SPEAKER 1: So welcome. I'm the chair of the Department of Mathematics. It's my pleasure to welcome you for this lecture about mathematics and juggling. It's also my pleasure to introduce Allen Knutson, who is going to give this talk. He obtained his PhD at MIT in 1996. He then went on the West Coast in California. But finally, he wanted to see some winter, so he came back here.
So he's going to talk about mathematics and juggling. I am somewhat worried about the juggling part. I've studied his CVs, so I can assure you that he knows how to prove theorems. If I was invited to a talk about the structure of French gastronomy by a mathematician, I would be very worried of tasting the cuisine. So I'm curious to see about the juggling.
I also thought something about the analogy between juggling and mathematics. If you learn one dimension, two dimension, three dimension, then it's really very easy to go to four, five, six, an infinite dimension. If you juggle two balls, you juggle three balls, it doesn't help so much, I think, with juggling more. And six, seven, that's a lot of balls, I understand.
So anyway, Allen, it's your turn.
ALLEN KNUTSON: OK, well, thanks for coming and seeing me here instead of being out enjoying this beautiful spring day.
So why should there be a mathematics of juggling? I think it's a little loud. Probably I don't need to convince this audience about this. I mean, there should be a mathematics of anything. Anything that's officially well understood should be the mathematics of that thing. And juggling is particularly mathematizable, as I'll show you in a minute.
So I think of this as an applied mathematics talk. I'm going to tell you a nice numbering system for juggling patterns that's good for recording patterns. And when it was invented in 1985, it wasn't as easy to transmit video clips around the net as it is today. So it was much nicer to have a nice, pithy numerical way of talking about juggling patterns, when you wanted to talk to people about [INAUDIBLE] juggling on Usenet.
So it's good for recording patterns. It's good for creating new patterns from old ones. And that's true if you have a difficult pattern you'd like to learn, and you'd like to come up with something slightly simpler, and then that's still too hard. So something slightly simpler, and that's still too hard, and at least find a bunch of stepping stones.
Or usually what jugglers want to do is to find something harder. The jugglers are not efficiency experts. The question is always, there's got to be a harder way to do this.
So I'll show you some of that. So before I make things numerical, let me give you the following idea of how you might draw a juggling pattern so that you have a picture of it instead of video. Imagine we have, where the lights don't work, a camera mounted in the ceiling that's looking down at my juggling these three glowing balls. Then we're going to take a long exposure photograph. What would that look like?
So it's only seeing any particular ball move from side to side. It doesn't see the up and down. And so from side to side, any particular ball is going to be looking like it's doing this-- that, that, that, that. So I have these three balls. One is doing that. One is doing that, and one like that. Is this visible, or am I going to have to draw it thicker?
AUDIENCE: It's good.
ALLEN KNUTSON: Visible? OK. So what's going on here, this is my left hand, this is my right hand. This direction is time. And so at any particular time, if we stop the juggler and take a flash photograph, what we'll see is these three balls.
So this is a pattern of what's called the three ball cascades that I was just doing, the standard pattern with three balls. Let me just make sure you know the structure of the three ball cascade. So it's this, the usual pattern with three balls. It's much, much easier than this one, which we'll discuss a little later. See how much slower and lower it is, because I'm not putting all the work on one hand.
Any particular ball here is getting thrown from the middle to the outside. And then caught and brought back in, and thrown from the middle to the outside. And I can give you tips about learning actually to juggle. Most of it is waiting. You come here and you wait. Oh, got to throw. Oh, OK, I'll wait. So most of the time, I'm sitting here waiting.
And I'm always throwing from the middle to the outside in this figure eight pattern. Actually, in Spanish, this pattern is not called the cascade. It's called el ocho. And this is el ocho con tres, so the 8 with 3, because of the figure eight pattern that any particular ball is doing.
All right, let's do a more complicated one. In the middle of doing three, I'm going to do two throws. So in all these throws, we're crossing from one hand to the other. They're all crossing throws.
I'm going to do two throws like this, that do not cross. And they'll have a little bit of time underneath that I used here to clap. And in that time underneath, I'll hand the other ball straight across. So let's try and get that so the colors are right. Well, I didn't. OK, but there's going to be up, up, and across. And the ones that go up are not crossing.
There are ways you could vary that. I could do it and not do any of the ordinary throws. But let's just stick with this up, up, and across. What does that look like to the camera in the ceiling. So we've got a couple of standard throws, let's say. And then this one doesn't go from the right hand to the left hand. 1, 2, ah, these were 5's, not 3's, sorry. There.
OK, this one doesn't go from the right hand to the left hand. It goes from the right hand back to the right hand. So that would really go straight on here. But I'm going to draw it out here so I don't get confused, so it doesn't collide with that, even on the picture-- hopefully not in the air either.
