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