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• Just as we have a notation for music,

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

• we came up with a notation for juggling.

• 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

• have a couple of alternative names.

• 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,

• they've got yellow, green, orange,

• 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.

• [tick tick tick tick tick tick tick]

• 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 so it has to be in the other hand.

• 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.

• [tick tick tick tick tick tick]

• 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.

• They literally form a braid.

• 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.

• [TICK tick tick TICK tick tick TICK]

• 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.

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• And that's for three balls.

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

• and white.

• 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.

• And I've got yellow pink green white.

• And look, the yellow,

• 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

• green white.

• 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.

• But now, the interesting thing is, if we have a look at the time that the yellow ball gets thrown here,

• and then the time the yellow ball next gets thrown is there, and look, how long is that?

• Well, I've got a beat here, and then I go, one two three

• four beats. And you can see that that really is

• four beats of time from a throw of the yellow ball to the next throw of the yellow ball. And they're all doing the same thing.

• So they will all have a four beat cycle.

• In a moment, I'm gonna talk about throwing the balls to different heights. At the moment, they're all doing the same thing.

• So at the moment, I can say that the pattern has four beats between throws. But shortly, I'll talk about an

• individual throw being four beats.

• Okay, so, having seen three balls and three beats, four balls and four beats,

• we can fairly obviously go on and do five and six and seven, and there's no real interest in that.

• We can go down and ask what does it actually mean, now, based on this kind of diagram,

• can I draw a diagram for two balls? And what does then imply, that then imply,

• for the physical juggling? And one ball, and perhaps even zero balls.

• I'm not gonna do that yet, because I really want to get to the payoff.

• I want to get to the actual notation, and why it's interesting and what happens.

• So I'm gonna go back, and I'm gonna look at four ball juggling again. Yellow

• pink green and white. So that's what's going on there. And the yellow ball will be thrown next over here,

• and the pink ball will be thrown next over here, and the green ball thrown next over here, the white ball thrown next over here.

• We know that that's what's going on. But now, let's cheat. Now, let's do something slightly different.

• I'm looking at the yellow ball landing here, and I'm looking at the pink ball landing here, and I'm going, well, they're good friends.

• Why don't they change places?

• So, in fact, what we can have is the yellow ball

• not come to here,

• but actually go to there, where the pink ball would go. So the yellow ball actually goes

• to where the pink ball would go. And the pink ball actually goes to where the yellow ball would go.

• So they exchange

• landing sites. They swap their sites. It is a site

• swap, which is where we get the name of the notation from.

• They are swapping the places that they go to.

• And that's great on the diagram.

• What does it mean physically? Well, if we look at this, the yellow would normally have this cycle time of four.

• But now, the time it's next thrown is 1 2 3 4 5 beats of time in the future.

• And the pink ball would be, normally 1 2 3 4, but in fact, 1 2 3, it's three beats of time in the future.

• So what's happening is I'm doing lots of cycle time four

• throws, so I'm doing 4 4 4 4 4. And then, suddenly and without provocation,

• I'm doing a five and a three according to my diagram.

• And what's really nice is that actually turns out to be exactly the kind of throw that you do when juggling five balls,

• followed by exactly the kind of throw you do in juggling three balls.

• So what's predicted from the diagram turns out to be the case in reality. And if I demonstrate that,

• I need the yellow pink

• green and

• white, in that order. So, yellow pink green

• and I'm gonna do the yellow and the pink as the

• high and low, as the five and three, so if I just juggled four for a while,

• and then I'm gonna do the yellow ball high, and the pink ball low, so we, counting down, so, three,

• two,

• one.

• Five three.

• And you'll see that they have changed hands, as is predicted by the diagram. The yellow did go high.

• The pink did go low. And they landed perfectly in rhythm, and I could just carry on.

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