Juggling by Numbers – Numberphile


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.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.


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