Math ain't about numbers.
If you think math is about numbers, you probably think that Shakespeare is all about words.
You probably think that dancing is all about shoes.
You probably think that music is all about notes.
Math ain't about numbers.
Math is about logic, it's about beauty, it's about connections, it's about
how you get from one place to another.
And for me,
the cool thing about math is a part of it, called topology.
I'm not even gonna tell you about topology because what I'm interested in
is a little section of topology called Non-Orientable Manifolds.
More specifically, I'm interrested in Klein Bottles.
This is a Klein Bottle.
Wait a second. Wait a second.
We haven't even touched upon a Möbius Loop.
How can I talk about a Klein Bottle without talking about a Möbius Loop?
I take a piece of paper
As everybody knows, if I have this piece of paper, it has two sides, four edges.
Turn around this way it becomes a cylinder, give it a half-twist, and a piece of tape.
Becomes a Möbius Loop.
Well, the cool thing about a Möbius Loop (then there is about fifty or sixty cool things about Möbius Loops)
is that, it's got one side, it's got one edge.
Take two Möbius Loops, one left-handed one, the other right-handed one, and I try to connect them.
Hey! This one has one edge, this one has one edge.
What happens if I take two Möbius Loops and sew their edges together?
Well, I know from experience,
that if I have four edges here and four edges here and I connect them.
Oh! I'll get something that has fewer edges.
So if I apply that to a Möbius Loop. I got one edge here, one edge here, sew them together.
I should get something that has... oh.
I'd lose this edge, I'd lose that edge. I ought to have something with no edges.
Cool.
If I take two Möbius Loops, hook them together. I'll get something without any edges.
Okay, I'm gonna try it.
HOT DAMN
far out
Here is a Möbius Loop, here is another Möbius Loop.
The cool thing is that if I take two Möbius Loops and try to sew them together.
I end up with a shape that requires four spacial dimensions to exist.
In that shape it's called a Klein Bottle
Named after Felix Klein! Felix Klein! This guy in 1882 came up with the absolutely nifty idea,
that if you can take a cylinder, turn it around, and orient one end of it, and loop it through.
So, his idea was: take a cylinder, bring it through itself, and have the end of the cylinder welded to a base.
That this would have the cool property of having only one side. What do I mean by one side?
If I have an ant crawling around here. [The] ant or caterpillar crawls around.
Oh, clearly it can cover this whole thing.
It can then slide continuously, over there, through this tube, and get to the other side of this piece of glass.
We never cross an edge to get to the inside. Absolutely nifty.
In an ordinary... Let me get an ordinary bottle.
Ordinary bottle.
Well, if I take—
if I leave the cap on the bottle, clearly it has two sides.
Here's an outside...
and the inside of the bottle is where I'm touching right now.
An ant walking around here couldn't get to the other side.
If I take the top off, watch this.
An ant walking along here can get to the other side, but he has to cross
a peculiar location called an edge.
If I make this bottle
arbitrarily thin,
this edge becomes sharp,
and the ant cuts— cuts herself when she tries to walk from the outside to the inside.
In other words, an ordinary bottle
has two sides: an outside that I'm touching,
and an inside that my finger is touching right now.
And they're separated by the lip of the bottle.
And, okay, we all know this.
A Klein Bottle, meanwhile... watch this.
You'd walk around here just like [you'd] walk around here,
but the other side of my finger
I can get to without ever crossing a sharp edge,
by carefully walking along, never crossing an edge,
going through this tube,
and getting to the place that's just on the other side of my finger,
without ever crossing an edge.
In other words: this is a bottle without an edge.
BRADY: Cliff, I'm comfortable with that, except you just told me that this wasn't
possible in three dimensions.
CLIFF: If we lived... Aw! If we lived in a universe with four spacial dimensions, four-space,
this tube wouldn't have to intersect itself
This weird place where there is self intersection, right... *hard k* ...there.
That would not self-intersect, and an ant walking along this way would go:
*boop* *boop* *boop* *boop* *boop* *boop* *voop* *voop* *voop* *voop* *voop* *voop* *voop*
and would go right through this piece of glass right here and keep walking to the inside
And an ant walking around this way:
*dg* *dg* *dg* *dg* *dg* *dg* *dg* *dg* *hard K*
would go right through this and keep walking this way.
BRADY: Cliff! So that hole you've created there so the tube can get through is like the floor?
CLIFF: This is worse than the floor. It's this place right here where the manifold... where the glass intersects itself
It's a figment of living in a three dimensional universe
If we live in four-space, we could go right through
On the other hand, that guy Felix Klein showed that in three dimensions you have to have this intersection
In a quality universe that has four spacial dimensions, Klein Bottles would be terrific!
Here, unfortunately, my Klein Bottles has to have this self intersection.
When I make Klein Bottle hats... Right? Here's a hat that is the same thing as that glass Klein Bottle.
Instead of welding it together, I've knitted this hat so that the peak of the hat comes through itself.
So that the hat itself can... the loop of the hat can be pulled through itself
and turn it inside out. But I'd rather not do this because it takes about five minutes to do
and I'm stretching things that shouldn't be stretched
but ultimately, I ought to be able to pull this sort of inside out. It's continuously deformable back to itself.
BRADY: Cool!
CLIFF: And, along with a matching Möbius scarf
Ah!
I'd like to say I'm ready for a California winter
Look! Look! Look! Look! Look! Look!
[A] friend of mine, Robert Leng, made a paper origami Klein bottle.
I mean, absolutely sweet. One sheet of paper: it's an origami Klein bottle
One of the neat things about mathematical topology is that you don't care what size something is.
I could expand this, contract it. I can make this... I could stretch it and it'll still have the same mathematical properties
Same number of sides, same number of holes, same number of handles.
Instead of making a big one, I could make it smaller.
I could make an even smaller Klein Bottle.
I mean... I can make a Klein Bottle even smaller.
To a mathematician, these are all the same thing.
And, of course, I can go in the other direction as well.
I can make 'em big!
So, I had to put a thousand Klein Bottles some place, so I put 'em under my house.
...and the wheels. You know? Their little motors drive the wheels around...
*whirring*