So let me first tell you what a braid is imagine that you have like two wooden planks and then there are some strings
attached
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?
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
Maybe I can show you on
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
Like this, they are
they are kind of
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
So that's how if you add a dimension
things that were
uncrossable before are suddenly
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]
exactly yes, exactly, so
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 that's a very good point
So let me try to show you why this ant picture is relevant to the braids and the fourth dimension?
So you have a braid in three dimensions?
And you can't untangle it
Because somewhere you have two strands that you want to move past each other and you can so this is one strand
and
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
But it's not working
Imagine the plane in which
this front strand lives and
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
You can braid something besides strings to show you that first. I like to show you a different way
To see just regular braids in three dimensions and since we were talking about ants
So you have some ants and I'm making a movie in which the ants start out in a neat straight line
and then they crawl around and
Eventually they get back to the neat straight line
I will film them doing this and then there will be a whole bunch of frames I could imagine just stacking the frames
In the first frame I have the four ants in a straight line
and then in the second frame
They will have moved slightly in the third frame
They will have moved again and then so on and so forth until I get to the last frame in which
they are again in a straight line and in between if I trace all the
Ways that they had moved what we got there is a braid okay?
so if I make a movie of ants crawling on a plane
Then that's the same as having a braid in 3D just by stacking all the movie frames on top of each other
So what does my previous statement mean in this context so my previous statement?
Is that braids in four dimensions are uninteresting? What would be braids in four dimensions?
Well, it would be like flies flying around in
Three-dimensional space, and then I make a movie of the flies flying around and I stack them on top of each other
so the fourth dimension is the time and the frames of the movie each frame is a
Three-dimensional thing because they are flying in space, so what I get is braided strings in four dimensions
So that is still uninteresting to a mathematician for the same reason why it's uninteresting to you
Because it wouldn't hold your hair together. It can always be undone
So there's no very interesting analysis that can be done the only point. I want to get out of it
Is that you can think of this
Uninteresting object as a movie about flies and now I can tell you what the interesting object is instead of making a movie about flies
You could make a movie about other things moving around in
three-dimensional space
The one that I actually think about in my research is rings flying around in three-dimensional space
So if you have some rings they start out in a straight line, and then they can fly around in
three-dimensional space and
Eventually get back to the same position if I make a movie about this and take all these three-dimensional
movies frames and stack them on top of each other in four dimensions
I get something like a braid so your braiding tubes. As opposed to rings, tubes are two-dimensional
So you're braiding two-dimensional things in?
four-Dimensional space so wire rings more interesting than flies
I have some slightly mangled rings here. So if they just fly around
not interacting ever they are exactly like flies and they are boring, but
Since they are rings. They have a hole inside something they can do is
fly right through each other and flying through each other that's
Fundamentally different from just being flies, and that's what makes these braided tubes
something interesting as opposed to
Just braided strings which in four dimensions are uninteresting so they are allowed to
Expand and and get smaller and bigger as they need to further purpose of flying through each other as
they fly through each other they create the analogous kind of
Tangled nests kind of things that you can move past each other right within a braid yeah
It's a Tube going through another Tube which would be impossible in three dimensions for a tube to go through another Tube without
intersecting it but in four dimensions
It's possible, and it catches the same way that strings would catch on each other in three dimensions
Can I only go completely inside each other or can they overlap like?
No
they can't intersect so they are still you know there you can think of them as
Physical objects physical object they are not phantom rings the material doesn't go through each other
It's just the rings flying through the hole
inside the other ring
This you can generalize to any dimensions, so in three dimensions you can braid
One-dimensional string with a one-dimensional string one dimensions with one dimensions one plus one equals two
which is three minus one in four dimensions, you can braid a
one-dimensional with a two-dimensional, so that's the fly when you take the
Trajectory in time that is the ring when you take the trajectory in time. That's three and
three equals four minus one
you can also
Braid A
two-dimensional with a two-dimensional which is four which is bigger than
four minus one and similarly in five dimensions, you couldn't braid a one-dimensional with a one-dimensional or a
one-dimensional with a two-dimensional
but you could you could do two with two or three with three as long as
dimensions of your two strings add up to at least the dimension of the
Containing space minus one then you can break them together
Well in my research, I mostly think about one in two dimensional things
Braided or tangled in four-dimensional space or the simplest thing of one-dimensional things
braided or tangled in
three-dimensional space and the relationships between them
So my favorite way of visualizing. It is is to think of it in terms of movies in three dimensions and
That is actually a little bit restrictive
because
Visualize the fourth Dimension is time and time only moves forward