Maybe this is "Mandelbrot set back to basics". I need your help though, can you think of a number?
(Brady: 7) - Good. Can you square that number?
- (49) - Can you do this while filming? You've done it correctly so far, okay, next question you're gonna get upset - square 49.
- (Nup)
Okay, good answer. The thing is I don't really care about the answer. What I'm caring about is an iteration.
I'm gonna keep asking you to do the same thing again and again
and if I ask you to pick a number and keep squaring it, you already know what's gonna happen, right?
It's gonna get big. - (Brady: It'll blow up.) - Is it? Square 1.
- (Okay, except 1.) - Okay good, now square a half.
(Ok, it's gonna get smaller)
So sometimes it blows up, and actually that happens when it's bigger than 1, and it becomes quite obvious when you try it.
But 1 doesn't blow up, so 1's different. And numbers less than 1, like 1/2,
actually become a quarter when you square them and then become a sixteenth and so on. But then numbers much less than one like minus
six...
and that one's blown up as well. So what I'm really talking about, and what iteration is quite often talking about, is when iterations are stable,
when they sort of head towards somewhere you can see, and when they're not. And it's best to see this on a diagram,
so have a look at my screen here.
You picked the number seven, which is kind of off my screen here, but these red arrows, I'm moving them around.
This is not the greatest diagram, let me get this clear in this to start with, it's just a line.
But the arrows are heading that way because when I square this number
I'm moving around they go that way and they never come back, much like your seven did.
But if I go less than this number one, they don't go that way anymore, they go that way.
In fact, they head towards zero and they're stable there.
So for the sake of moving it around, it's kind of nice to see the difference, it's a really sharp line
then it completely changes behaviour, but that way they're unstable, that way they're stable. Stable, unstable, stable, unstable,
stable, stable, stable unstable - you get the idea.
In fact if you go negative
they're still stable. The head positive because when you square a number that's negative you get positive.
Still head toward zero until you hit minus 1 and then
unstable. Stable, stable, unstable, stable. Now despite all my commentary
it's not that exciting, yet. And if you know anything about the theory of
iterations and the consequences of them, or if you've watched any other video about the Mandelbrot set you know that
we're not just talking about real numbers, i.e the numbers on the line
I've just been showing you, we're talking about numbers in two dimensions. So
complex numbers. And I'm not gonna
give you an introduction to complex numbers here because I don't think you need to hear it and there are other places you can get
them.
Holly on the Numberphile Channel has done excellent videos about how this happens,
but I want to show you the same comments that she's made on brown paper. I'd like to show you
moving. Here's my two-dimensional number line. Here is the number that you get to pick randomly. We can still do it in one dimension
just by sliding along here and the other dot that's moving around this tiny one is the square of the other dot.
So 1 squared is about 1, bigger than 1 it heads off
that way, lower than 1 it goes that way; but I can move off the line now. And it has a weirdly circular flavour to it,
which I kind of like. It's like
chasing the number around, it moves quicker than you expect and that's already quite pleasing if you're a nerd like me.
But what I'm really interested is the iteration if you keep doing the instruction: square it, square it, square it.
We saw what happened on the real line. In two dimensions it looks different. So it looks like this.
So this line indicates if we start there it goes to there when you square it and then to there and there. And so if
I move this around the picture is kind of pleasing, kind of wobbles around,
but it's always heading towards the centre - which I'll say is stable.
And there's a reason there's a circle on the screen because anytime I'm inside this circle, it looks like it's stable,
even if it's kind of pretty as it moves.
But if I go near the edge of the circle
it's less obviously stable, it takes a lot longer to get to the centre and then if I go outside the circle,
unstable.
Stable, unstable, stable, stable, unstable, stable.
This is a really nice demonstration of why complex numbers are helpful in the, instead of
your entire image of what's happening being on a line you've got a picture. But it gets better, right. Instead of just squaring it
what would happen if you did square it, add something; square it, add something; square it, add something?
Now we could go back to the real line and see it happening on just one dimension
but I feel like we should just go straight into two dimensions.
This is just the square it, square it, square it, square it. This letter C is indicating a number I could add every time. So
let me just slide it away from zero, at the moment it's not adding anything. But if I add this
number, this red number, which is a complex number, it's two dimensional. You can see something very different has happened.
If I start with this number and I keep squaring it and adding that number; square it, add that number; square it, add that number; square it, add
that number - this is the path it goes and it makes a lovely star shape.
What's interesting though is not just the star shape, although I kind of like it, it's the stability.
So all around here it's stable,
it always gives you the same sort of stability, like the starry stability, even if it's expanding or shrinking
but you kind of get the feeling that sometime it's gonna blow up. And it's not obvious where
because if I go near the circle you think that's gonna blow up, it's gonna blow!
But it doesn't!
