What's so special about the Mandelbrot Set? - Numberphile

was coloring before, it is stable. You can see that wherever you go in the black region you get a nice stable pattern with lovely orbits. And that's

what

I love to see move. But if I go over the limit: unstable. Stable, unstable. Stable, unstable. Therefore, it is even stable in this region. It piles up like two piles, but if I leave the region it disappears. If I come up here, you will pile up in three piles. And in fact, Holly has made a lovely video about how many stacks you start making depending on which of these little bulbs you go into.

But the first time Mandelbrot saw this he had no idea about any of that structure. He simply saw a shape that surprised him. First of all, it's beautiful, it's a little strange, but to get a viewer to show it, like I have it on the screen now, you can go and watch any number of YouTube videos and zoom in, but you can see

what

Mandelbrot couldn't see. That, he saw this little blob and thought, I wonder what that is and he had no idea until he wrote his code better that it was another version, not exactly the same.

And you can zoom in forever and now, I don't think YouTube needs any more videos of me zooming in on Mandelbrot's set; However, I will do it anyway. (Brady: Ben, why are there different colors? Like it's not always black and blue.) Absolutely. So the original Mandelbrot set that Mandelbrot saw had two colors: stable, unstable; black, white or whatever, I did black and blue. The colors you see in this are an arbitrary decision, but they're not as arbitrary as you might think. So let's go back to the original here. Black for stable but the colors indicate it is unstable, but the color indicates how unstable.

So if you remember the first question I asked you, I said square, square, square, and within two iterations you like no. Because is big. And that's an indication that you don't need to do any more iterations to know what it's going to do. With complex numbers, it's less obvious what it's going to do, it's much harder to predict, but we finally prove it, mathematicians, I mean we. Not me: Mathematicians showed that if you leave a circle with a radius of 2, you will never return. But if you're inside you can't guarantee it. It could be bouncing near the edge and could stabilize again.

And you see that some of the orbits are complicated. So all you do with the computer is ask it to check, let's say 200 times and if you're still inside the circle of radius 2 after 200, you say: let's color it black. He's probably stable. But after about 50 iterations, if you just walked outside you think, well, now I know it's gone. But maybe the next time you check it, it will take sixty iterations and you'll think, well, that's less unstable, maybe I could color it differently. That was a long way of saying that the different colors in the image are the different levels of instability.

It's really about how many times you checked before you knew they were going to explode. And what matters when you zoom in is that the colors change very, very quickly. All the colors on the screen here indicate that a small movement of the original point gives you vastly different behavior. And that has become what we now call the hallmark of chaos theory: a small change produces fundamentally different behavior. In the Mandelbrot set you move, your change, each point you achieve depends on which C you choose. But Julia configured you fix one and I can convert this software into Julia configuration mode.

I'll do it a little first. If I press J here, there's the circle that is Julia's original set, and it's here in the middle. But if I move the mouse, I can see that the Julia sets with different C values ​​start to change beautifully. I mean, it's just a quick, sketchy animation, but it's already much nicer than the outlines you saw before and what I love about this is that Mandelbrot's set is like a map to Julia's sets. So if you go and find a little bit of Mandelbrot down here, this is called Seahorse Valley, because it looks like you have a lot of seahorses spiraling around.

But if I switch to a Julia set at this point, it looks the same. But if I move away from the Julia set, you don't degenerate into Mandelbrot, you just get the Julia set, the seahorse. There is. That's the biggest you can get. So if you zoom in on the Mandelbrot set, you get similar little regions that are like the Julia set of that area. It's like a map. It is a geography of iterative stability. When you say it like that it sounds complicated but you just do something over and over again and find out what happens in the long run.

And the fact that it makes something so beautiful, I mean, it's become a cliché for mathematicians to get excited about, but I still get excited because it's lovely, I didn't design it, it's just there to explore. and we can just sit and watch it mesmerized for hours. And you can schedule it in ten seconds on a spreadsheet if you want. So as you said, it's not in the Mandelbrot set, right? And what that means, so let's call this number, let's say, well, I've already used C. But let's call it C anyway.