 people who do it – that’s a Fibonacci number – will get 20% off an annual premium subscription.