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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

  • is that you follow the arrow and write down the number, and that tells you the cycle time of the throw you're doing.

  • So this tells you that I can do three three three three three three, and that's good because that's the native three ball pattern.

  • [tick tick tick]

  • From there, I could also do a four, and come back with a two.

  • [tick tick tick tick tick]

  • Now, I haven't talked about a two. But basically, if you've got two juggling balls,

  • then really, the only thing that you're gonna do is two two two two two two two two two two.

  • And if you draw the diagram, that really is how it looks. You're just hanging on to the balls.

  • You don't really even need to let them go. So that's green and orange.

  • Okay, so having done that,

  • if I do a four, it turns out that I can actually do a five after that, if I choose.

  • Two going there.

  • There's also a one coming back here. And if I stop for a moment and pull out some of the patterns that we've got here,

  • four four one

  • was one of the very very earliest patterns that we ever found by using these techniques.

  • And that's the one that I use to torture

  • the really good jugglers at the European juggling convention, because it's only three ball juggling,

  • it looks a lot like a pattern that they know, which is called the box.

  • But it's not.

  • It's much easier. It's much lower.

  • It's much cleaner. And every time I showed it to them, and they tried to do it, they couldn't do it.

  • So this was a great way of getting really good jugglers hooked on this,

  • where did you find this pattern? And so I could then start to talk about this.

  • Brady: "What does a 4 4 1 look like?"

  • 4 4 1, okay, so, 4 4 1.

  • What I have to do is I have to do a 4, and then a 4, and then a 1. Now I'll do that.

  • I'll start with three ball juggling, and then I'll start, I'll just do a single shot, starting with the orange ball,

  • and so next time round, so we get 4 4 1.

  • And I'll do it again, one shot, 4 4 1. And now if I do it constantly,

  • We get 441, 441, 441, 441.

  • And if you watch, each ball goes everywhere.

  • And that's the really cool thing about this pattern, is, all three balls are doing the same thing,

  • but out of phase,

  • in such a way that they all meet together. And then, in the middle of doing all of that,

  • you can simply drop back to doing three balls, and it just ... disappears.

  • It's just a really cool neat clean pattern, and, and I love it particularly, 'cause it's one of one of the ones we found.

  • Now there's an interesting thing that happens here is that you get patterns that start and finish here,

  • but don't travel over there, and patterns that start and finish here, but don't travel over there. And so, within the juggling world,

  • there's a thing known as

  • excited patterns, that you can't do just starting from the three ball cascade.

  • And they're a ground state, which are patterns that you can. So, for example, five three one, five three one,

  • you can just patch that into the middle of a cascade.

  • But five one five one, which is the three ball shower, where the balls go in that cyclic pattern, that everybody thinks

  • they know is the way you juggle,

  • you can't actually do that, instantly starting from the cascade, starting from the three ball

  • figure of eight. You have to do a preparatory throw. And there are different preparatory throws that you can do.

  • So, for example, you can do a four,

  • five one five one five one five one.

  • What people used to do is a five two,

  • five one five one five one, and they have, each has their own merits, and so forth. But now,

  • every traversal of this, of this diagram, every way of wandering around on this diagram and writing down the black numbers as you go,

  • gives you

  • a jugglable sequence of numbers.

  • Brady: "How many different sequences are possible? How many different tricks are there?"

  • It becomes really messy if you try and do it by saying, how many are there of a certain maximum cycle time,

  • what we would normally call height, but there is a lovely formula that's slightly

  • unexpected and slightly unusual in its terms.

  • But if we ask for a repeating pattern,

  • and ask for a repeating period of, say, length n,

  • and now this is the unusual bit that people don't expect, and then we say, up to

  • b balls, but not including, so that's strictly less than b balls,

  • then the number of patterns,

  • the number of possible patterns

  • is b to the power of n.

  • Which is just unexpectedly simple.

  • Brady: "What does a zero mean? On your diagram, what does a zero mean?"

  • On my diagram it means at this point I'm scheduled to do a throw, but all the balls are currently in the air.

  • So I have zero juggling balls to throw.

  • Brady: "And what does a one mean?"

  • A one is this one that transfers directly from one hand to the other. So if I've only got one,

  • juggling ball, I actually just transfer it from one hand to the other. Now again,

  • my hands are full for exactly half the time.

  • Brady: "A two?"

  • A two is doing this.

  • Which is surprisingly difficult, because you have to throw and catch.

  • But in truth,

  • in the middle of juggling and having lots of balls in the air all at the same time, you would never let that go.

  • You actually just hold it up and show it to display, rather than releasing it.

  • You'd never do that.

  • Brady: "And a three?"

  • The three is actually when we're juggling three balls.

  • It's, it's the first what people think of as proper juggling.

  • But when you look at the the, the notation that we've got, the structure,

  • the diagrams, the mathematics underneath it of how the patterns and timings work,

  • this fits neatly and cleanly into the whole thing, as does this for one ball, as does this for zero balls.

  • Brady: "What proportion of professional jugglers are into the math and the notation, and what percentage of them think this is just

  • "a nerdy way of looking at their art?"

  • It's fairly clear that

  • almost everybody who's heard of the notation,

  • and it really is split fairly evenly, you know, it might not be fifty-fifty, it might be sixty-forty,

  • but, but there's a good proportion of people who actually can use this to communicate with each other, a slightly smaller

  • proportion who can use this to devise notations, and of course we've got rules underneath this, and the idea

  • then is how do we break these rules to do things which are more artistic potentially?

  • Then you get the jazz musicians, for whom

  • writing down music kind of kills it. And they don't want the music written down, they want to ornate around a theme.

  • And as you get out into that side of the, the juggling world, they don't like this. They know it's there.

  • They don't want to use it. If somebody comes to them and says, this is a really cool pattern.

  • What can you make of this? They want you to demonstrate

  • what it is, so they can then elaborate on top of it, do variations on top of it,

  • mutate it into something that's, that's not as

  • technical as this.

  • Brady: "Can you watch live juggling and sort of almost see the numbers?"

  • Yeah, absolutely. And again, live juggling, you get the, the superb fluid

  • dynamic creative dance performance. But then, you look at it and go, okay, that was four four one,

  • done in a gorgeous way. And that was just four two done in a very clever way.

  • So I do pick out what the patterns are.

  • And it's really interesting, because it does then mean that I don't have to concentrate on some of the aspects of it.

  • It frees up my brain to, to think about the rest of it.

  • It's like chunking in maths.

  • Once you get a particular skill really under control, it frees up your brain to think about the next phase.

  • It's like playing a sport, dribbling the ball in football, playing a forehand drive in tennis, it has to be totally

  • mechanical, has to be totally

  • as easy as breathing, so that it frees up your mind to think about the structure of the game, the structure of what's going on.

  • And so it is when I watch juggling.

  • I see the pattern and I have the siteswap in my head of what they're doing, and that then lets me see more about what's going on,

  • and more of the detail underneath.

  • Riffle them together is the way we say it. They sometimes do it on the table this way, you know,

  • casino dealer will do, will do that. That's riffle. This is called overhand, and the other one I call smushing.

Just as we have a notation for music,

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A2 初級

数字のジャグリング - Numberphile (Juggling by Numbers - Numberphile)

  • 6 1
    林宜悉 に公開 2021 年 01 月 14 日
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