Video with Transcript: Fibonacci Numbers hidden in the Mandelbrot Set - Numberphile
0:00 So, today I'm gonna tell you about how the Fibonacci sequence appears in the Mandelbrot set.
0:04 Hopefully, your mind will be blown by the end of that sentence.
0:06 So the Fibonacci sequence, the rule is that you take the previous two numbers,
0:10 and you add them together to get the next. Right? So we started with 1 1. Their sum is 2.
0:15 The next number will be the sum of 1 & 2, which is 3.
0:19 The sum of 2 & 3 is 5. The sum of 3 & 5 is 8. I'll do one more.
0:24 And then, you can continue on.
0:26 Brady: "Easy."
0:27 Easy, right? And this occurs everywhere that has interesting connections to things in nature, and all of that,
0:31 but I just want to show you where it appears on the Mandelbrot set.
0:35 So, slightly less easy. So, the Mandelbrot set is a special object inside of the complex plane. So the plane of complex numbers. And
0:43 the way you cook this thing up is by
0:46 considering a certain type of dynamical system. So if you give me a complex number c,
0:52 so here's a picture of the complex plane, so these are the real numbers and these are the real numbers
0:57 times i, which is that square root of minus one, if you give me a complex number c,
1:01 it's in the Mandelbrot set if, when you take the function z squared plus c
1:06 and you look at what happens to zero when you plug it into this function repeatedly, if that number doesn't get large,
1:13 then c is in the Mandelbrot set.
1:15 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,
1:21 right? Then you look at the function z squared minus one, you plug in the number zero,
1:28 zero, if you plug it into this function, gives you minus one.
1:31 Minus one, if you plug it into this function, gives you one minus one, which is zero again.
1:36 And so you're just gonna repeat this pattern. And so this number doesn't get large, no matter how far out we go.
1:42 And so this number is in the Mandelbrot set
1:44 It's about right here. So for each complex number, you make this decision based on what happens to zero under iteration.
1:51 So it looks something like this. So there's this big piece in the middle, all this interior is included, by the way,
1:58 and a little disc around minus one, and there's some more pieces coming off of here,
2:04 and some kind of funny tendril-y stuff goes on out here, and a few more pieces this way, another heart-shaped piece.
2:11 Mathematicians love this thing. Even non-mathematicians love this thing, but maybe for different reasons.
2:16 All right, so where is the Fibonacci sequence?
2:19 I mean first of all, this thing has nothing to do with whole numbers and addition, and
2:25 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.
2:30 But so, let me show you how to find it. I didn't draw too many of them,
2:34 but there's a bunch of these little components coming off of this main piece.
2:38 Brady: "What are they called?"
2:39 They're called the hyperbolic components, but let's not get into that.
2:43 To these components I'm going to assign a number,
2:46 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.
2:52 So like here, this component, there's this part where we have these tendrils, and there's three different directions
2:58 you can go in the antenna. And so this component will have number three. And similarly, over here, it actually turns out
3:03 there's only two directions
3:04 you can go in that antenna, so this is component number two. Now if we look for the next biggest one,
3:11 the largest component between the two and the three component, I've drawn it here.
3:16 I haven't drawn the tendrils,
3:18 but I'll try. It turns out
3:20 that there's exactly
3:22 five directions that you can go from that antenna. And the next biggest between these two, if you draw the antenna,
3:28 I'm not sure I can fit it in here, since I think you already know what the answer will be,
3:32 is eight. So if you go through the Mandelbrot set, and you start with these two components, the two and the three
3:39 components, and you look between them for the next biggest component, the next biggest one will be the next Fibonacci number.
3:46 So I want to explain why.
3:49 Brady: "You'd better."
3:50 All right. I'll explain at least part of why. How about that?
3:53 I called this big piece, the main component, it's called the main cardioid.
3:57 Brady: "What's its number?"
3:58 Its number is one, actually. That's a really good question.
4:02 But it's not so obvious to see from antennas, so.
4:05 So the main cardioid here,
4:07 number one,
4:09 it turns out that there's a very natural way to stretch this thing back into a disk,
4:15 which is something we understand really well, right?
4:17 So this is always useful in sort of geometry or that kind of study, if you can
4:21 change something just a little bit and get back to something you understand really well.
4:24 So there's a natural way to view this thing, just by some stretch,
4:29 so this disk maps under this stretch to a cardioid.
