HOLLY KRIEGER: So what I want to talk about
is what is maybe
the least efficient way possible
to approximate the number pi.
BRADY HARAN: Well, that's not a good sell!
HOLLY KRIEGER: It's not a good sell
but it's really interesting.
So I wanna talk about approximating pi in the Mandelbrot set
Maybe it's best to do a bit of a reminder about
what the Mandelbrot set is.
So, remember, the Mandelbrot set
was this object that we have in the
complex plane.
So, here are the real numbers.
And here are the imaginary numbers.
And the Mandelbrot set was a collection of
complex numbers, so I'll try and sort of
draw a nice version of this here.
And-- it's complicated, right?
And it was this collection of
complex numbers that had the property
that when we look at the function
z squared plus c and we apply
it to zero, so zero plugged into this function
is c, and then we do that again
with the output.
So c plugged into this function is c squared plus c .
And so on.
A number c is in the Mandelbrot set exactly
when when we do this process and repeat and repeat and repeat
The number stays smaller than or equal to two.
But today I'm not too worried about complex numbers.
I just want to talk about real numbers in the Mandelbrot set.
BRADY HARAN: Does that mean we're just talking about things along the line?
HOLLY KRIEGER: Exactly, exactly.
And in fact I'm just gonna be interested in what happens
On this portion of the real line here.
So, one thing that I should say first
is that this point, this point is called the cusp
of the Mandelbrot set.
And this point is exactly the value where
c is one quarter.
BRADY HARAN: That's the last point on the real line-- in that direction.
That's in the Mandelbrot set. And pi is here somewhere presumably.
So pi is not in the Mandelbrot set.
HOLLY KRIEGER: Pi is not in the Mandelbrot set.
That's absolutely right.
And when I say that I'm going to approximate pi in the Mandelbrot set
I don't mean I am just gonna approximate pi on this picture here
I mean that I'm gonna cook up a bunch of numbers that have to do
with the Mandelbrot set
that approximate pi.
BRADY HARAN: So you're going to use the Mandelbrot set as a tool.
HOLLY KRIEGER: As a tool to approximate pi, exactly.
And, of course, you know, looking at this you can say "well, it looks
like there's a circle in there".
Which, indeed, this is a circle.
Inside of the Mandelbrot set.
But I'm not gonna use that one,
that's a little too straightforward of a way to get pi.
So, here's the idea.
What happens if we take a real number
That's larger than one fourth.
So, as you said, it's not in the Mandelbrot set.
Right?
And what that means-- let's call this number...
Say... uh....
Well, I've already used c, but let's call it c anyways
Whatever this number c is, when we start doing this process
for this number c, we look at the function z squared plus c
and we plug in zero and we output c.
Then we plug in the output
and we get c squared plus c, then we plug
in that output and so on.
At some point, these numbers are gonna be larger than two.
Right, because that was our restriction
that you're not in the Mandelbrot set if these numbers
are eventually larger than two.
Okay?
So what we're gonna do is to each
number c
we're gonna associate this number, say, N of c.
And this is equal to the number
of steps of iterating zero under z squared plus c
--So exactly the process I was describing before--
Until we get something larger than two.
BRADY HARAN: Any number after the cusp?
HOLLY KRIEGER: Yes.
BRADY HARAN: Is a killer? It's gonna kill the Mandelbrot set?
HOLLY KRIEGER: That's right, it eventually gets big.
BRADY HARAN: The thing you're interested in now
is how quickly that number will kill us?
HOLLY KRIEGER: Exactly. And this is something
that's totally natural to be interested in
This is how people draw pictures of the Mandelbrot set
for example, is exactly counting these kinds of steps.
So if we start with the real number c that's larger than two
then after the first step
of iteration here we get c itself which is larger than two.
And so we know that after just one step
that that parameter cannot possibly be in the Mandelbrot set.
So in that case, N of c is just equal to one.
And we have this very small value of N.
BRADY HARAN: And that applies to all numbers above two.
HOLLY KRIEGER: That's right, but as you might
imagine, the closer you get into the cusp, the more
steps it takes to get larger than two, right?
We're starting with the smaller number here and so it takes
more steps to get there.
All right. So let's make this a little bit precise.
We start with a number really close to one fourth, say...
Say, c equals one fourth plus epsilon.
Where this is some very small positive and real number.
BRADY HARAN: So we're just after the cusp.
HOLLY KRIEGER: That's right, you're just to the right of the cusp.
On the real axis there. And then we
count the number of steps it takes us to escape
This is something you can do in, say, you know, Wolfram|Alpha or Sage or whatever.
Can be the program you wanna use. So if you draw
a table of how this number N of c depends as epsilon
is coming in and getting smaller and smaller,
we're coming in towards the cusp with c
you can actually count, just by iterating, how long does it take to have
a number that's larger than two.
So for example, if epsilon is one, then c is one plus one fourth, which is one point two five.
Then N of c is just two.
It only takes two steps to get larger than two.
BRADY HARAN: Well that was, uh, that was over quickly.
HOLLY KRIEGER: That's right. On the other hand if we
take some much more reasonable value of epsilon, say like
One one-hundredth, and so c is point two six, say...
Then the number of steps it takes to escape turns out to be
Thirty, actually.
And if we take epsilon to be, say
Even smaller, and so c is even closer to one quarter, then the value we get in that case is three fifteen.
BRADY HARAN: Oh, wow, that's a lot of steps.
HOLLY KRIEGER: It's a lot of steps, right, but we're really
close to the cusp here, it should take a lot of steps to escape.
And if we take epsilon even smaller, so that c is even closer to the cusp
We get something even better.
Even larger value of N of c.
BRADY HARAN: Ooooh...
HOLLY KRIEGER: But you can see something suspicious happening here, right?
BRADY HARAN: I can see it. We can all see it.
HOLLY KRIEGER: Everyone can see it.
So, what's happening here is as long as you put the decimal
point in the right place
And as you can see, I've changed my epsilon sort of regularly here, so the decimal
point kind of moves regularly, too.
As long as you put the decimal point
in the right place, these values N of c
are actually converging to pi.
BRADY HARAN: That's cool!
HOLLY KRIEGER: Yeah, yeah! So you get these approximations of pi
right, okay, three point one five is not such a good approximation
three point one four is a little better, it turns out the next one
is something like three one four one four.
Which is even better
and so on.
And so, this tremendously inefficient method of taking some value
of c that's really close to the cusp of the Mandelbrot set and iterating
it many thousands of times until you get a number that's larger than two
will give you approximates of pi.