So there, with a moment there, so outside the circle still stable, but somewhere else outside the circle unstable. And
in fact, it's not obvious where this happening; stable, unstable,
stable, stable, unstable? No stable still, and - but over here inside the circle always stable,
unless I go there. I keep saying the circle as if it's important,
it's really not important anymore, the boundary is clearly not a circle, but it's kind of nice to move this around
and think that some places are unstable that some are stable. What I'd really like to see,
and I hope you were too, is the boundary of stability. So
let's see that. In fact
let's look at the boundary. If the number I choose to add each time is 0 you get the circle. That's like the simple iteration,
but if you choose to add something different - by the way, I'm going I'm gonna write something down here.
What I'm capturing is this iteration, which always looks a bit technical,
and if I cover that bit up, it's what we did at the beginning.
I said square a number and that's your new number; square it, square it, square it.
But instead of just squaring it you do square it add something, then you use C for it's a constant, but it's a complex number.
Now we're ready to look at that. If C is zero you get a circle. If it's not zero - I
see where my brain melts a little tiny bit. If C is not a zero the shapes become these shapes.
And I always find this a little bit of a shock. There may be some familiar looking shapes,
I don't know but, sometimes they're not even connected together. They're like dusty little particles. And
these regions, particularly when there's obviously a region which is not a circle necessarily, they're called Julia sets.
And again Holly's done a really lovely video about Julia sets.
A bit of history about Julia sets: found by Gaston Julia,
hence the name, who was a guy, a French mathematician, I think. Early twentieth century.
He did it without a computer, like he wasn't
able to just do this and move it on GeoGebra and see the boundary,
he was doing this by hand and realising there was some beautiful structure. Like, I mean every time I tell thar story I'm like
fair play.
He also is really interesting in that he had no nose.
Genuinely go and find a picture of Gaston Julia, he has a leather thing across his face.
His nose was damaged in, I think the First World War. Before I leave Julia entirely,
the sets that you can see give you a little bit of a hint about what's coming, so
let me show you two things about the Julia sets. First of all, the simplest one is a circle, it's not that exciting.
But they are exciting, but they're not always obvious regions. Sometimes they look like they're separate regions. These don't look like they're connected.
The way I'm animating it here, they're all slightly dusty looking.
That's just the way I'm making it happen quickly.
But sometimes they're definitely not joined together and sometimes they look like things you might recognise the outline
as looking vaguely like something I've hinted about in
this video may be looking like. But there is something important that Holly mentions in some of her videos about which ones are joined together,
connected mathematically, and which ones are
disconnected. Separate like dust. Let me show you what they end up describing.
So if we go back to this diagram when I did the iteration, we were
originally starting by you picking a number and then squaring it. What happened in 1979 is a guy called Benoit, that was his first name.
Didn't get better for him, his surname was Mandelbrot. He's a very famous figure in mathematics these days, but it was 1979,
1980. It's kind of recent in the grand scale of mathematics. Benoit was like,
hmm. I wonder what happens if I always start at zero? Instead of me saying pick a number,
I would say like pick a number as long as it's zero.
And then all I really care about is what's the constant I add each time? And which ones of those
give me stable, and which ones don't? Now he didn't have a good computer,
he was doing it on an IBM - working for IBM at the time and having to print out on a dot matrix or
probably even pre-dot matrix printer.
But we can kind of shortcut that, so let me show you what it looks like. Always start at zero and
move the constant around each time.
And if you do that all you'll see is sometimes it's stable like this, and sometimes it's different stable. I just really love this,
I'm just going to move it round.
They're kind of
predictable and yet not predictable. You get spirally things, you get spider webby things. And sometimes it's not stable. It's unstable.
Sometimes it's got like, how many arms is that? Seven arms? Sometimes it's got three arms,
sometimes it's got something else. And sometimes it's just rubbish.
(Brady: This is just the squaring and adding part?)
Squaring and adding. These are, what they call these are the orbits of-
for every constant I pick which, you mentioned I picked one arbitrarily, happened to be up in the top left there, this one in fact.
That's its orbit.
But any other constant you pick, anywhere on the screen, has an orbit and some of them are boring.
They're just gone and some of them are not.
They're stable and some of them are not quite sure what the hell's going-
excuse my English there. The,
the idea that maybe some points are special and some are not is what Benoit Mandelbrot first began to investigate. Other people were
investigating it but it ended up with this idea getting his name on it.
So let me simulate what he first saw. Remember, he didn't have a monitor on his computer
he had a
printer. You'd like type some code in, if
you want a picture you print it. Like ten minutes later after you've solved all the printer problems - which still occur, weird -
you then have go, to go and look is - like have I, have I broken the printer or have I messed my code up or
is what I'm seeing actually real? And what he found was that what he saw - the technicians got their first and they were like
well this he's, he's got it wrong again. And they cleaned up his images and took all the sort of dust, that they assumed were printer artefacts,
they were like: there's this smudgy bit here, clearly not meant to be there
we'll just like chop that off or tell the printer. And he eventually had to go the technician and say my prints keep coming back
different from what I expected. And they were like well we're just cleaning them up. He's like
stop cleaning them! I think what I'm trying to see is a weird messy dusty looking thing.