And this one is good. So that was from my right hand, back to my right hand, then from my left hand back to my left hand. And then there's this one, which is the one straight across. So notice that this slope is much higher than that slope. It's covering more space in less time. That ball going cross is going very fast. And then I go back to these ordinary throws, that we would ordinarily do in three in the three ball cascade.
OK, so that's that trick I was doing a moment ago, just once in the middle of three. So this, you could imagine drawing pictures like this and shuttling them around the net in order to convey what pattern you want to be talking about. But we can actually compress this a whole lot, too, if we don't keep track of all this picture.
I just want to keep track of the basic thing. So here's the idea of-- and I'll explain the name later-- these juggling patterns. This notation is called site swaps. I'll tell you why later. Record only the order the balls are thrown and caught.
So let me show you a couple of patterns that are different as juggling patterns, but are the same if all you're keeping track of is the order. So here was three that I was doing before, that we have pictured up there. And here's also three, but now all these throws are from the outside to the inside, so sometimes called the reverse cascade. Or I could do it reverse in one hand and not in the other.
And none of these are different in terms of the order that balls are thrown and caught. Here's another one that has the same order that balls are thrown and caught, I hope you see. But people would often call this a different pattern. So that one actually has a name. It's named after Steve Mills. It's called Mills' Mess.
So that's what we're going to keep track of. And I'll say that numerically here. So what I'm going to keep track of, whenever I do a throw-- so a throw number, or let me say an n throw, means that, after it's thrown, n minus 1 other throws happen. Then it lands.
OK, so here, I do this throw. And while it's in the air, I throw my other two balls. And then it's the third one to land. So that's a 3. And in fact, all these are 3's. So this pattern, not too surprising, you might call it 3, 3, 3, 3, 3. So what's different here-- 3, 3, 3-- this guy, while it's in the air, I do 1, 2, 3 other things, and then it's the fourth to land. 4, 4.
This guy, I throw it, and it's the very next thing. Nothing else happens in between. That's a 1. And then I go back to 3's. So this is what jugglers actually say aloud to one another at juggling conventions. Oh, yeah, so in the middle of that, you do 4, 4, 1, and then you go back to 3's. Or you just run 4, 4, 1 for a while.
So I was showing you that before when I was doing 3, 3, 3, 4, 4, 1, 4, 4, 1, 3, 3, 3, 3, 4, 4, 1, 4, 4, 1, 4, 4, 1, 4, 4, 1, 3, 3, 3, 3, 3, 3, 3. So in another slight variation-- 4, 4, 1, 3, 4, 4, 1, 3, 4, 4, 1, 3, 4, 4, 1, 3. So in this one, the green ball is only doing 1's and 3's. The red and silver are only doing 4's.
So remember, 4 is actually the sort of throw you do when-- if I only did 4's, I'd get this pattern. All the throws are from the middle to the outside, but they never cross hands. Every throw here is from a hand back to itself, two in one hand and two in the other hand.
And it's the same if I did them like this. It's the same order. Those are still 4's, even though you might call this a different pattern than this one. So that's another way to think about an n throw. It's the sort of throw you might do if you were doing n balls and just n balls.
So let me make a couple of simplifying assumptions on what sort of juggling I really want to keep track of. And these are called vanilla site swaps, with Rocky Road to come later. So vanilla site swaps are alternating hands and only one ball in a hand at any time.
So it's easy enough to come up with patterns where that's not true. So here's a perfectly good pattern with four balls, where I'm throwing two throws at the same time. Again, it's two in each hand. But it's going to be a little more complicated to notate than if I only have one throw to talk about it at a time. So I'm not going to talk about it yet.
Another possibility is patterns where you have two balls in a hand at once-- or not in hands. There we are. And so two balls in a hand at once, this is a fairly simple one. Here's a more complicated one.
So there are perfectly nice, very interesting patterns that violate these. And we should talk about them, but just not yet. Let's start with these assumptions. So what that means is I'm alternating hands, therefore even throws go to the same hand. And odd throws cross.
If I'm going throw, throw, throw, then I do a 3. And it comes down 1, 2, 3 later, it'll come down three throws later. It'll be in the opposite hand. But this 4-- 1, 2, 3, 4-- that's the fourth one to come down, I've alternated to come down back in the same hand. So we saw that in the 4, 4, 1. We had the 4, 4, 1. So in 3, everything's crossing, but the 4's in the 4, 4, 1 weren't crossing.
So I wanted to say, what are some basic throws you may talk about? Let's not admit the existence of that one yet. So this, this you've seen the usual cascade. And here, two in each hand. What do 5's look like? They're the throws you have if you're just doing five. So this again-- this would be el ocho con cinco-- every ball is in this figure eight pattern. So any single ball is in this figure eight.
But there are just more of them doing it than there were before. What do 6's look like? So five, again, it's the cascade, five ball cascade. What do 6's look like? They look like three in each hand. So that's like that. What do 7's look like? What do 10's look like? I'm sure you can imagine it at this point.
