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Just as we have a notation for music,
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we have a notation for language, we have a notation for dance,
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we came up with a notation for juggling.
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And the really cool thing was that there was some unexpected
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mathematics underneath that then let us predict the existence of juggling tricks that, as far as we knew, had never been done before.
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Well, the one I'm going to talk about is called siteswap. If you go on the Wikipedia page, it does
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have a couple of alternative names.
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But really the place to start is with a little bit of juggling, so that you can see what it is
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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.
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But, in fact, the easiest thing to do is to throw them
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so that they all come down in the same order. And if you do that,
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they've got yellow, green, orange,
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and then it will be the yellow one's turn again.
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But if my hand's taken in turns, that means the yellow has to change hands. So let me show you.
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Yellow with this hand, green with that hand, orange with this hand.
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So yellow green orange yellow green orange yellow
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green orange yellow green orange. And you'll see that they have to change hands
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because the balls are taking it in turns, and so are my hands.
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And so it just works out that that's the way it goes.
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So the first thing to be able to do is at least to be able to describe that juggling pattern
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in any notation that you use, and then, think to yourself,
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well, can I then also describe other ones?
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So what I'm gonna do is I'm gonna have my left hand here, and my right hand here.
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And I'm gonna have time running in this direction. So imagine that I'm walking
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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.
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I'm gonna leave out some of the detail of my hands moving from side to side.
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But I'm gonna say, as I'm walking forwards,
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I'm gonna throw with the right hand from here, and that ball's gonna go over to the other hand.
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I don't know when that balls gonna come down, but it's gonna go over to the other hand. And a moment later,
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I'm gonna throw with the left hand, and then a moment later,
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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
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right hand, and then with the left hand. And I've got this this,
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this rhythm about the whole thing, this, this constant metronomic beat going on here.
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[tick tick tick tick tick tick tick]
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So I'm gonna put the, the colors onto here now. So this one was with the yellow ball,
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and this will also show you that they, they have to change hands, because if I throw the yellow one here,
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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.
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And so it has to be in the other hand.
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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,
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and then it's in my hand for that moment there.
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So there's the yellow ball going across to that hand there.
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And then the green ball does the same thing. It goes over here, in that direction,
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and then it's in my hand for a while, there. So that will be the green ball there.
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And then the orange one gets thrown. It goes across to the other hand,
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and so that's over here in this hand, and then that's the orange ball being thrown, and then the yellow one again,
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and then the green one again, and then the orange one again.
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[tick tick tick tick tick tick]
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And, in fact, what we end up with here is a plait.
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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.
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They literally form a braid.
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It's ephemeral, because the balls don't leave, actually, leave the trails behind,
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but if you could imagine that. And a guy called Henry Segerman has made 3D printed models of
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juggling tricks that are then pushed through time, pushed through space, so you can see the intermingling, interleaving of the balls.
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But here, what's happening now is, you can see that with three ball juggling, what's happening is
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I throw the yellow ball, green ball, orange ball, then I throw the yellow one again, so the length of time
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from that throw of the yellow ball to that throw of the yellow ball is
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three beats of the clock, three ticks of the underlying metronome.
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[TICK tick tick TICK tick tick TICK]
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So, if you've got this tick, and it goes yellow green orange, yellow green orange,
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then the yellow ball will be thrown every third time. So it's every third beat that the yellow ball gets thrown.
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Another little comment about this is that the actual flight time, the time that the ball spends in the air, has to be less,
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to allow for some time in the hand, what we call the hold of the juggling ball,
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and that's the dwell, the, the time that it's in the hand, and then we have the flight time here,
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and you'll see that flight time is a little bit less
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than this time that it takes the balls to cycle around, and this is what we call the cycle time.
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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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[tick]
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[tick]
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[tick]
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[tick]
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[tick]
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And that's for three balls.
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Now let's move on to four balls, and see what happens. Yellow pink green
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and white.
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So yellow pink green white, yellow pink green white, so here we go. We go,
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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
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pink green white. And now, you'll see that the balls are actually staying in the same hands.
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So let's actually draw the diagram for four balls.
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I've got left hand and right hand. Throw it here, throw it here, throw it here, throw it here, throw it here.
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And I've got yellow pink green white.
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And look, the yellow,
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it's the yellow's turn, and it's the right-hand's turn again.
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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
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green white.
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So the yellow ball is thrown here, and next it has to be thrown there,
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so it stays in the same hand. I'm gonna have it bob around, so it comes around. Again, again,
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it has to come down that little bit early,
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I catch it before I throw it. Obviously, it has to spend some time in the hand.
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For simplicity, we often assume that that catch is exactly halfway between. In practice,
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it's not. In practice, your hand is full for more than half the time.
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Because the only time you can control the juggling prop is while you're holding it.
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So you tend to hold it for as long as you can to give you the maximum control.
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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
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over the time that I'm holding the yellow ball.
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and then it's in the hand, then the green, then the yellow ball gets launched again.
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And this is actually how it feels. It feels like you get these two balls
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bouncing over each other in the hand. And if I actually do yellow and green in the same hand, here,
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you can see that they, they really do feel like they're bouncing, one over the top of the other.
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It's almost like they're playing leapfrog. And, in fact, if you turn this side on, you can see that
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they're playing leapfrog over each other. Although, it's leapfrog through time, not leapfrog through space.
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But now, the interesting thing is, if we have a look at the time that the yellow ball gets thrown here,
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and then the time the yellow ball next gets thrown is there, and look, how long is that?
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Well, I've got a beat here, and then I go, one two three
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four beats. And you can see that that really is
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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.
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So they will all have a four beat cycle.
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In a moment, I'm gonna talk about throwing the balls to different heights. At the moment, they're all doing the same thing.
