>> PROFESSOR SÉQUIN: This really is the galactic concentrator, OK. If you hold that
in the right spot it will take the power that comes from the black hole at the centre of
the galaxy and you focus upon you and it gives you eternal beauty and intelligence and all
the good stuff. But don't tell that to anybody else.
Hi, were going to talk about the topology of twisted toroids.
Now that sounds a little frightening, does it?
>> BRADY: It does.
>> PROFESSOR SÉQUIN: Right, it is going to be simple. I'm Carlo Séquin and this
is a bagel and if you're familiar with bagels you know we don't just eat a bagel dry. What
you do is you basically slice it, right? You kind of cut it open and then you put cream
cheese on it. For a mathematician, this is a torus. Anything that has a smooth shape
with just one hole, that's called a torus. To a mathematician you can smoothly deform
that in to any other shapes. Interestingly enough if you take you coffee cup with a handle
like that, and even though it has this big void and all of that, topologically it is
still a torus. So to mathematicians, you know, shape is not important, it's just the connectivity.
And something that has one tunnel here or one handle, that essentially is a genus one
surface and they call that a torus.
Now, if we cut this bagel, what do we get? We're basically getting two tori, each one
individually even though the shape is different, you know, still has essentially one ring,
one handle and one hole through it so it's still a torus.
But now, what if we're kind of crazy and rather than slicing this in one nice plane,
we essentially start stabbing through the torus like that and then as we go around the
perimeter we rotate our knife around. And if I'm doing that very carefully and kind
of cutting through and I'm rotating my knife all the way around and then doing it on the other
side, the doughnut will still fall apart in to two individual pieces.
>> BRADY: Is that two pieces professor?
>> PROFESSOR SÉQUIN: It's two separate pieces but they're now linked. Individually,
both of them are still tori. They're still essentially rings with one handle, but now
I can hold one and the other one will not fall on the floor, which is a great advantage
if you're eating those on the train and it's kind of shaking.
And there's another benefit. If you compare that surface in here, compared to the surface we had before,
this surface here is actually bigger. That means we can put more cream cheese in
to this bagel and then when we've put it back together it'll make a much juicer bagel.
To a topological oriented mathematician they're just two tori because in topology we're actually
allowed to have sliding surfaces through one another. Now, if you're a knot theoretician
and you're talking about knot and links then this is different because these two rings
are linked, whereas the other two were not linked. You can separate them, you can move
them apart but you cannot separate those two things.
Now, this geometry is not my invention. This geometry is actually due to Keizo Ushio. He
is a well-known stone sculptor in Japan, well known over there, not so well known in this
particular country. I first met him in 1999. It was at a conference on art and mathematics
in San Sebastián, Spain, and before the conference he had carved a really large, about
three foot diameter, granite torus in a perfect shape like that. And then, during the conference
he was drilling a sequence of a couple of a hundred holes in this spiral fashion through
this granite doughnut, just like I have sliced through the bagel before.
Now, in this case we have a nice hole and that diameter of that hole is just about the
same diameter as the diameter of the rim. So now, when I take these two apart, they
come apart much more nicely and they rattle, and not only that but I can put them into
this configuration which you can see is locking up very nicely. And he then rearranged those
two rings and then essentially left a sculpture there that looks something like that.
And yeah, I wanted to better understand that because I saw it and I understood it but I
felt until I have one in my hand that I can manipulate I really don't know. So, I went
home and within one month I made this object on our own fused deposition modelling machine
which is, you know, a very sturdy rapid prototyping machine. Built it a little bit apart so, just,
it wouldn't touch, just so in this particular shape. And there's kind of a filler material
that would be in the gap when you build it, but the filler material can easily be then
scratched out and you can wrench these things apart and then clean out the surface, all
the filler material. So, I get the really perfect replica of this object and now I finally
understand what is going on.
So, we have two of these interlinked. Could I somehow get three of them interlinked?
And so, I had to think about it for a while, but then it suddenly became obvious. Well,
if you take a different kind of knife, not kind of like, you know, the straight knife
that we sort of rotate around the middle of the blade as we go around the torus. But we
instead take a knife like that, so now we have essentially three sort of half blades
coming together in the middle and I slowly rotate that knife as I push it forward along
the rim of the torus, and going once around the torus I will make sure this rotates exactly 360 degrees.
It's kind of important to do it that way and then
you get this, and you can see those spiral cuts and you don't know what to expect until you kind of shake it loose.