What do 2's look like? Well, they are the lowest throws you could imagine doing. And they're going back to the same hand, because 2 is an even number.
So let me now tell you about a bunch of patterns so we can see some examples. So we already saw 3 and 4, and 4, 4, 1, and 4, 4, 1, 3. Let me show you 5, 1, 5, 3, 1, 5, 3, 1, 3. And then we'll think about how, in general, to take a pattern and figure out what to-- take a list of numbers and figure out what the actual juggling looks like.
So OK, remember, the difference between these, this is odd length, 4, 4, 1. And so it was this left-right symmetrical pattern. 4, 4, 1, 4, 4, 1, 4, 4, 1, 4, 4, 1. Because it's odd length, and so sometimes it starts in my right hand, sometimes it starts in my left hand. Whereas 4, 4, 1, 3 is even length.
So it's always going to be 4, 4, 1, 3, 4, 4, 1, 3, 4, 4, 1, 3. It's this asymmetrical pattern, because it's always starting in the right hand. 5, 1, asymmetrical pattern, it's going to be all 5's from my right hand, all 1's from my left hand. So my left hand thinks that it's in this pattern, whereas my right hand thinks that it's doing 5. And put together is this three ball shower. All these things end up with the water terminology-- the cascade, the fountain.
So two in each hand is called the four ball fountain. Three in each hand is called the six ball fountain. But this one, this way you might try to learn to juggle, if you didn't have somebody around to tell you, don't because this is so much easier, that's the three ball shower. Another one like that, and I mention the 7, 1-- so 7, 1 is the four ball shower. And there's also 9, 1, and 11, 1. But you can guess what those look like.
So 5, 3, 1 is an odd length pattern. I'm just going to do it once in the middle of three. Out of my right hand will come a 5. And then my left hand will just do another 3. And my right hand will do another 1.
So it'll be a 5 and a 1. 5, 3, 1. 5, 3, 1. So you see the one underneath. Hear it, at least. If I kept doing that-- this is an odd length pattern-- I will end up with 5's coming from each side.
5, 3, 1, 3, an even-length pattern, the 5's are always going to come from the right. Or supposedly the left, but I'm right-handed, and so they come from the right. 7, 1, you saw.
Let's look at how to-- I guess, this I got hide. OK, let's look at how to take a sequence of numbers and try and figure out what's actually going on for a juggler. So here's a sequence of numbers.
And when I say 6, 6, 1, 5, 1, 5, you should think, oh, he means, 6, 6, 1, 5, 1, 5, 6, 6, 1, 5, 1, 5. And in the infinite past, 6, 6, 1, 5, 1, 5. So I'm just going to list one cycle of it, and you have to interpret that as a periodic thing.
OK, I'm going to throw a ball at the very beginning, this 6. So that's this three in one hand throw that comes back to the same hand. And that's going to come down 1, 2, 3, 4, 5, 6 steps later. But because I'm only drawing this with one cycle, instead of drawing this, I'm instead going to draw that.
And then this 6 would come down on that 6, six steps later. But back in the cycle, it's here. 1's OK. The 5's going to come down 1, 2, 3, 4, 5 steps later on the 1. And the 1's going to come down here, and the 5 is going to come down 1, 2, 3, 4, 5, 6 steps later, back on the 1. So let me make this different colors, actually. There will be that 6 and this 6, and this 1 followed by 5, and this 1 followed by 5.
So you should expect to see-- and so the 6, 6 is going to look like that. Those are with the two balls going way up high and not crossing are going to look like. The 1, 5, 1, 5, each of those is a ball that's going 1, 5, 1, 5, 1, 5. And that's all that ball's going to do forever. So on the outside, we've got the 6, 6. And in the middle, we have the two 1, 5's chasing each other. 6, 6, 1, 5, 1, 5. 6, 6, 1, 5, 1, 5, 6, 6, 1, 5, 1, 5, 6, 6, 1, 5, 1, 5.
Now, how did I know there was only one ball doing these things? In this particular case, there only one ball. In general, actually, how did I know there were only four balls here? Let's state and prove a theorem. Theorem-- the number of balls is-- is there anybody who has figured out since the beginning of this talk-- not who already knew-- but who has figured out, since the beginning of this talk, how to determine the number of balls? I should have stated it as a problem. Yeah?
AUDIENCE: It looks like that the number of balls per hand is half of the number? Or [INAUDIBLE] three of the numbers, and the odd one--
ALLEN KNUTSON: If there's only one number in the pattern, absolutely. But if it's a more complicated pattern, like 5, 3, 1, 3, if there's a bunch of numbers, how many total balls? Well, I'll tell you. Oh, yeah?
AUDIENCE: Is it the number of even numbers plus half the number of odd numbers?
