Name: Juggling by Numbers - Numberphile
Uploaded: 2020-03-27T18:08:57.000Z
Duration: 24 min 59 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...

we have a notation for language, we have a notation for dance,

Past simple

And the really cool thing was that there was some unexpected

mathematics underneath that then let us predict the existence of juggling tricks that, as far as we knew, had never been done before.

Well, the one I'm going to talk about is called siteswap. If you go on the Wikipedia page, it does

But really the place to start is with a little bit of juggling, so that you can see what it is

we're trying to capture. I'll just do three to start with, and a lot of people think that they'll go in a circle.

But, in fact, the easiest thing to do is to throw them

so that they all come down in the same order. And if you do that,

and then it will be the yellow one's turn again.

But if my hand's taken in turns, that means the yellow has to change hands. So let me show you.

Yellow with this hand, green with that hand, orange with this hand.

So yellow green orange yellow green orange yellow

green orange yellow green orange. And you'll see that they have to change hands

because the balls are taking it in turns, and so are my hands.

And so it just works out that that's the way it goes.

So the first thing to be able to do is at least to be able to describe that juggling pattern

in any notation that you use, and then, think to yourself,

well, can I then also describe other ones?

So what I'm gonna do is I'm gonna have my left hand here, and my right hand here.

And I'm gonna have time running in this direction. So imagine that I'm walking

forwards as I juggle, and this, more or less, is where my left hand is, and this, more or less, is where my right hand is.

I'm gonna leave out some of the detail of my hands moving from side to side.

But I'm gonna say, as I'm walking forwards,

I'm gonna throw with the right hand from here, and that ball's gonna go over to the other hand.

I don't know when that balls gonna come down, but it's gonna go over to the other hand. And a moment later,

I'm gonna throw with the left hand, and then a moment later,

I'm gonna throw with the right hand, and then I'm gonna throw with the left hand, then I'm gonna throw with the

right hand, and then with the left hand. And I've got this this,

this rhythm about the whole thing, this, this constant metronomic beat going on here.

So I'm gonna put the, the colors onto here now. So this one was with the yellow ball,

and this will also show you that they, they have to change hands, because if I throw the yellow one here,

and then I'll throw the green one next, and then I'll throw the orange one next, and then I've got to throw the yellow one here.

And in truth, it comes down a little bit before that, so if we look at when it comes down, it might come down there,

and then it's in my hand for that moment there.

So there's the yellow ball going across to that hand there.

And then the green ball does the same thing. It goes over here, in that direction,

and then it's in my hand for a while, there. So that will be the green ball there.

And then the orange one gets thrown. It goes across to the other hand,

and so that's over here in this hand, and then that's the orange ball being thrown, and then the yellow one again,

and then the green one again, and then the orange one again.

And, in fact, what we end up with here is a plait.

And if you juggle and walk forwards, or, more particularly, juggle and walk backwards, look at the paths that the balls leave in the air.

It's ephemeral, because the balls don't leave, actually, leave the trails behind,

but if you could imagine that. And a guy called Henry Segerman has made 3D printed models of

juggling tricks that are then pushed through time, pushed through space, so you can see the intermingling, interleaving of the balls.

But here, what's happening now is, you can see that with three ball juggling, what's happening is

I throw the yellow ball, green ball, orange ball, then I throw the yellow one again, so the length of time

from that throw of the yellow ball to that throw of the yellow ball is

three beats of the clock, three ticks of the underlying metronome.

So, if you've got this tick, and it goes yellow green orange, yellow green orange,

then the yellow ball will be thrown every third time. So it's every third beat that the yellow ball gets thrown.

Another little comment about this is that the actual flight time, the time that the ball spends in the air, has to be less,

to allow for some time in the hand, what we call the hold of the juggling ball,

and that's the dwell, the, the time that it's in the hand, and then we have the flight time here,

and you'll see that flight time is a little bit less

than this time that it takes the balls to cycle around, and this is what we call the cycle time.

These names are not universally agreed. Some people call it the beat time, some people call it the underlying native time.

Now let's move on to four balls, and see what happens. Yellow pink green

So yellow pink green white, yellow pink green white, so here we go. We go,

yellow pink green white yellow pink green white yellow pink green white yellow pink green white yellow pink green white yellow pink green white yellow

pink green white. And now, you'll see that the balls are actually staying in the same hands.

So let's actually draw the diagram for four balls.

I've got left hand and right hand. Throw it here, throw it here, throw it here, throw it here, throw it here.

it's the yellow's turn, and it's the right-hand's turn again.

So the yellow has to come back to the same hand, and that's what we saw when I was actually doing the juggling. Yellow pink

So the yellow ball is thrown here, and next it has to be thrown there,

so it stays in the same hand. I'm gonna have it bob around, so it comes around. Again, again,

it has to come down that little bit early,

I catch it before I throw it. Obviously, it has to spend some time in the hand.

For simplicity, we often assume that that catch is exactly halfway between. In practice,

it's not. In practice, your hand is full for more than half the time.

Because the only time you can control the juggling prop is while you're holding it.

So you tend to hold it for as long as you can to give you the maximum control.

But you can see here, the yellow ball comes, and it's in air. And then the green ball does the same thing. It bobs

over the time that I'm holding the yellow ball.

and then it's in the hand, then the green, then the yellow ball gets launched again.

And this is actually how it feels. It feels like you get these two balls

bouncing over each other in the hand. And if I actually do yellow and green in the same hand, here,

you can see that they, they really do feel like they're bouncing, one over the top of the other.

It's almost like they're playing leapfrog. And, in fact, if you turn this side on, you can see that

they're playing leapfrog over each other. Although, it's leapfrog through time, not leapfrog through space.

Subtitles ListPlay Video

Juggling by Numbers - Numberphile

rhythm

physical

slightly

demonstrate