Name: Braids in Higher Dimensions - Numberphile
Uploaded: 2020-03-27T18:12:29.000Z
Duration: 14 min 27 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

So let me first tell you what a braid is imagine that you have like two wooden planks and then there are some strings

In a neat row to the first wooden plank, and then they are all tangled up

But then when they arrive to the top they are again in a neat

Straight line where they started so like one goes to one two goes to two. Yes

So that's called a pure braid I'll only talk about pure braids today

It doesn't hugely matter you can also consider ones that that crisscross and end up somewhere else

But let me try to draw one that actually goes back to where it started from it goes from

Bottom to the top in between you have some tangled mess

So when you have some big tangled mess, what's your what's the first question that comes to mind?

How would you untangle it or can you even untangle it so sometimes you have a big tangled mess

And it's entirely possible to untangle it

What does it mean to untangle it it means to make it into this braid. This is like the non braid

So how do you untangle something? So imagine that the strands are say made of rubber?

They are stretchy you can move them around as much as you want

You can stretch as much as you want, but you cannot cut them once you cut them the game is ruined

So the question is if you see a tangled mess like this

Can you make it into a an untangle? This has some practical?

Applications for example if you made this video a year ago, I had long hair and when you braid long hair, oh

just this little bit so the standard thing you do is you divide it into

Three parts this will make me look rather silly

And then you do this braid one goes over the middle one and then the other goes over

but became the Middle one and you just continue doing this until

You reach the end of your hair and then you put an elastic on it. Now it's not two wooden planks

But at the top. It's attached to your head and at the bottom It's fastened in an elastic

so the top and bottom are kind of fixed the same as in the picture that I just showed you and

The good thing about that braid is that it cannot be untangled

Because if it could be untangled then as you move your head around it probably would become untangled and the one that I just drew

Probably cannot be untangled for example if you look at just the first and the last string you can see that those are

wrapped around each other in a way that you can't imagine untangling, so

Why is it that some braids cannot be undone when you move the strands around move the strings around?

What's the problem that you might run into it's that two strands catch on each other?

And you can't move them past each other without breaking them

So that's the main reason that some braids cannot be undone if the strands would just go through each other magically

It ghosts then any braid could be undone. This is in

Three-dimensional space that we all know very well because we live in it

I'm claiming that if you think of braids in four dimensional space then they become completely

Uninteresting the main reason is that if you have two strands in four dimensional space they can basically go through each other like ghosts

So why is that let me try to give you an analogy first?

So there's a little ant that lives in the plane

So this ant is two-dimensional all it ever knew was this

Two-dimensional plane that it lives in and this two-dimensional plane has a wall the wall is one dimensional

it's just a line in the plane, so this little ant

Can't go through the wall and will never know

What's on the other side of the wall it just hits a red line

It just hits a red line, and it can't go through it. What happens if One day

a miracle happens and all of a sudden the world of this ant becomes

Three-dimensional so all of a sudden there's height that day

The ant could just go up to the line and say wait

But this line has no height at all so I can just jump over it and discover the other side

Crossable because you can use the extra dimension to cross them

[Brady] is that assuming that when we kick up to the fourth dimension

Our hair doesn't take on an extra dimension with it?[/Brady]

This wall that the ant that that has kept the ant on one side of its plane

For its entire life when you added an extra dimension the wall didn't suddenly become infinitely high in two-dimensional

So let me try to show you why this ant picture is relevant to the braids and the fourth dimension?

Because somewhere you have two strands that you want to move past each other and you can so this is one strand

Behind it. There's another strand and you

Want to move it past what you can't because it gets stuck on this strand. That's in front of it

Yeah, so what you want to do is to move this

Strand behind that you're trying to move past

Will only intersect that plane in one point right so that point will basically be the ant

So if you're just thinking of the picture that you can see in this one plane

Then you have the ant which is where the strand behind?

Intersects the plane and you have this other strand which lives in the plane

And that's the wall and then when I try to move this strand behind and get stuck on this strand

That's basically the ant running up against the wall that becomes the hitching point of the two strands that becomes the hitting point

But if I add an extra dimension now this is hard to imagine because I mean the the braid is already in three-dimensional space

so then I have an extra fourth dimension you can imagine that extra fourth Dimension by

Imagining that this playing all of a sudden becomes a three-dimensional space, okay?

so if I add an extra dimension to the whole picture

Then there will be an extra dimension added to this plane

So in this new space the bottom braid can simply jump over the other strand in the extra dimension

So not in the three dimensions that it lived in before, but when you add the extra dimension it can jump over

[Brady}Could you still braid your hair if you were a four dimensional being would your hair have a new property that's still made it braidable?[/Brady]

Well it depends, so if I stayed the same, but I was placed in four dimensions then no

For the reason that I just told you because each strand of my hair is basically one Dimensional

[Brady] your DNA would probably all fall apart as well.[/Brady]

but if when I was placed in four dimensions

I myself became four dimensional and my strands of hair became two-dimensional then I could so I

Said braids in four dimensions aren't interesting and now I will show you why they are

The key is that it's not enough to Braid strings

I just showed you that if you're braiding strings that can always be undone in the fourth dimension

So I have to show you what to braid instead

Subtitles ListPlay Video

Braids in Higher Dimensions - Numberphile

stick

eventually

stretch

physical