ALLEN KNUTSON: Well, that would be the same for 2, 4, and 6, right? So it's going to have to actually-- it's going to grow as the numbers grow. It's not going to just be how many there are.
AUDIENCE: It's the number of loops you have in that one cycle. So you have four loops in that 6, 6, 1, 5.
ALLEN KNUTSON: OK, but what if my pattern had been 4? And then you say, oh, that's 1, 2, 3, and then it would look like that. There would be one loop, but that loop would actually have four balls in it, not one ball. So each loop has a certain number of numbers in it.
This particular example is silly. It only has one ball for each loop. If I had made this a 7-- which I'm not going to show you-- if I had made this a 7, then there would be two guys in this orbit of this juggling pattern. So that's a hint. I increased this number from 1 to 7, and that made it come down in the same place it used to.
I increased this from 1 to 7, which was the length of the pattern, and the number of balls went up by 1. The answer is, it's the average of the numbers of the throws. So 3, 4, 3, 3, 3, 3, 3, 4, for these. This is now 5. And this is 4.
So I'm not going to write out this proof so much in words as give you an example of how it's proved. So the idea is, it's easy if all the numbers are equal. If not, find a decrease mn, where m is more than n, and replace with n plus 1, m minus 1. Stir until done.
OK, so let's see an example of this. Let's start with 5, 2, 5, 1, 2, which is actually a pattern. I could tell. What pattern is it? Well, let me copy it here and we'll see. 5, 2, 5, 1, 2-- so it's got the 5 and the 2. And it's going to the 1, going to the 2 and the 5.
So it's got these throws we know, these 5's. It's got these 2's that I told you about. But I'm going to be lazy. I'm not going to do the 2's like this. I'm going to do them like this. And so actually, it's time for me to admit what goes on in a couple of the holes I left there. 2 I'm going to use to denote a hold.
So 1 you've seen a bunch of times now is this handoff. 0 I'm going to use to denote an empty hand. So it's easy to mistake 2 for 0 because, in both cases, nothing happens. But they're different. Here's the pattern, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2. And the average is 3. And here's the pattern 4, 0, 4, 0, 4, 0. And the average is 2.
OK, so here's 5, 2, 5, 1, 2. And here are the orbits of 5, 2, 5, 1, 2. There is a ball that's doing 5. So I'll explain in a moment how to soup this up to tell you how many balls there are in an orbit. There's a ball that's doing 5. There's another ball that's doing 5, and that's all they do forever.
So they're like this. They're chasing one another as these 5's. And then there's another ball that's doing 2's and 1's. Here it is. So that's only getting handed off and waiting. So if you want to see this performed with a small baby in place of this green cup, then you should look on YouTube for Knutson and giocolieri, jugglers in Italian, where I juggled a small child in front of 700 Italians.
So this is called the baby juggling pattern for that reason. I've also seen it done with a monkey. I didn't do it. I've also seen it done with 170 pound man. It was really impressive. He'd be hoisted from here to there a couple of times.
OK, so that's the baby juggling pattern. And I want to prove to you that the number of balls in it is three. So what I'm going to do is look for a decrease. Say, here's this 5 and then this 2, and say, well, instead of throwing these balls so that the 5 comes down then and the 2 comes down then, let's do this little surgery on the pattern and have them swap sites.
This is the origin of the name. So that means that, instead of the 5 coming down then, it's the 2 that's supposed to come down then. I mean, instead of the 5 being thrown from here to come down then, it should wait one step and be thrown from there. So instead of being a 5 to get there, it should be a 4 to get there. That's the m minus 1.
And instead of this 2 being thrown from then to get there, it should be thrown from one step earlier, which means it needs to be a 3 to get there. So that's the n plus 1. So what I've done is I've made a slightly different pattern. The numbers are a bit more even. Which you could believe that, if I kept doing this, I could get all the numbers to be the same.
So the numbers get a little bit more even. And the average doesn't change, because the sum doesn't change. And it's still a perfectly valid pattern. I know for sure I don't have two balls coming down at the same time, because I know who's coming down at what time. They've just swapped sites.
So we repeat this. 3, 4, here I've got this decrease, so I can turn that into a 2 and a 4 and a 2. And then I can turn these into-- so that was swapping these. And then turn these into 3's.
And when do you get stuck? You only get stuck if there are no decreases. So you might wonder, what if the pattern you get stuck at looks like 2, 3, 3, 3, 4? You say, well, my numbers aren't all the same, but there are no decreases. But that's the same thing, really, as 3, 3, 3, 4, 2. It's the same pattern, just starting in a different place.
And now there's a decrease. If there isn't a decrease somewhere, it's because all the numbers are equal. And at that point, we're in the easy case, QED. I see. Interesting. Does that really not go behind there? Oh, OK, good.