And they were like, oh you want that. I don't know exactly what happened. There's a really good book by James Gleick called Chaos
which kind of tells that story, worth a read. But I'm gonna simulate the dust that he saw. I'm sorry about the lack of brown paper
but there's a reason for the colour.
This is the same orbit thing I was showing you just now, can you see there's a, like a
spirally three there.
But I'm leaving a black trail because I'm telling GeoGebra to colour the screen black if it's stable. And if it's unstable like
that one it goes blue.
Black, blue. Stable, unstable. And I'm just gonna scribble on the screen. Now
this is not gonna make a good picture
but I want it to, I want it to kind of to feel what Mandelbrot felt like. So he was seeing black and blue dots, or in his
case probably just white and black dots. Black for stable and in my case blue for unstable. And he started to wonder
why there's a black region here this, that's not over there? It's not symmetrical.
I really think it must have felt something like this when you have a rubbish printer and you don't trust your code and you wonder
wonder what the hell are you seeing?
So this is going to take ages, right?
I'm going to shortcut the process. What he eventually saw, in higher resolution than he saw, is this picture?
And the black regions, like I was colouring earlier, are stable.
You can see anywhere I go in the black region you get a nice stable pattern with with lovely orbits. And that's the thing
I love seeing move. Bbut if I go over the edge:
unstable.
Stable, unstable.
So it's even stable in this region. It goes into like two piles,
but if I go outside the region it disappears. If I go up here you get into
three piles. And in fact Holly's done a lovely video on how many piles you start making
depending on which of these little bulbs you go into.
But the first time that Mandelbrot saw this he had no idea about any of that structure. He just saw a shape which
shocked him. First of all, it's beautiful,
it's kind of weird,
but getting a viewer to display it, like I've got on the screen now, you
can go and watch any number of YouTube videos and zoom in on it
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 quite the same. And you can just zoom in forever
and now, I don't think YouTube needs more videos of me zooming in on the Mandelbrot set;
however, I'm going to do it anyway.
(Brady: Ben, why are there different colours? Like it's not always black and blue.)
Absolutely. So the original Mandelbrot set that Mandelbrot saw was two colours:
stable, unstable;
black, white - or whatever, I did black and blue. The colours you're seeing in this are an arbitrary decision
but they're not as arbitrary as you might think. So, let's go back to the original which is out here.
Black for stable but the colours indicate that it's unstable,
but the colour indicates how unstable. And so if you think back to the very first question
I asked you, I said square it, square it, square it, and within two iterations you like
no.
Because it's big. And that's an indication that you don't need to ask 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 eventually we proved - mathematicians, I say we. Not me -
mathematicians proved that if you go outside a circle
radius 2,
it's never coming back.
But if you're inside you can't guarantee it. It might be bouncing round near the edge and it might stabilise again. And you see some
of the orbits are complicated. So all you do with the computer is you ask it to check,
let's say 200 times and if you're still inside
radius 2 circle after 200 you're like:
let's colour it black. It's probably stable. But after like 50 iterations, if you've just crossed outside you think, well now
I know it's gone. But maybe the next time you check it takes sixty iterations and you think, well that's less
unstable, maybe I could colour that differently.
That was a long way of saying the different colours on the image are the different levels of instability.
Really, it's how many times you checked before you knew they were going to explode. And what matters when you zoom in
is that the colours change really really quickly.
So all the colours on the screen here indicating that a tiny movement of the original point gives you massively different behaviour.
And that's become what we call the sort of hallmark of chaos theory now, is where a tiny change gives you
fundamentally different behaviour. On the Mandelbrot set you move, your chang- every point you made is
dependent on which C you pick. But Julia set you fix one, and I can turn this software into Julia set mode.
I'll do it slightly first. If I press J on here, there's the circle which is the original Julia set, and
that's in the middle here. But if I move the mouse around you can see the Julia sets with different values of C
start changing beautifully. Like, I mean, it's just a quick sketchy animation
but it's already much more nice than the outlines you saw earlier and
what I love about this is the Mandelbrot set is kind of like a map of Julia sets.
So if you go and find a bit of the Mandelbrot set down here, this is called seahorse Valley,
because it kind of looks like you've got loads of seahorses spiralling around. But if I switch to a Julia set around this point
it looks the same.
But if I zoom out of the Julia set
you don't degenerate into Mandelbrot, you just get the seahorse Julia set. There it is. That's as big as you get.
So if
you zoom in on the Mandelbrot set you get little sort of like regions which are like the Julia set from that area.
It's like a map. It's a geography of, of iterative
stability. When you say it like that it sounds complicated
but you just do something again and again and find out what happens in the long term. And
the fact that it makes something so beautiful, I mean
it's, it's become so cliche for mathematicians to get excited about but I still get excited about it because it's lovely,
I didn't design it, it's just there we can explore it,
and we can just sit and stare hypnotised at it for hours. And you can program it in ten seconds on a spreadsheet if you want to.
[Preview] So as you said, it's not in the Mandelbrot set, right? And what that means, so let's call this number say,
well, I've already used C. But let's call it C anyways