DR HOLLY KRIEGER: So what we're gonna talk about today is something pretty close to my own work,
and it's this question that has arisen about
asking about iteration, right?
So you and I have talked about like the Mandelbrot set and
all of these things that I'm really interested in that have to do with iteration,
and how it interacts with fractions with rational numbers.
Alright, so, here is where we start,
which is where we seem to always start,
which is z² + c.
So I'm interested in the function,
which takes in a number z, squares it,
and then adds some constant c.
And so, we've looked at this kind of thing before
but just to take a quick look at some of the behavior that occurs,
so, for example, like if c is equal to 1 and let's say we start with the number 0,
then if I repeatedly apply this rule, if I iterate the function,
then let's see what happens.
So 0 maps to 0² which is 0, plus 1,
so that's the first iteration.
Now I take 1 and I put it right back into the formula,
so I square it and I add the value, which is 1,
so 1 + 1, that's 2.
I take 2, I square it, and I add 1,
that's 5. I take 5, I square it,
I add 1, that's 26, and so on.
BRADY: Oh, you're not gonna do the next one.
HOLLY: I think it's 677, but... [laughter]
BRADY: You check... alright...
HOLLY: You can check later. [laughter]
HOLLY: Okay... BRADY: Okay, so that's "getting big" time?
HOLLY: So it's getting big, right,
yeah, so this is actually a really good point.
This is exactly what I want to talk about,
is that if you just randomly do this, choose some number c...
Choose some numbers that to start with and iterate under this formula,
kind of what happens a lot of the times with a lot of the ones you try,
is these get really big,
or maybe they get really small, but in any case,
what often happens is when you do this process
you end up with different numbers at each stage.
You never come back to 0, for example, and so...
If you do though, so let's see an example where you do.
So if c is equal to 0, for example...
That's supposed to be a 0...
So c is equal to 0 and I put in 0,
so now we're looking at the map which takes that,
and just maps it to its square.
Right, then if I put 0 in,
Well, I square it, I get 0 right back out.
And so I don't get this nice infinite list of numbers.
I just end up back right where I started.
So this is called being periodic, by the way.
A number is periodic for this function
if when I put it in and then iterate like this,
I end up back where I started.
BRADY: I mean you gave a really easy example there.
Do we get examples where it can be like 30 steps and then it comes back?
HOLLY: So this is exactly the question I want to ask!
So, okay, just to do a few more examples before I get to 30. [laughter]
Hmm, looking at this case of c equals zero, just to make it clear,
I don't have to choose 0 here, right?
Like if I put the number 1 into "z maps to z²", then 1 maps to 1², which is still 1.
So it's possible for something to be periodic,
even if it's not straight starting at this point 0.
Okay, so here's the question,
which is exactly what you asked, which is
can we keep doing this but in a more interesting setting?
So not just returning right away,
but after, say, 5 steps or 30 steps or whatever
coming back to a number that we started with.
And to make this question interesting,
I want to be able to do it with a number that's easy for me to write down,
i.e. with a fraction.
So that's the question I wanna be able to answer.
BRADY: Isn't an integer easier than a fraction?
HOLLY: An integer is easier than a fraction,
but we're going to run into trouble right away with integers. [laughter]
So there are not that many examples with integers,
and you can kind of see why if you look at the z² example.
So what I did works for 0 and 1,
but for any other integer...
okay, forget minus 1 for a second...
any other positive integer,
if I plug it into this, I'm gonna get a number
which each time is larger and larger and larger and larger,
so I can never come back, I can never have a periodic integer other than these ones.
So, yes, it would be simpler, but it's so simpler that it's not going to happen. [laughter]
BRADY: Alright, you can have fractions then.
HOLLY: Alright, thank you. Okay,
so the first thing to point out is that it can happen with fractions.
So let's say c is equal to 1/4.
Now if I start with 0, what happens is, well,
I've got 0² + 1/4,
(1/4)² which is 1/16, plus 1/4,
which I guess is 5/16.
And anyways, it's not as easy,
it's not as obvious in the case when c was equal to 1,
but 0 is not going to come back to itself.
We're gonna get more and more complicated fractions each time.
And so we can't get back to zero,
but if instead we start with z equals 1/2,
then if we take 1/2 and we square it,
and we add 1/4,
we end up right back where we started.
So we found another example of something that
just comes right back to itself
but is a fraction,
so it can happen for fractions,
and the good news is that it's not just
coming right back to yourself immediately that happens.
So if, say, c is equal to...
-1, then actually zero does this
because 0² - 1 is -1,
and (-1)² - 1
takes us right back to 0.
So the number of integers that we see here or fractions that we see here
can be higher than 1, at least.
So here we just have a number going back to itself,
so we only see a single number when we iterate.
Here, we see two numbers when we iterate.
It's possible to see 3.
I think it's c is equal to -29/16,
and you start with
z equals -7/4,
which we've actually talked about this number before.
It's kind of a tricky one in dynamics.
BRADY: That is your first video!
HOLLY: I know, I know, reminiscing about the days.
Alright, so,
if we start with, say, z equals -7/4...
Oh man, I really don't want to do these computations, but let's do it.
So -7/4, when I square this I get 49/16,
and then I subtract 29/16.
That's 20/16, otherwise known as
5/4.
Now when I take 5/4, and I square it,
I get 25/16, and then I subtract 29/16,
I get -4/16,
otherwise known as -1/4.
And if I remembered the values correctly, [laughter]
when we do this to -1/4, we should end up back where we started.
So if I take -1/4 and I square it,
I get 1/16, and then I subtract 29/16,
what I end up with is -28/16,
which hey, okay, good, really is -7/4.
BRADY: Where you started. HOLLY: Where I started.
So, for this particular value of c,
we can have three fractions when we iterate:
-4/7 goes to 5/4, goes to -1/4,
which takes us back where we started.
BRADY: Alright. HOLLY: Okay.
BRADY: Four. HOLLY: Four, [inaudible]. [laughter]
That's it. We can't do it for four,
we can't do it for five,
we don't think we can do it for six,
and we have no idea about more.
BRADY: Also, four and five...
proven can't be done.
HOLLY: Proven can't be done. BRADY: Right.
HOLLY: But for higher than that,
we don't know for sure.
So your question initially about 30,
I have no idea how to answer it.
If I had to take a guess, I would say no way.
BRADY: Two and three, are there an infinite number of ways to do it, or...
HOLLY: Oh, that's a really good question, and the answer is yes.
Yeah, so there are an infinite number of ways.
There's a really complicated formula in the case when you're taking three,
that tells you exactly which values of c and exactly what values of z to achieve this,
but there is a way to do it.
BRADY IN VOICEOVER: Have you caught the Numberphile Podcast yet?
Our latest guest is Holly, who you just saw in this video.
You can find out more about her career life and mathematics;
how she got to where she is.
It's a really interesting listen.
HOLLY IN PODCAST: Try not to read YouTube comments. [laughter]
The more important thing is I always tell my family members
not to read YouTube comments. [laughter]
BRADY: What do they do, do they email you and say...
HOLLY: They did on the first video and then they promptly stop. [laughter]
BRADY IN VOICEOVER: And if you'd like to see more videos with Holly,
there're links to that on the screen as well.
HOLLY IN INTERVIEW: And the fact that we just don't know.
I mean, I get frustrated by that fact. [laughter]
But I think it's really interesting about, like, sort of,
what is a hard problem in mathematics.