All right, let me show you a few more patterns. 5, 2, 5, 1, 2. 5, 6, 1 is my favorite pattern. 7, 2, 7, 2, 7, 1, 2. 5, 0, 1 is a surprisingly good one. 5, 5, 2, 5, 5, 5, 5, 0, 4, 5, 3. Let's see some of these. So let me start with this one, 5, 0, 1. So that's got one orbit.
I should've told you how to generalize this. The number of balls in an orbit is equal to the sum over that orbit, divided by the length of the pattern. So if you believe this theorem now, then how do you prove that? Well, you say, I can break this 5, 2, 5, 1, 2. I could break 5, 2, 5, 1, 2 as being this orbit, plus this other orbit, plus this other orbit.
Each of these three orbits, they don't talk to each other. They don't need the other balls to be there. Those other throws could just be 0's. So I've broken this pattern as a sum of three other patterns, where each of those patterns only has one orbit, and the hands are empty otherwise. Then you apply the previous theorem.
OK, so the total in this orbit is 6. I divide by length of the pattern, 3. And I find out there's two balls doing that. And in the other orbit, there's 0 balls. So that is this two-ball pattern. Each ball here is doing 1's and 5's. So each ball here thinks that it's in a three ball shower.
Remember, that was this pattern, 5, 1. But one of the balls thinks it's in this shower, and the other one thinks it's in this left-handed three ball shower. So this pattern, this two-ball pattern is much harder than doing three. This pattern is almost as hard as doing five. It's one of the ones that I teach people if they already know three and they already know four and they want to learn five.
This is a good thing to practice. And you saw it before. You saw it with this other ball in it. If I take the purple ball out of here-- can I do that? I think I should be able to. If I take the purple ball out, then I get the 5, 0, 1, instead of the 5, 3, 1 I was doing before.
OK, I wanted to show you the difference between 0's and 2's. These are two different ways to think about four out of five. So if I'm juggling five balls, and I drop one-- no, I drop one.
And I drop one, there are two ways that I might go into doing four out of five and trying to hide that fact. Dropping two, it's harder to hide the fact. One is, I could always throw. I can throw even if there isn't a ball coming into my hand.
So this is 5, 5, 5, 5, 0. I have this empty hand, there, there. And it's much more difficult than 5, 5, 2. And the 5, 5, 2, whenever I don't have a ball coming to a hand, I don't throw. I only throw if I have to to avoid collisions. So this, the rhythm you hear is-- oops, I did a 0 there.
The rhythm you hear sounds like two throws, but it's two out of three. It's 5, 5, 2, 5, 5, 2, 5, 5, 2. I can show you the full hands there. 2, 2-- oh, dear. 2, 2, 2, 2, 2, instead of the empty hands that I had with 5, 5, 5, 5, 0.
So that's the difference between 2's and 0's. 5, 6, 1, so that's like 5, 0, 1 and like 5, 3, 1. It's got the same orbit structure. But instead of having no balls in the middle, or having one ball in the middle-- like I had in 5, 3, 1, that I showed you before, that's always doing 3's-- I'm going to have two balls and they'll be doing 6's.
So it'll be on the outside doing these 6's, like I had in 6, 6, 1, 5, 1, 5, from before, that guy. But the order is slightly different. So there's going to be two balls in the middle doing the 5, 0, 1, you've seen, and two balls on the outside doing 6's. So that's my single favorite pattern.
7, 2, 7, 2, 7, 1, 2, that's length 7, so I've got three balls that are always doing 7's. And here I have the baby. So this is a baby juggling pattern with three balls and a baby instead of two balls and a baby. Sorry, baby.
This I did not perform for the Italians with a baby. 4, 5, 3, should I really have written that one there? So 4, 5, 3, it's a very beautiful pattern. So it's got four, and you know what that looks like, two in each hand, or what it looked like 10 minutes ago, two in each hand.
And it's got 5 and 3. So maybe I should just show you 5, 3 first. So that's like 5, 1 that we saw before, where my right hand is doing something harder than my left hand. Except my left hand isn't just doing these handoffs now. It's doing 3's, these little 3's. So my right's doing all 5's, and my left hand is doing all 3's. And we get this sort of Star Trek insignia picture.
That's called a half shower. And I told this one's called a full shower. So 4, 5, 3, I'm going to do that in the middle of 4. I'll be in the middle of 4. And I will have a 5 and a 3 in there. So there's the 5, 3. So that's doing it just once in the middle of 4.
So one thing that's clear with the numbers, you could think of it, oh, there's 4's, and there's the 5, 3, so it's related to 4 and it's realed to 5, 3. Or you could think, I could knock this 4 down by the length of the pattern to get 1, and say it's like 1, 5, 3 that we've already seen. So with the numbers, it's really very easy to start seeing relations between patterns, which is part of the point of having this numbering system.
