Hi my name is Henry Segerman. Hi I'm Saul Schleimer. We're going to
tell you about these patterns which are called Cannon-Thurston maps. Or rather,
these are approximations to Cannon-Thurston maps. "Cannon-Thurston map" is not a brilliant name
because it really doesn't tell you what the thing is. But it's the name
we're stuck with unfortunately - so let's try and tell you what it is.
So these are space filling curves, or rather they're approximations to space filling curves.
Maybe there's a simpler one to talk about first which is the Hilbert curve.
The Hilbert curve also is a very complicated looking squiggle. We're showing you how it's
built up as an approximation of less complicated squiggles.
Right, so each of these squiggles - you start with just three sides of a square and then each
bit gets more squiggly and more squiggly and in the limit in the limit - So none of those
approximations are the actual Hilbert curve but the limiting object is the Hilbert curve.
So the Hilbert curve is not any of these piecewise segment things it's the limit,
and in the same way these are not these Cannon-Thurston maps, these are approximations.
Okay so to show you a less squiggly version:
here is an earlier approximation to a Cannon-Thurston map and this here is the
later one. And if we got them the right way around - we did - you can kind of see that
that this corner and that corner matches up and this corner and that corner matches up.
Now, you might be thinking I'm an idiot because there obviously has a symmetry around the center
but if you look very carefully this is dark in the inside and that's light in the inside.
There's also the question of what space do these tile - or what space do they fill?
So the Hilbert curve from before - that filled up the square whereas these curves seem to be filling
up this really complicated looking object with a fractal boundary. But in fact if you kind of put
things together you can kind of see that - put it next to it next to each other - you can see
that this wants to plug into that and that wants to plug into that so these they actually tile.
So what these things want to fill is the whole plane not just the square.
So the Hilbert curve has a nice sort of symmetry: if you cut it into quarters
and you zoom in on one of those quarters what you get is an exact copy of the original thing.
So for the approximates that's not quite true. If you take one of the approximates and do the same
trick - divide it into four pieces and then rescale - you won't get the same approximate
back. You'll get the previous approximate. But in the limit what you actually see is that the
entire square is filled up but the way in which it goes through that square is exactly the same on
the original and in this cut down rotated version. Now the C-T maps, they're also space filling -
they're surface filling curves, and they do have a similar property to the Hilbert curve. It's - it's
harder to see so if you look at the detailed one and you look at the pieces you can see that these
colored pieces they kind of have the same shape. And those - they're not the same as the big piece.
Well it's not the same just under a euclidean scaling. It's a different kind of transformation
called a Möbius transformation, and that's sort of a hint that maybe there's some hyperbolic
geometry going on here - which there is and which is really where these come from.
So there's an important point there which is - so the Hilbert map, the Hilbert curve - it's
a little bit artificial. Right you make some choices, you're careful, and you get yourself
a space-filling curve right - and there's plenty of other examples which are sort of cooked in the
same way. Whereas for the Cannon-Thurston map you make one initial choice, which is "I'd like to
think about the figure-eight knot today, please", and then you get a single Cannon-Thurston map
which is forced upon you by hyperbolic geometry. So yes so these Cannon-Thurston - not that
one these - we'll get back to this one, these ones come from the complement of
the figure-eight knot. So the figure-eight knot is a knot and as usual in mathematical knots,
the ends are fused together so it's just a circle - it's a loop of string - and we're interested
in the complement of the figure-eight knot. So everything that isn't the knot.
So on the outside, if we want to try and understand the knot and understand its complement,
one way to do that is by a spanning surface. Think about the knot as being made out of some stiff
wire, we dip the whole thing in soap film, and then we poke out some of the panels until we get
a nice surface - a nice surface whose boundary is that thing. It's exactly the knot, yeah. And this
particular surface you can sweep it all the way around the entire complement and get back to the
start. Right so this isn't always going to happen - right if you just have some knot or whatever and
you have a surface that spans it, it may be that you get stuck somewhere. In this case, you don't.
So what we want to do with the surface and the complement of the knot is we
want to lift them to the universal cover. We want to unwrap the topology, both of the
surface and of the knot complement, to get no topology left and just have geometry instead.
