Name: スーパーボトル - Numberphile (Super Bottle - Numberphile)
Uploaded: 2021-01-14T10:15:35.000Z
Duration: 16 min 41 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

A 2-manifold is a thin piece of surface, like this piece of paper, or

法助動詞 B1

this piece of thin plastic. But a 2-manifold must not have any branches. A book that has many pages,

but has a spine where they all come together, that would not

But even a weird thing like that would qualify as a 2-manifold.

2-manifold may have borders such as this Möbius band.

Or it may have holes, like this particular Klein bottle. Or it may be

completely closed, like this surface of an icosahedron, or the shell of an egg. Now for topologists,

all objects are made of infinitely stretchable rubber. And so the geometry really doesn't matter. And if I deform this Möbius band,

you know, it stays a Möbius band, and the topologist would still classify this in exactly the same way as the original shape.

From this point of view, all of these 2-manifolds can be classified by just three integer numbers.

The first one is the number of borders. This particular Möbius band has just a single border. I can run my finger around the border

here, and if I go around the loop twice, I'm coming back to where I started.

This is the edge of the thin surface. This particular plastic piece has two borders,

you know, one is a circle here, and the other one is this particular edge here.

Brady: "So this is the part where if an ant was crawling on it, it would cut itself in half."

That's right. And if we look at this piece of paper,

this one has three borders. One is the rectangular frame on the outside, and then we have, you know, a circular border here,

and sort of kidney shaped border over on this particular case. So this is the first number.

The second number is the sidedness of a surface. So this piece of paper is clearly two-sided. As a matter of fact,

It has a blue side, and it has a white side, and you cannot get from one side to the other without

an ant would cut itself to death. The Möbius band, on the other hand, has a single side.

spreading your paint without ever crossing one of the edges you would come around, and you would notice that by the time you're done,

you have actually painted both sides of this particular surface.

So that's a single-sided surface with a single edge. On the other hand, the Klein bottle here is also single-sided surface,

but, as you can see, it has no edges. However, we cannot really create a

Klein bottle in three-dimensional space, without having either some opening, some punctures, or a

self-intersection line. And you can see here, here is one of those self-intersection lines.

Now if I don't like these self-intersection lines, I could cut a slightly larger hole into the wall of the thick arm,

so that the thin arm then can be passing through. So in this model, rather than living with the self-intersection,

I have cut a large enough opening into the green body so I can bring out the thin arm without creating a self intersections.

But for the price of creating an opening, a so-called puncture, which has its own border.

So this would still be a Klein bottle, but now it is a Klein bottle with one puncture.

And that means that simple holes that you cut in a surface do not change the type of surface for a topologist.

Even though this little thing has a whole lot of little holes, it would still be considered a Klein bottle,

but with many many many many many punctures.

Now from that point of view, this little cylinder

is really just a sphere with two punctures.

And this becomes a little bit more obvious if you try to cap off these openings.

Now it would be a sphere with only one puncture, and by the time I'm adding the second cap, it becomes more obvious that

There is a third number that is important for the classification of all 2-manifolds,

and that has to do with the connectivity of the surface. It's called the genus of a surface.

A sphere, or equivalent topological shapes, would be of genus zero. A donut would be of genus one. A two-hole donut

and the surface of that would also be a 2-manifold of genus 2.

The genus is clearly related to how many holes you have in your donut,

or how many handles that there are. A more precise definition would be, how many closed loop lines

can you draw on that surface, that if you were to cut along them, the surface would still hang together?

If I take my piece of paper, cut a circle loop into it, then the inner part of

that hole falls out and is clearly no longer connected.

On the other hand, if I take my simple torus, and I cut along this red line

then I simply get some kind of, you know, tubing, but it still all hangs together.

So this is of genus one, because I draw one such line.

If I had a two hole torus, I could draw two such lines, and I can cut along those,

and after both those cuts the surface still all hangs together.

So that's a surface of genus 2. If I take my Möbius band,

I maintain that this Möbius band will not fall apart.

I get a loop twice the size. It has a twist of 360 degrees in it,

but it is still in one piece. And since I could do one such cut, that means that Möbius band

had a genus of one. Try to cut it again in the middle,

then it actually would fall apart. There's only one way of cutting the Möbius band along a line and then we're done.

to see whether it's not perhaps genus two.

and I cannot get from one piece to the other, so the surface is no longer connected.

So clearly Möbius band is only genus one. The Klein bottle is actually of genus two,

and I've tried to indicate that by drawing two such lines which I could cut along. The red line on the one side,

and the green line on the other side, and if I did that, the surface would still be hanging together. So, the Klein bottle

with no borders, and it is of genus 2. Now I would like to make

super Klein bottles in some modular way. The first possible way of making a module is what I'm showing here.

I'm essentially taking the top half of the classical Klein bottle that has the important mouth.

The way it is, this is still a two-sided surface. To make that clear,

I essentially painted the inside here in silver, so the inside of this green toroidal body is silvery, and then it is being brought out

here in this silvery stem, which by itself, on the inside, is still green. I want to use two of these

modules to make my super Klein bottle, and I'm contemplating on essentially bringing together

into a ring underneath these curved connectors.

But I can see if I try to put them together like that, then silver meets silver in the upper branch,

and green meets green in the lower branch. And so I never really stepped from

green to silver, and so that cannot possibly be a single-sided surface.

It's a perfectly good, you know, two-sided surface. Ah! But what if I turn this thing around?

So maybe I want to connect it like that. Let me actually try to do that. Some assembly required.

And it looks good, you know, I'm getting from green to silver over this branch

But then wait. If I go through lower branch, I get, once more, a green silver transition.

And that means I have an even number of changes between the two surfaces,

and so I'm not really getting a net single-sided surface. So again, it's two-sided. That's disappointing.

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スーパーボトル - Numberphile (Super Bottle - Numberphile)

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