So 5, 3, 4-- the other thing you could do with it, of course, is look at the orbit structure and see that the 3 is its own orbit. That's the silver ball that's its own orbit. 5, 3, there. The other three balls are all in the same orbit, though. So it's much harder to figure out what they're doing.
Each one of them is doing a 4, then a 5, then a 4, then a 5, then a 4, then a 5. It's only the ball that's in the 3 orbit in this 5, 3, 4 pattern that is doing something simple. Oops. There we are, approximately.
ALLEN KNUTSON: Yeah.
AUDIENCE: So is there a way to estimate from the numbers, how complex or difficult is a certain pattern to do? Let's say that we fix the average, that was the best way to estimate how complicated it is.
SPEAKER 1: So the question is, how do you look at a pattern and figure out how hard it's going to be from the numbers? Well, so a number of things come into it. One is, the highest throw is really the most obvious thing. More than the average, really the highest throw is going to be the single biggest determiner of the difficulty.
And it may well be that, like me, you might be very right-handed and much better at putting a high throw out of your right hand than out of your left hand. And so you could look at a pattern and say, well, 7, 1, that only has 7's from the right hand, not from the left. 7, 7, 1 has 7's out of both hands, and so it might be much more difficult.
Another thing is, if you have a high throw and a low throw, like a 7 and a 3, then you can't see them. And so that's sort of hard to capture from a mathematical theory. But it's really clear from the juggler point of view how difficult that's going to be if you're mixing those heights.
But it's not as simple as saying, oh, I don't want the throws all to be roughly the same so I can see them, because 0's, 1's, and 2's, you don't have to look at. It's only once you get to a 3 you have to look at it. So you're very much asking an applied mathematics question here, and so it doesn't have a very simple and clear mathematical answer. Any other questions at this point? Yeah.
AUDIENCE: Yeah, how can you determine what the rhythm is going to sound like? Can you do that just by looking at the numbers and being like, bang, know what the rhythm is?
ALLEN KNUTSON: Well, the single biggest determiner is going to be whether it's odd length or even length, and that an odd length 1 is going to sound more even. But the other thing is just, certain throws are noisier than other throws. Namely, 1's are noisier than other throws. And 0's and 2's don't make any noise at all. So that's really I think the most of it. Yeah.
AUDIENCE: Do you keep the balls all in the same plane when you do it, and the same orbit?
ALLEN KNUTSON: Pretty much I keep them all in the same plane. And that's very much by choice. When you see people who are learning two in one hand, for example, and they're doing this, where they're throwing them back at themselves, that really doesn't extend if you want to start doing anything harder. It's much harder to avoid collisions by depth perception. So absolutely, if you want a clean, smooth pattern where you can see what's going on, keeping them on the same plane works a lot better.
ALLEN KNUTSON: Yeah.
AUDIENCE: Did you create the number code, or is that just how you learned juggling?
ALLEN KNUTSON: No. So it was created by three different groups of people working independently in 1985. So I can easily believe it got created by a number of people before then and they didn't tell anyone. And I certainly know it's been recreated by a bunch of people since then before it really got popularized. But basically, we give credit to a couple of guys in California and this one group in Cambridge for creating it.
It's very non-standard-- I mean, more and more so as time goes on. 1985 is a long time in the past now. It used to be very non-standard for people to learn this as part of learning to juggle. Really, most of things I'm showing you are so much harder than doing three. They're only going to be appreciated by somewhat advanced jugglers.
And most of them are not at all what you'd want to work on if you were juggling for money. I go to juggling conventions, and about half the people who come to them are professionals, the street performers, cruise ship jugglers. And they'll do this for fun in their spare time, but absolutely, it has nothing to do with their work. Yeah.
AUDIENCE: How does juggling for money differ?
ALLEN KNUTSON: That you want to do showy things. Like when I'm juggling clubs, for example, and I'll want to be working on clubs in a way-- say I'm juggling clubs with somebody else. I like to have throws that are pretty high and lofty and go slowly and such that things are as easy as possible.
Whereas, if you want to do that for a paying audience, then you want to throw it really fast so the whole thing looks really, really hard, instead of just a basic thing. And there's going to be a lot of this sort of stuff. There's just a lot of things that are very orthogonal to this as an intellectual pursuit. Yeah?
AUDIENCE: You mentioned juggling with other people. How do you extend your notation here when you have two or more teams?
ALLEN KNUTSON: Let me get to that when I talk about even juggling with oneself and having two hands. All the stuff I've been telling you secretly has been one-handed juggling. Mathematically, this is all really just the same as talking about one-handed juggling. So I'll get to that. Yeah?
AUDIENCE: I wanted to know if there are-- there are clearly some patterns which cannot be realized with alternating hands, because you have two balls falling at the same angle.
ALLEN KNUTSON: Again, you're getting beyond vanilla site swap now. It's the same issue, I'd say.
