Today, M stands for Mandelbrot set We've all seen it, right? Let's just zoom in. What you find is all these amazing pictures here, like, these little baby Mandelbrots Uhh, remember those we'll need them for later Okay, now, the Mandelbrot set is actually just the black bit That one here. The halo that you see around it- this one here, you get when you have a very very close look at the data that you generate when you make up the Mandelbrot set Now, the inside's completely black- the dark side of the Mandelbrot set Umm, it's actually not that dark when you also have a really really close mathematical look and that's what we're going to do today and When you have a close look, what you get to see is, for example this guy here or even better this guy there the mysterious Buddhabrot fractal. Uhh, well, I mean you can kind of see where the Buddha comes from, pretty obvious. There it is Umm, okay, so it's going to be about the dark side and I thought well, maybe, today my audience is going to be, umm Darth You know, pretty obvious Darth's interested in the dark side, so I'll tell him about the dark side of the Mandelbrot set and [well], let's see Okay, so what's the Mandelbrot set well? It looks like a set of points, but actually it's a set of numbers Okay, what are these numbers? Well, here are the real numbers. There's zero, there's one. You know, so every point that you see actually corresponds to a number But of course there's a lot more than the real numbers here for example. What's that point up there? What number is that? Well, that's i, the square root of -1. It's a complex number. And You know everything else that you see here is just complex numbers So for example this guy over here is just 1 over here, 0.5 up here. So that's 1 plus 0.5 times i Now you may know or may not know complex numbers Doesn't matter Only thing you really need to know for today is that they are beautiful, very Important extension to the real numbers you can add them you can multiply them and that's actually what you [need] to do to figure out Whether one of those complex numbers is inside or outside the Mandelbrot set okay? now darth is really interested in figuring out for example whether the number 1 is inside the Mandelbrot set So what is that you have to do? Well, he has to run this scheme here infinitely often. So what it does is it takes the number 1, sticks it in there where it says number and That gets you a formula here, which is x squared plus 1? Now what we do is we initialize by sticking in 0 into this formula And that gets us 0 squared plus 1 is equal [to] 1 [now] [we] take what we get out here and stick it back in again. We get 1 squared plus 1 is 2, 2 squared plus 1 is 5, 5 squared plus 1 is 26 and So on and actually in this [case] it's pretty obvious that what's going to happen is that the magnitude of the numbers that we're getting here is going to get bigger and bigger than it's actually going to approach infinity as you kind of push this further and further and Whenever that happens you actually figure out that the number that we're talking about That one here in the green rectangle is outside so magnitude exploding to infinity means Outside the Mandelbrot set. Sorry Darth, 1's not for you Okay, well that's fine. It's fine. Now how can it not be? Well somehow the sequence has to be contained in a finite region. Let's give an example: so that minus 1 for example supposed to be inside the Mandelbrot set so we replace the 1 by a minus 1 and It gives us a new formula x squared minus one we initialize Put 0 in, 0 squared minus 1 is minus 1. Minus 1 squared is 1, minus 1 is 0. back to the beginning, and then of course things repeat Minus 1, 0, to infinity, it's not going to go anywhere. It's going to stay confined All right, and now we have to do this for every single number that we see here and See whether it's inside or outside. That is a lot of work and actually How are we going to do this? We're going to do this infinite sequence? We can't really do that, right? I mean, you can't, I can't. I can't do this infinitely often What has been shown, what has been proven is that well you don't really have to wait To the end of times to figure out whether we're going to infinity or not the only thing you need to Know is whether a sequence strays at any time outside this yellow circle, if it does, it explodes to infinity, if it doesn't Well, you can forget about it. And actually the whole Mandelbrot set is Contained in this yellow disc. It doesn't doesn't go outside or Darth is not so happy because that obviously means well it's actually not that big our Mandelbrot set, our dark side, but anyway let's continue So what we actually do is we we set a [bailout] value to be able to approximate the Mandelbrot set so we set the bailout value for example to 500 so we iterate every single one of our sequences 500 times and If by that time the sequence exits the yellow disc, we declare Whatever we're looking at at the moment to be outside and otherwise we declare to be inside, so we you know We're actually declaring some points to be inside that are not inside. Just because it takes them a lot longer to get outside The yellow Circle, but it's fine if you want a better picture, We'll just crank up the bailout value for example to [5000] or 50,000 Okay Let's have a look at one of those points. Right so here we go So we start and we want to figure out whether that thing is inside, or outside We'd set our bailout value to 500 we start okay? Second ones here third guy's here fourth guy's here fifth guy's there sixth guy's there seventh guy's outside Perfect. We know this point is outside the Mandelbrot set fantastic And now we also know [that] it took us seven steps to get outside so to actually get this halo that you see in all The mandible pictures what we do is we color the outside points? According [to] how many steps then it takes them to to get outside the yellow circle here Okay, so that was for example seven so all points Corresponding [sequences] take seven steps to get outside get color with the same color. That's how you get the halo now You can do something else. You [can] actually just plot all those escaping paths, okay so all those escaping paths for lots and lots and lots of points outside the Mandelbrot set and that sort of gives you a density plot of points escaping to infinity and When you do that, you get these Buddha-brot pictures. They will also look different depending on where you set your bailout value So for example that one here corresponds to the bailout value of five hundred if you go for five thousand It looks slightly different if you go for fifty thousand it looks slightly different and now to get the color picture actually what people [have] done is they've taken these three pictures and made them into the blue green and red Channel of A Color picture, so this one here really amazing now. This was actually invented by one of our regular viewers Melinda Green it's a Amazing fractal, but actually hardly anything has been you know investigated here So there's a lot of things that need explaining Nobody's really looked at this So if your budding mathematician this would be a really really nice object to look at to explain all these features here that you see So far what we've done is well. We've we've seen some light happening inside here, but that illumination was all by Basically