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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.

  • [tick]

  • [tick]

  • [tick]

  • [tick]

  • [tick]

  • 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.

  • [tick]

  • [tick]

  • [tick]

  • [tick]

  • [tick]

  • [tick]

  • [tick]

  • [tick]

  • So the brilliant thing there is that the diagram lets us predict that this should work. And in fact, this is a juggling trick

  • that's been known.

  • And so now, we can say, okay, well, it's possible to do a four, and it's possible to do a five three.

  • We could have a look at pushing two of them one later,

  • and the other one has to be brought back more. So, instead of having a four four four,

  • it becomes five five,

  • and then the four has to become a two.

  • So we end up with

  • five five two. And that can also be done.

  • And now, you can see a pattern building.

  • So we get, look at these last numbers here, four three two, and you predict that there's going to be a one here.

  • All of these are five, so we get

  • five five five, and we predict that this should be possible. And we can do it just from the numbers.

  • We can do it by drawing the diagram. And when we first did this,

  • this was a juggling trick that we didn't know. And as far as I know, had actually never been done.

  • When I took this to juggling conventions, people did not know it.

  • After four months, people from America were trying to show it to me,

  • because it was a juggling trick that had just gone right round the world, from juggling club to juggling club to juggling club.

  • It's called the fake five, because it feels and sounds a lot like juggling five balls.

  • And if I actually do it, here we go, so we get yellow,

  • yellow pink green white. So we do a single shot at this, starting with the yellow ball,

  • counting three, two, one,

  • five five five one.

  • And those three high throws are exactly the kind of throw that you would do if you're juggling five balls.

  • But hang on, what's this one? What am I doing with the one? And you may have noticed I did a zip across underneath.

  • And we can actually show that on the diagram, why that becomes inevitable.

  • What's this one business?

  • Well, once again, if we very very quickly sketch this, I say I'm throwing here, throwing here, throwing here, throwing here, throwing here, throwing

  • here. I throw this ball,

  • I've only got one, I have to throw it... here.

  • It's got to be the same ball. So that means that the ball

  • basically has to zip across to be held in the hand for a short time,

  • to be zipped across to be held in the hand for a short time.

  • So, in essence, if, if the, if the catch is happening in the middle of those two throws, then basically, the ball gets zipped across,

  • spending no time in the air at all. And we can start to talk about the flight time of the ball, and how that compares.

  • And if the catches are always in the middle, then we always have the ball in the hand for one beat.

  • So you've got your cycle time, you take up one beat of that with the dwell time, and the flight time is always, therefore,

  • one less than the total cycle time.

  • That has interesting implications if you look at zero ball juggling. Because that would mean that the ball actually has to go backwards in time.

  • But there aren't any balls. But there are some interesting things that you can say about that.

  • But here, we can do a one. A zero is basically having an empty hand.

  • And so now, we can start to say, well, what, what are all of the possibilities?

  • We can start to draw the diagrams.

  • Are there other ways of inventing juggling tricks? Are there other ways of knowing that we've got them all?

  • How does this help us find every possible juggling trick?

  • When jugglers talk about the height of a ball,

  • there are two possible things that they could mean. One is the physical height.

  • But we also talk about the natural height of a five ball pattern.

  • And we'll call that a height five. The natural height of a six ball pattern,

  • and we'll call that a six. That varies according to the speed.

  • But when you talk about the height of a ball,

  • then you'll tend to mean the, the number of balls that you're juggling for that given

  • physical height, so.

  • The, the term is often used interchangeably,

  • which can be confusing.

  • But once you use it often enough, it, it stops being something that you worry about.

  • There are weight diagrams that help us, which are finite state diagrams.

  • But I can show you a finite state diagram for every possible three ball juggling pattern,

  • where the height of throw never exceeds five.

  • So let's do that.

  • What I do know is that there are

  • ten places to be, and as I draw this, you'll think, hang on,

  • that's only eight. But I'm gonna put an extra one here, and an extra one here.

  • And I'm gonna start in a strange place.

  • I'm gonna start here,

  • and I'm gonna label that arrow as a three. And the way you read this diagram