AUDIENCE: Yeah, I just wanted to as if there is an easy-to-check condition, such that the list of numbers gives you a [INAUDIBLE] pattern.
ALLEN KNUTSON: Oh, right, so that's an issue. Yes. So the basic point, of course, is that you shouldn't have two balls landing at the same time. If my pattern has 4 and 3 in it, then-- I shouldn't have 4 and 3. So that's no good.
AUDIENCE: Numbers with difference 1?
ALLEN KNUTSON: No, it's that, if I had, say, a 5, and then three steps later I had a 2. So you don't want to have a distance d between things, and have a throw, and have another throw where these guys are both coming down at the same time. So you have to make sure there are no collisions of this sort. You have to look through the numbers and look at every pair and see that there aren't going to be collisions. That guarantees you half of what you want.
For the pattern to be jugglable, you want that there are never two balls landing at the same time. And the other is, you actually want a ball landing. You do want to say, OK, I'm trying to make a 5 now. But I don't have a ball. But the pigeonhole principle says, if you've got this finite periodic pattern and no two balls and at the same time, then balls land at every time, so there will be a ball. So the only thing you have to check is one direction.
AUDIENCE: It's actually a [INAUDIBLE] problem, because you are [INAUDIBLE] to check [INAUDIBLE] a lot of conditions.
ALLEN KNUTSON: Well, I said you have to check every pair, so it's quadratic in the length of the pattern.
I guess we don't need that stuff anymore. I guess I want to get to now about how to generalize this if you want to deal with multiple throws at the same time, or you want to deal with multiple hands throwing at the same time. So a multiplex is when you have many balls in one hand at the same time.
So let me give a basic example. I'm doing three. And at some point, I catch a second ball. Maybe that was too simple. I'm doing four. At some point, I catch a second ball. Oh, yeah, which example did I want? I'll jump straight to this example.
So in this one, what's going on? Well, it's a symmetrical pattern, so that means it's odd length. If you listen for a little while, you'll figure out that that length is 3. You'll notice that I have a ball in a hand, and I put another ball in there. So that means that this ball is waiting, and another ball is going into there.
So I'm waiting with one hand. And then I'm handing off another ball into there. And now I have two throws to do. And the way I'm going to denote that is I'll put brackets around it to say that these two throws are happening at the same time.
So what are those two throws? Let's see. What are those two throws in this pattern? So they look like that. So there's a low one and a high one. And it's the high one that is crossing. So there's an even number and a larger odd number. And what they are, it turns out, is 4 and 5.
So this is how I'm going to denote a multiplex pattern, where I've got several balls I want to throw at a given time. And now what do you still need to check? You need to check not that at any given time there is only one ball coming in. You need to check that, at any given time, there's the right number of balls coming in.
So this, when I get to this point, I don't want to look down and see my hand is empty or my hand has one ball. I want to know that I have two balls coming in. And I do. So I'm going to draw the same orbit picture for this pattern as I was doing for non-multiplex patterns.
And at this time, I throw the 4. And it's going to come down 1, 2, 3, 4 steps later. And I threw the 5, which will come down 1, 2, 3, 4, 5 steps later. And here, that's good. I have one ball coming in, one ball coming out, one ball coming in, one ball coming out, two balls coming in, two balls coming out. So this is a valid multiplex pattern.
I showed you a simpler one before, where I had just doubled one of the balls. So instead of having one green, one red, one silver, I've doubled the silver ball. So what's that pattern? That's 3, 3, and then I doubled that one. And the orbit structure is very simple.
So that's one of the things you need to generalize beyond vanilla site swap. And the other is synchronous throws. So at this point, say I want to say, at any given time, not the throw that I'm doing, but the throw that I'm doing from my right hand and the throw that I'm doing from my left hand.
And so this is the standard notation for it among jugglers, that we'll use parentheses, not brackets. And you can mix these two. You can have synchronous patterns that are also multiplex, so that inside here, seeing what your left hand is doing, you might have several throws going on.
And let me say here, there's a convention which is to use all even numbers. So let's say I have this pattern called 4, 4. So that means I'm going to be repeating that forever. So 4, 4, then 4, 4, then 4, 4. Just like I was repeating 6, 6, 1, 5, 1, 5 here.
If I were using the same rule as I was here of, OK, you take an arrow out of a number, count that many numbers forward, that's where the arrow is pointing-- so the 5 is pointing 1, 2, 3, 4, 5 steps forward-- then you might think these 4's are going to be pointing one 1, 2, 3, 4 steps forward. But the convention is that there's nothing here. You count the gap in between. So this is actually going 1, 2, 3, 4 steps forward, and this one also.
So why do we do this? Because this pattern, as drawn, is this one. And it seems silly to call those 2's, even though, mathematically, it would make more sense to call these 2's. They're landing two steps in the future.