>> BRADY: That's brilliant.
>> PROFESSOR SÉQUIN: And now, we can see we have three of those pieces all inter linked.
Of course, we can do it with three blades, what about if you used four blades? I could
put them at 90 degree and add a fourth one. It's the same as basically using two cuts
with a knife, one going around that way, the other one going around that way. But in either
case we would still turn the whole contraption through a full 360 degrees so every piece
of material winds around the torus and connects back to itself. Well, what would you get?
Probably something that has four rings and in order to make it more visible I gave this
one, not a round doughnut shape cross-section, but I basically made a square cross section.
It's topologically still a torus, just the fact that it is rigid and has flat faces doesn't
matter, and now after the cut, yes indeed, you get something that has four intertwined rings.
You immediately ask, what about five, what about 6? Well, six is a nice one because
now we can start out with a hexagon cross-section, a perfectly regular hexagon, and we cut the
hexagon we say on three diameters. Then each one of the pieces is itself a small triangle
and these triangles are perfectly equilateral, so we can see a perfectly equilateral cross-section
and then six of those essentially form the hexagon that makes this shape and by cutting
it apart, yeah, you of get now, six rings. And that makes an even more intriguing kind
of potential sculptures that we might have and that it makes an even harder puzzle.
That's where I stopped, so now I tried to build one of these, OK, like we've seen before, rotating
the knife a full 360 degrees as we go around the doughnut. Well, if you do it wrong and
you rotate your knife only 180 degrees, and what does that do? Well, somehow this doesn't
want to come apart here. If you look at that then, it still hangs together. It seems to
be like one band, actually it's a band that goes around the loop twice and seems to be
somewhat intertwined. And-
>> BRADY: Is that one continuous piece of bread then?
>> PROFESSOR SÉQUIN: It's one continuous piece of bread, so it's a little harder to,
you know, cram the cream cheese in there. And again, we can look to some sculptures
of Keizo Ushio who has made this kind of shape but much more beautiful and much more regular.
I've exaggerated what he has done in some of his sculptures and you can see, it's basically
a doughnut and you get this one groove that spirals around, it goes to the inside. Now,
I really would like to make a really giant sculpture of this. It would be awesome to
have a sculpture like that where you could step inside and see yourself reflected again
and again in all the different parts of this inner space. So, let's focus on that inner
space. What have we taken away from the torus? We have cut something out and that thing that
we have cut out is actually a Möbius band.
>> BRADY: Aaah.
PROFESSOR SÉQUIN: Ah, what's a Möbius band? Oh, let's see.
If you take off your belt here, right, and you wrap your belt around, you know, that, in
the ordinary way you close your belt is just a cylinder. But, if we give that belt a 180
degree twist so the dull side gets to the shiny side and connect it in that way, now
we have a Möbius band. And what's special about the Möbius band is that if we start
hiking on the shiny side and we go around, we go around, we go around, we go around,
and suddenly oops! We're ending up on the other side. So we have stepped from one side
of the band to the other without going over the edge, by just going once around the loop.
So that's a Möbius span and that's of course a very famous structure. But before, I already
mentioned that we might want to rotate the knife at different rates, so what happens
if you rotate it three times as fast? So as it goes around it swings through three times
180 degrees? And you get this.
>> BRADY: Wow.
>> PROFESSOR SÉQUIN: But now we've taken out a Möbius space that makes three flips.
I call it the Tripley-twist-it Möbius space and I certainly sent pictures of that to Keizo Ushio
and asked him whether he would carve one of those, and so far he hasn't done it
yet but I think one of these days he might come up and create a big stone sculpture that
has this particular shape.
>> BRADY: That looks like kind of a propeller or something, the one on this side,
it looks like that should be on the front of a plane or something.
>> PROFESSOR SÉQUIN: Yeah, or it could be the head of a tunnel boring machine going
[imitates machinery noise] or, but you don't know, this really is the galactic concentrator, OK
If you hold that in the right spot it will take the power that comes from the black
hole at the centre of the galaxy and you focus upon you and it gives you eternal beauty and
intelligence and all the good stuff. But don't tell that to anybody else.
[Both laugh]
>> PROFESSOR SÉQUIN: Yeah, you're not the first one that thinks I'm crazy. Now, you were
asking is this any good? And you know, some architects were fascinated by the concept
of Möbius something, so obviously its 'can you build a Möbius house?'