4:32 And I want to look at what happens to, first of all, my center point, it turns out it goes to zero, this ray
4:40 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,
4:47 it maps to this line inside the main cardioid. If we go, say, a third of the way around the circle,
4:54 it maps to a kind of a funny curve
4:57 inside the main cardioid. And same for two thirds, and so on. So you can track
5:02 what happens to these rays under this stretch.
5:04 And here's the thing. Is that the place where the rays end up in the main cardioid
5:09 are exactly the places where it connects to these components.
5:13 So here is the connection to the two component,
5:17 and here is the connection to that top three component,
5:21 and that bottom three component. We have a five component up here in the Mandelbrot set, so there must be some
5:29 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.
5:37 And the point is that, under this stretching map,
5:40 the number of antennae, the number we assigned to each of these components, is the
5:44 denominator of the fraction that tells you how far around the circle you went. This question of, what's the largest
5:51 component between any two other components,
5:53 turns into this totally separate question of, what's the fraction with the smallest denominator between these two fractions?
6:00 Brady: "Why is the five component at two fifths, and not one fifth or three fifths? Two seems arbitrary."
6:07 So the one that I drew is between one-third and one-half, and so that's the two fifths.
6:12 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.
6:18 But the point is that you can read off these numbers in two different ways, right? The antennas or the
6:23 denominators of the fraction for the way around the circle. So we're closing in on Fibonacci. So I said that, okay,
6:29 we changed this question completely to a question of, what's the fraction with the smallest denominator between two fractions?
6:35 One third, which is less than some fraction,
6:38 which is less than one half. And the question is, I want a smallest denominator fraction here, how small can it be?
6:44 Well, it can't be 4, right? Because 2 over 4 is the same as 1/2, and 1/4 is too small.
6:50 But it can be 5, because 2/5 really is between these two numbers.
6:54 Okay, so 2/5 is the answer here.
6:56 Now, what if we were to do the next one? The next one was asking, between the three and the five
7:00 components, right? So we want a number that's between
7:02 1/3 and 2/5 that has the smallest possible denominator. And again
7:07 you can check, six and seven aren't gonna do it. It turns out that the answer is 3/8.
7:13 And in general, the crazy thing is that if your fractions are close enough together, the way to find
7:19 the one between them that has the smallest denominator, is just by taking the mediant, or the fairy sum, or,
7:25 as some people refer to it, the freshmen sum,
7:29 because you get it by adding together the numerators, and adding together the denominators.
7:32 Brady: "Why is that a freshman sum?"
7:34 Well, because, I think it's mean, actually. I mean, maybe a kindergarten sum? Is that better?
7:40 Brady: "Because why? Because people who don't know maths would think that was a legitimate..."
7:43 That's right, that's right.
7:44 Brady: "Well, it is a legitimate thing."
7:45 It is a legitimate thing. I can tell you why it's bad, right?
7:48 So why is it a bad way to add fractions? Because it matters how you represent your fractions.
7:52 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?
7:58 So, it's not so good. But.
8:00 Brady: "It works for this."
8:01 It works for this.
8:03 Right? So 1/2,
8:04 here's the symbol usually used for mediant. The mediant of these two fractions
8:08 is just 2/5. The mediant of these two fractions is 3/8, and so on. So where is Fibonacci?
8:14 Fibonacci is because,
8:16 look at the fractions I started with.
8:18 I started with the first two elements of the Fibonacci sequence,
8:22 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.
8:28 And so it's exactly the rule which defines the Fibonacci sequence
8:33 coming up in these fractions,
8:36 coming up in their mediants,
8:37 and so coming up in the Mandelbrot set.
8:55 If you'd like to better understand the Mandelbrot set, and I mean really understand it, then why not check out this?
9:01 Look, it's a Mandelbrot set quiz, and a Julia set quiz, as well.
9:05 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
9:12 by not just watching stuff, but doing stuff.
9:16 There's a lot to like about these curated sequences.
9:18 But what I like is that they guide you through step by step, help you understand.
9:23 But it's not all about
9:25 scoring your work, or making you feel silly. I mean, you can look up the solutions,
9:29 you can look up hints. They help you along the way.
9:31 And they also help you understand just how beautiful mathematics can be. It really made an impression on me.
9:36 I really feel like these people know what they're doing. They've made a good thing. To check it out,
9:40 go to brilliant dot org slash numberphile. I'll put that down in the description. You can sign up for free, but the first
9:48 233 people who do it - that's a Fibonacci number - will get 20% off an annual premium subscription.