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So at the moment, I can say that the pattern has four beats between throws. But shortly, I'll talk about an
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individual throw being four beats.
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Okay, so, having seen three balls and three beats, four balls and four beats,
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we can fairly obviously go on and do five and six and seven, and there's no real interest in that.
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We can go down and ask what does it actually mean, now, based on this kind of diagram,
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can I draw a diagram for two balls? And what does then imply, that then imply,
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for the physical juggling? And one ball, and perhaps even zero balls.
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I'm not gonna do that yet, because I really want to get to the payoff.
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I want to get to the actual notation, and why it's interesting and what happens.
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So I'm gonna go back, and I'm gonna look at four ball juggling again. Yellow
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pink green and white. So that's what's going on there. And the yellow ball will be thrown next over here,
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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.
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We know that that's what's going on. But now, let's cheat. Now, let's do something slightly different.
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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.
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Why don't they change places?
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So, in fact, what we can have is the yellow ball
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not come to here,
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but actually go to there, where the pink ball would go. So the yellow ball actually goes
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to where the pink ball would go. And the pink ball actually goes to where the yellow ball would go.
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So they exchange
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landing sites. They swap their sites. It is a site
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swap, which is where we get the name of the notation from.
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They are swapping the places that they go to.
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And that's great on the diagram.
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What does it mean physically? Well, if we look at this, the yellow would normally have this cycle time of four.
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But now, the time it's next thrown is 1 2 3 4 5 beats of time in the future.
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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.
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So what's happening is I'm doing lots of cycle time four
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throws, so I'm doing 4 4 4 4 4. And then, suddenly and without provocation,
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I'm doing a five and a three according to my diagram.
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And what's really nice is that actually turns out to be exactly the kind of throw that you do when juggling five balls,
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followed by exactly the kind of throw you do in juggling three balls.
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So what's predicted from the diagram turns out to be the case in reality. And if I demonstrate that,
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I need the yellow pink
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green and
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white, in that order. So, yellow pink green
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and I'm gonna do the yellow and the pink as the
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high and low, as the five and three, so if I just juggled four for a while,
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and then I'm gonna do the yellow ball high, and the pink ball low, so we, counting down, so, three,
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two,
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one.
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Five three.
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And you'll see that they have changed hands, as is predicted by the diagram. The yellow did go high.
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The pink did go low. And they landed perfectly in rhythm, and I could just carry on.
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[tick]
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[tick]
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[tick]
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[tick]
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[tick]
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[tick]
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[tick]
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[tick]
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So the brilliant thing there is that the diagram lets us predict that this should work. And in fact, this is a juggling trick
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that's been known.
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And so now, we can say, okay, well, it's possible to do a four, and it's possible to do a five three.
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We could have a look at pushing two of them one later,
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and the other one has to be brought back more. So, instead of having a four four four,
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it becomes five five,
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and then the four has to become a two.
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So we end up with
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five five two. And that can also be done.
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And now, you can see a pattern building.
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So we get, look at these last numbers here, four three two, and you predict that there's going to be a one here.
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All of these are five, so we get
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five five five, and we predict that this should be possible. And we can do it just from the numbers.
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We can do it by drawing the diagram. And when we first did this,
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this was a juggling trick that we didn't know. And as far as I know, had actually never been done.
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When I took this to juggling conventions, people did not know it.
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After four months, people from America were trying to show it to me,
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because it was a juggling trick that had just gone right round the world, from juggling club to juggling club to juggling club.
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It's called the fake five, because it feels and sounds a lot like juggling five balls.
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And if I actually do it, here we go, so we get yellow,
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yellow pink green white. So we do a single shot at this, starting with the yellow ball,
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counting three, two, one,
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five five five one.
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And those three high throws are exactly the kind of throw that you would do if you're juggling five balls.
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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.
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And we can actually show that on the diagram, why that becomes inevitable.
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What's this one business?
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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
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here. I throw this ball,
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I've only got one, I have to throw it... here.
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It's got to be the same ball. So that means that the ball
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basically has to zip across to be held in the hand for a short time,
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to be zipped across to be held in the hand for a short time.
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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,
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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.
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And if the catches are always in the middle, then we always have the ball in the hand for one beat.
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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,
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one less than the total cycle time.
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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.
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But there aren't any balls. But there are some interesting things that you can say about that.
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But here, we can do a one. A zero is basically having an empty hand.
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And so now, we can start to say, well, what, what are all of the possibilities?
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We can start to draw the diagrams.
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Are there other ways of inventing juggling tricks? Are there other ways of knowing that we've got them all?
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How does this help us find every possible juggling trick?
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When jugglers talk about the height of a ball,
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there are two possible things that they could mean. One is the physical height.
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But we also talk about the natural height of a five ball pattern.
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And we'll call that a height five. The natural height of a six ball pattern,
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and we'll call that a six. That varies according to the speed.
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But when you talk about the height of a ball,
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then you'll tend to mean the, the number of balls that you're juggling for that given
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physical height, so.
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The, the term is often used interchangeably,
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which can be confusing.
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But once you use it often enough, it, it stops being something that you worry about.
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There are weight diagrams that help us, which are finite state diagrams.
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But I can show you a finite state diagram for every possible three ball juggling pattern,
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where the height of throw never exceeds five.
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So let's do that.
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What I do know is that there are
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ten places to be, and as I draw this, you'll think, hang on,
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that's only eight. But I'm gonna put an extra one here, and an extra one here.
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And I'm gonna start in a strange place.
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I'm gonna start here,
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and I'm gonna label that arrow as a three. And the way you read this diagram
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is that you follow the arrow and write down the number, and that tells you the cycle time of the throw you're doing.