Brady! The title of this video is:
Tentacles Akimbo!
Just because I've wanted to do something that
sounds not mathematical, but is!
Glassblower, Lucas Clarke, comes over to my house and says,
"Cliff, I made something and I can't understand it."
And he shows me this.
It's two glass tubes, put in a ring.
And you know, you...
treat it as eyeglasses, or you treat it as two rings linked together like this.
He says to me, this is two rings connected.
I said, yeah, it's mathematically a two-hole torus.
He says "okay",
then he reaches over his
bag and he pulls out
this.
And he says,
"Isn't this the same thing?"
Then I said,
"Yeah, you've just taking this,
stretched it,
and so, this is the same as this,
you know, it's still a two-holed...
torus, with two holes in it."
Topology does not care about distance, it doesn't care about size;
it's as if you make it out of stretchy rubber.
Okay, so he says, "What if I...
inflate the middle of this?"
"So I'm going to, just, blow air in, inflate this."
And I said, "It should make no difference in its properties."
So he pulls out this.
Okay, from here to here.
It still has
one, two holes and it's connected here.
Well, it would be called a two-hole torus,
or could also be called a
handlebody with two handles on it. Okay, no problem.
So,
I'm looking at it, thinking about it.
He pulls up the next thing, he says,
"I can take this
handle over here, go...bzzzz!
and bring it over there."
And I say, "Yeah, that should be no problem."
So,
he does.
So it goes...
over to there.
So the next thing this glassblower does is to say,
"What happens
if I move this
connection here? Just carry it around over to there, and this connection over to here?"
That's perfectly legal according to operations of topology.
You could move this thing where you want as long as it's still connected.
So he says, "I'm gonna take that
connection and go...ziiiip!
and this one go...ziiip!
like that."
Okay.
So he does.
So he moves those two ears from being next to each other, parallel, to...
so that they're connecting like this.
(Brady: Okay, yeah. All legal so far?)
Everything's legal. No problem at all.
He says,
"Suppose I shrink this sphere part now."
Well, that's easy.
Shrink it from that to this.
(Brady: Oh, what have you done now?)
So that,
becomes that.
And this one,
becomes this one here.
Helps to have a big nose.
And so this
sphere part becomes this wide part here.
(Brady: Where'd that new hole come from, though?)
What new hole, this new hole here?
Where is it from?
He starts asking me where is that from,
and then he goes one step further and fully shrinks that
into this.
Wait a second,
he's telling me that this
can be transformed into this,
following ordinary rules of topology.
I'm looking at all this he's saying can you explain it? And I'm saying: well,
go back to this.
This is a lot like a human being - it's like me
standing with
my arms like this
There's a hole there and a hole there.
This is
akimbo. Here I am akimbo.
And this,
is the same as me with my fingers linked together like this
Yeah, there's a hole here,
there's a hole here, and there's this space here that my head's waggling around in is the same as this space here.
Right. Here's that space,
is the same as
this space here. - (Brady: Okay, yeah)
My fingers are like this
And so I say, somehow
this has been transformed to this,
which is the topologically the same as going from akimbo,
to
interlinked fingers.
So I'm sitting there, I'm thinking I ought to be able to go from here
to this. And that night I'm in bed trying to go...like this.
Then my wife nudges me and says "cut it out Cliff, go to sleep already!"
And I keep going like this, waggling my fingers around and it's not working
right and I'm saying what's wrong with this? And, and, and, and, and -
a couple days later, visitor comes by - three visitors - from Denmark. Oh, people are oftentimes coming over to my house to check out
topology and things, and I show this to them and these three grad students -
I think -
from Denmark came through and I showed this to them and I said I'll give you a Klein bottle if you can solve it.
They sit around thinking about it, talking about it, and a couple days later
they show me the answer and I'll show it to you now. I'm going to need a little bit of help.
I'm going to need a couple
tentacles.
Can I go from one arm akimbo,
to this?
Well, I'll try it. The way I'll do it is I'll put a - put my finger here and I claim that: going like this
is the same
topologically, as going like this.
This,
this tentacle is
to just give me a longer finger, so I can go like this, and
round like this,
over to here,
to my thumb, and then move my thumb along
to this.
Got it. Thanks to the wonderful miracle of tentacles,
I can show that I can go from here, to here.
Okay, on my left hand I can do the same thing
Got it. With two hands I can go from here, to here. So this is a preserve. You know, the number of holes -
one hole, one hole - has been preserved.
So, what about -
what about
with two hands going to this? I'll start with this.
And,
I'll start over here and
this hand here. I'll go around
over to my nose.
Right, there's my nose, I'll now take this finger over here and bring it to my shoulder
like that.
I'll now reach around,
around this way...
See the end of this?
See, I have to keep this guy touching.
I can move it around,
till it gets to my thumb,
and get it like this.
See how I've done that? Now, I can move my fingers together, and I've got -
get rid of that - I now have this shape right here. I've managed to bring my fingers around
to enclose this one finger. Similarly, having done that I can take this finger,
my left index finger,
around
and, go round this way.
Bring it like this, before you're there - like that.
No, is that cool or what? In other words, I can transform from
arms akimbo
into
interlinked fingers, with the aid of
a
couple tentacles.
And by extrapolation I can do it, if only I had longer fingers. If I had two longer fingers
I could do it entirely. All of which to me, to me says
topologically, oh, there's cool things you can do with
tentacles akimbo. But wait, there's more! Brady, there's more!
Go
[Student: ] Okay first I'll show the lemma. If you start with your finger on your chest,
you can extend to your thumb, by following along your arm,
and this pointer finger can go all the way down here, up here, and connect right here with this thumb, to make this.
This is very useful. Watch. You have this, we can extend this to just fingers - just like the ball, and get to this.
Then, applying the lemma I just showed you can go all the way up your arm,
to get this.
This can then be applied on the other arm
by following your finger all the way up and then connecting with your thumb and then you have the desired result