So we had this video on cohomology fractals which sort of goes through what the universal cover
is in detail, so go and check that out. But yeah in brief it's like - anytime
you had some - so you can wander around in your space and come back to the start
and what you want to do is sort of make a new space where it's unwrapped - so whenever you think
you're getting back to the start you're actually in a whole other place - walk to some other place.
So you see lots of copies of stuff. Right so, we're unwrapping both the component of the knot
and this surface. When you unwrap the complement of the knot you just get
three-dimensional space - it's kind of like a big ball. A big ball, right. And when you
unwrap the surface you get a big disk. And the big disk lives inside the big ball,
sort of as a - you know you might imagine as being an equatorial disk. But it turns out that
this disk is kind of crinkly right - it's super crinkly. It's super crinkly. So in fact
it crinkles so much that, well here's a way of talking about the crinkling. If you're standing on
the center of one of these disks and you look out like distance, five or something right, then you
see a certain amount of crinkling. And if you look out distance ten then you see much more crinkling.
So that's one way to think of these approximations right - it's like how far
out you're going to go - and when you go when you're looking all the way out to the boundary
of the disk the boundary of the disk is going everywhere in the boundary of the three-space.
Infinitely crinkly, and so then it just naturally fills up the whole two-dimensional
boundary of this three-dimensional ball. And there's some sort of delicate relationship
between the fact that the knot complement was fibered - the surface went all the
way around - and the fact that the curve, the boundary of the disk, goes everywhere.
Right, you could imagine taking some different kind of surface which is just not
as crinkly and then you would expect that as you take the disk further and further out you're just
going to see a circle on the boundary sphere. So there's a beautiful sequence of pictures from
a paper of Bill Thurston's where he shows this kind of process - where you start with a disk in
hyperbolic 3-space and it's boundary is a round circle. And then he makes the disk a little
more crinkly, and a little more crinkly, and a little more crinkly, and in the final figure the
curve actually is trying to go everywhere. But you'll see that his picture doesn't quite
look like our picture. Right, I mean he was doing it 40 years ago.
The computers have nothing to do with it. So first of all he was an amazingly good programmer. It's
not that we have more powerful computers - it's we have a different algorithm for
examining where the disk goes - where the boundary of the disk goes. And you can compare his picture
to ours and you can see that we are investigating in a different way. So what he's doing
is he's investigating the boundary of the disk where each segment of the boundary of
the disk has length more than epsilon. Whereas what we're doing, well, what are we doing?
Well we're doing something else, involving something called "veering triangulations" which
I think we're not going to talk about. But it sort of gives you better control of -
one of the problems with generating these sort of fractal images is if you're not careful about
how you're doing the work you're you spend forever trying calculating stuff inside of a single pixel.
Whereas with these sorts of things were sort of being careful to spread out the work that
we're doing so that we fill up the space. To give him his due, Thurston did the same,
he also was spreading out the work but he wasn't spreading out
the work in the same way that we are. So these ones here, they come from the
figure eight knot, but that one - does not. There's still a three-dimensional manifold
but it's not a knot complement and there isn't even a surface inside of it. So there's no disk to
get the universal cover of to get the boundary of. Okay, there's kind of a way to think of this
example, s227 as it turns out - that's the name of the manifold - in a similar way to these examples,
because what's going on here is there's a surface in the knot complement which sweeps around and
then you lift it to the universal cover and you get a collection of surfaces, a collection of
disks - which sweep through the whole thing. So here there's no surface downstairs in the
manifold but there are a collection of disks in the universal cover. So you have to carefully
build those and then you get these boundary patterns - via the same sort of process.
If you want more details on that then - we have an upcoming paper with Jason Manning which will say
why this all works - and we'll explain where the pictures come from.
My brother Will made these carvings using a CNC mill with a v-bit. The "v"
is referring to the cross-section of the bit. The material is a white and blue PVC laminate,
and strictly speaking, the curve which separates here the white from the blue is the approximation
to the Cannon-Thurston map. Thanks for watching!
um