points outside doing something now what I want to do is really Mathematically drill into the into the inside and really show you What's going on? Not just talk [about] it, but [actually] show you what's what's going on? Okay, here we go we're going to focus on the real numbers because there we can draw pictures and Actually, it's just kind of parabolas which is going to be parabolas, so let's just go for an obvious point [a] zero So if you stick this into the formula, we will just get x-squared you can draw that that is and we can iterate Not very interested in this case. So what we do is, we make it a bit more interest by first going outside Okay, so going outside so we go to this point here, which is 0.3. The picture, how does the picture change? Well, we just raise the parabola by 0.3. Okay? Now, we initialize with zero Okay, out comes 0.3. That's just this distance here Now how do you see in the picture what the next value is going to be when we stick point three inside? Visually that goes like this go up And we get the next value and we repeat and repeat and repeat and repeat And we get all the functional values This is a very nice way of visualizing things, but there's actually an even better one and so for that one we just put in the diagonal here and Then just watch watch this so we're going from zero okay from zero going up to the Parabola Horizontal over to the Green up to the Parabola horizontal over to the green up to the Parabola horizontal over to the green, and you just keep on going like this and We actually get exactly the same sequence of numbers happening But we can kind of see at a glance how they happen, right? So we can kind of just in our mind See that kind of zigzag between the Parabola and the green lines is really really nice Now your homework is going to be to explain why this works? Once we've got this picture. We can dynamically change it and actually observe what happens to the zigzag So first let's let's go up okay, so first we go up here we go, and you can see we're basically just raising the parabola up here and What does exit path still escapes now? Let's go the other way to the critical point here. [that's] point two five Let's see what happens there were we lowering the parabola You can already see what's happening the bottleneck here kind of gets squished together and at some point in time You know we get the parabola meeting the line and so what happens here? Is that the whole sequence kind of gets sucked into this point. Now Darth gets really excited at this point -- Tractor Beam Tractor beam all right now what happens next well. We're kind of going through this interval here to [the] next critical point Which is minus 0.75 lets us go Okay, so we're going all the way down, and you can see tractor beams happening all the way along, so Things get sucked into this this point in attracting fixed point so attracting fixed point all The way along here all the way along here, and actually not only along here is this real part of it But it's happening everywhere here inside that main bulb this cardioid, your main cardioid of the [Mandelbrot] set the same sort of behavior here things are being sucked into one point at minus 0.75 the Parabola intersects the [Green] [line] at exactly 90 degrees and Things start splitting up and one attracting fixed point starts splitting up into two. So in this [dis] that's coming up here And it's actually perfect [discs] around minus [one] The sequence is going to be attracted [by] two points oscillate between two points. Let's just see how [that] works [on] The way to minus one you see being attracted between two points at minus one We've got an extreme case happening where we flip between two points? [we're] not only attracted by two points but we flip between two and then well that same behavior you get inside This circle here so basically two tractor beams kind of taking turn attracting things now We move into next circle here here things are going to start splitting up into four different attracting points For different attracting points there is another circle attached to that one and they were getting into eight different attracting things and Then as another circle sixteen and thirty two and so on that continues forever well Not forever all the way over here, [but] up to a point. [I'm] just going to draw Right there, and let's just go there And if doubling up doubling up doubling up at that point in time things Get chaotic so chaotic and you kind of stay chaotic all the way to -2. Except [there] are islands of order, so let's just Go for a while these islands of order correspond to these many, Mandelbrot sets that he kind of come across here Like on the way. We saw kind of one thing where the Chaos kind of goes into a -- whoo! we've got something nice happening here - something nice happening here period Three and actually When you have a really really really close look at this region kind of zoom in you'll also find that There's a lot of this nice doubling and regularity Happening whenever you come across one of those little Mandelbrot sets we're not going to go into detail [here] Let's just keep on going from here [to] -2. And let's see what happens Here's Chaos Chaos Chaos Chaos something interesting is about to happen Palm So we're going down here over there up here, and then we're exploding to infinity That's what happens at [minus] [two] or beyond [minus] [two] and so we escaping to infinity from then on Really nice picture, okay? So this this statement that the sequence is contained and when its containers in the Mandroid said that statement's actually Hiding a lot of really really nice complicated behavior and that can [be] summarized nicely in a color picture like this so You know we choose one color to designate the region that has like one of those attracting fixed point one tractor beam? We have another one where I've got two tractor beams one for four tractor beams and then all kinds of other [stuff] here So all these bulbs that you see here have a characteristic behavior like everything in here has for example three Tractor beams lots of three tractor beams and the points are flipping back and forth between three different things just like in the main bulb of this baby-Mandelbrot set that we saw before there's other nice behavior here, so it [isis] three four five or all the natural numbers Counting up here by the main bulbs that you see here. There's the odd numbers going there You can see lots of other stuff [in] fact. There's a lot of nice nice mathematics in here. So there's one guy There's another guy in between there's the largest one. What's the number that Corresponds to that phone which is three plus four really nice? What about here [that] guy here? That's the largest one between those two guys it's just three plus five [is] eight and Other things that you see here, so I come to you see these antennas here one two three four things coming [out] We've got a four here here seven things coming out I've got a seven here there are three things coming out with one of three here But I think I'll leave that for another video is already too long okay, so at this point Darth is probably pretty disappointed and Figured out that well actually there is no real dark side to the Mandelbrot set So we have to go and look for for that somewhere else you you
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