Okay, I want to show you how to fill a Klein bottle.
Can a Klein bottle be filled? Does the Klein bottle have volume?
You ask a mathematician, and they'll say: Well...
...a Klein bottle doesn't really have any volume. Uh, it's a topological shape.
This doesn't have volume! But wait a second. It looks like it has volume
Well, if you ask a topologist, does a measuring cup have volume?
They'll probably tell you, no
Does a sphere have volume?
Yep, no matter how... (STRUGGLES WITH WORDS): you know, you know...
4* PI* R^3. Whatever. It has a volume
It divides the universe into an outside...
[EXCITEDLY TAPS THE PLASTIC BALL] ...and an inside.
And no matter how do you stretch and contort it. If you shrink it if you make it bigger,
It'll always have a volume
Oh. A piece of paper.
I can make it into sort of a cup somehow or another
But it'll always [TAPS PAPER] occupy very little volume.
Well... (STRUGGLES WITH WORDS): you know...
in the limit none!
Likewise a bag right? I can put stuff into
I can put stuff into it
I can even put stuff that has volume into it
but
(Brady: It doesn't divide the universe) it doesn't divide the universe into two parts if I were in outer space
all of these grapes
would flow out
A bag doesn't divide our universe into two parts
This is related closely
to a two-dimensional problem called the Jordan curve problem which asks a cool question of
is there a difference between
That curve and
this other curve? This encloses an area
this doesn't enclose an area. Turns up to be a
surprisingly difficult thing to prove and the mathematician Camille Jordan who finally solved it was a real smart
character. In the same way
to a topologist the question is
does something divide the universe into the outside and the inside?
A Klein bottle
even a
three-dimensional Klein bottle an immersion like this. Well the air on the inside of it can easily escape to the outside
sort of and
this cup, this measuring cup which says Oh
500 milliliters over here, a physical volume
to a topologist
they just melt it down, flatten it out and say oh this is homeomorphic to a disc
It has no volume. A disc has no volume
So be careful when you talk to mathematicians in the kitchen
Don't don't don't don't let them near your recipe book
having said that you know a sphere has a volume
a doughnut, a torus has a volume
but a Klein bottle doesn't have a mathematical volume. Got that. What about a physical volume?
Well this obviously, you can put, you know
In America, it's possible to put 16 ounces of water in it, everywhere else in the world
you could put 500 milliliters into it. Got that so if a Klein bottle
doesn't have a mathematical volume. Does it have a physical volume? Well the way I make these
The answer is well sort of yes. You can fill a Klein bottle
with water and
throw some dye in it also. You can't pour that out? Well, yeah, I can pour it out
Let's pour some water out go through here. Glug glug glug
Pour it out. Out it goes drop by drop
Essentially it's vapor locked it won't. So yep
I can get water in I can get water out
So I lack the measuring cup your Klein bottle in the real world only has volume depending on its orientation in gravity
That's right. Another way of thinking of it is this
measuring cup depends on gravity to hold
to hold water.
In the absence of gravity or if the gravitational field is inverted, it doesn't hold water anymore. Well, I got that
likewise a Klein bottle
but a sphere
Hey, this is if you put water in here, it'll stay in there no matter what, so
How do you fill a Klein bottle?
Well
step one
For me, I'll put a drop of food coloring in Boop
Which is way more than you need and a little bit of water in, a few drops
It'll go right through. Notice my fingers have already been dyed
So, okay. I got some, got some water in there
Problem is every time I pour water it vapor locks
And I don't only get sort of this much water in so come on over. I'll dunk it underwater and show you
how to fill a Klein bottle with water along with a little bit of blue in it
dunk it under water and
wiggle it around. Each time a bubble of air comes out
and with each bubble of air
a little bit of water comes in. Let's get a bubble of air
And held under water long enough. So you're doing like a sort of a 360 yeah, to get the bubble out. Yeah
And if each bubble is about a milliliter or two
eventually
with each bubble leaving
Conservation of something tells me that a bubble-sized bit of water is going in
It's still pretty slow. Oh, come on. It's not slow. This is fast compared to the age of the universe. This is speed compared to
even something is rapid as global warming. This is fast stuff here
not just fast, but you know compared to the rate it takes to make a Klein bottle it's
enormously fast
You could use an aquarium pump you could use a water pump and so on but. Here's a Klein bottle that's completely filled with water
And if I lift it up
at first it'll
Notice now when I take my hand off the bottom a little bit dribbles out but not too much
Can you put water in a Klein bottle? You betcha in our
physical universe
problem, of course is
You know water
can also be
taken out
the real problem though is this
Once you get water in there
Yeah, you can get almost all of it out by tilting it and sort of doing the opposite of what you did before
Tilt and pour tilt and dribble out. Yeah that works so you get water in
Kind of shake it out. The problem is the last few drops of water. Ah
There's not much air circulation inside of
a physical Klein bottle as a result the last few drops of water that surface tension holds there
Do I have an example? Probably not very visible on the video, but there's droplets of water in here. I can see that, yeah. You can! Yeah
Well getting the last few drops out
Means overcoming surface tension
Hard to do
so
to remove the last few drops
I'll let the sit in a dish drain for Oh
An hour or three. A little bit of water will collect there. I'll pour it out
Then you know, get it nice and dry on the outside
the little bit of water still in there, I'll either
pour acetone in swish it around pour it out and then let the acetone evaporate. Pour alcohol in
that'll work or
Intubate it with an aquarium air pump and pump air through it let the water evaporate
This
and he says
Isn't this the same thing?
That's why you do the astronomy after all. It's not gonna make you rich it's not gonna make you famous
It's not gonna get you girls didn't work for me
What it does do is it gets you closer in touch with the universe.