But since that pattern's so much like this one, we would prefer to call them 4's. So we make that choice. We take what's the natural thing to do mathematically and double all the numbers. So that you don't have to say, well, he said 4, but is that 4 like this sort of 4? Or is that 4 like this sort of 4, because he's doing something synchronous? So it's just a choice of mathematical convention.
AUDIENCE: It is a choice with the respect to [INAUDIBLE] theorem [INAUDIBLE]?
ALLEN KNUTSON: And it does that. So that is a good motivation to make this choice, that, again, the average of the numbers will be the number of balls. Yeah?
AUDIENCE: When you have 3, 3, 3, 3 that you just threw, the average there is 3, but you have four balls.
ALLEN KNUTSON: Ah, how does the averaging theorem generalize? You have to add up all the numbers and divide by the length, not by the number of throws, by the length of the pattern. So before, the length of the pattern and the number of throws was the same.
But now, the number of throws is 4. There are four total throws here I guess. Oh, sorry, the number of numbers is 4, and the length of the pattern is 3. So the total here is 3 plus 3 plus 3 plus 3, divided by 3 is 4, like it's supposed to be.
So I forget when we started. I have one more topic I could include. It'll be about another five, 10 minutes. OK?
SPEAKER 2: Do you want to stop and have questions, or do you want to hear one more topic?
AUDIENCE: One more topic.
AUDIENCE: Which is the topic?
ALLEN KNUTSON: I'm afraid it's really cool.
ALLEN KNUTSON: OK. So here's another way to think about what these numbers mean and how you might talk about a juggling pattern. Let's say we are watching a juggler on videotape, and we stop the videotape and ask, how much information do we have about the future?
We don't know what throws the juggler is going to make in the future. But we do know what throws jugglers made in the past. So we know how many balls there are in the air and when those balls are going to land. So another way to say that is, imagine that somebody is juggling, and they stop.
What does it sound like? Well, in this case, if I stopped just after I make this left throw, you hear, catch here, catch here, pause, catch here. So let me draw that as catch, catch, pause, catch. And I might say, so in the past, I've always caught balls, and in the future, I've got no more balls that are going to land.
So here's now. And the balls that I have in the air, they're going to land at those three times. So here's where the action is, of course. I know in the past I always caught a ball. And there's only three balls in the air, so let me just write this as the state of the-- so this is a juggling state.
OK, so what was I doing in that pattern? Which pattern is this?
AUDIENCE: [INAUDIBLE] shower.
ALLEN KNUTSON: Yeah, what are the numbers?
AUDIENCE: 5, 1.
ALLEN KNUTSON: 5, 1, yes. There's 5's from my right hand, 1's from my left hand, average of 3. OK so now I'm going to do a 5. This was just after I did the 1, and you heard the 1 land here. So now I'm going to do a 5.
And what's going to happen? I've got a ball in my hand right now. So the time will move forward one step. I'll throw this ball so it'll land 1, 2, 3, 4, 5 steps in the future. I'll still have this and this. But instead of that, I'll get an x there, that this ball has been through to land five steps in the future.
OK, what did I do after that? So this was my right hand 5. Now I do my left hand 1. The ball is in my hand. I'm going to throw it to land one step in the future. Time will march forward, and you'll get x, x dash x. That's back here.
So what if I were doing not the this complicated thing, but I were doing the three ball cascade? That's this state, where I just stop juggling in the usual three ball cascade, and you hear catch, catch, catch, no pauses. So from here, I could throw the ball I'm catching right now. And it'll land 1, 2, 3 steps later. And I'll get xxx again. And that looks like that.
How can I relate these? What if I'm doing a three ball cascade, and I want to go into the three ball shower. That's what this is for. It's for figuring out these transitions between patterns.
So you look at this and say, I want to get from this state to that state. I'm going to need to create a hole. So instead of throwing a 3 to land then, I will throw it to land one step later and make a hole there. That will be this hole. So I'll throw a 4.
OK, what does that look like? I'm in the middle of 3. I want to get to 5's from my right hand. And so I want to do this just before I do the 5 from my right hand. And you do the 4 for my left hand, as this thing, this two in one hand throw.
So I'm going to use the green one for that. There will be this green 4, and we'll see what color follows it. There. And it was followed by that 5 up above. Let's see it again. So the next time, green 4, 5. Oops. And I did the 1, which I wanted 2. OK, one more time. Green, 4, 5, 1, 5, 1, 5, 1.
What if I want to get back from the shower to the cascade? So I could, at this point, throw the ball that I'm holding as a 2-- I mean, hold it as a 2. So that's this 2. And so that means I'll be doing 5, 1, 5, 1, 5, 1. And instead of doing the 5, I do a 2, and I'm back at 3. So 4 to get in, and 2 to get out. 4 5, 1, 2.
Now, 2's are boring, so what could we do instead that would be more interesting to look at? Instead of doing this 2, we can't do a 3. That would cause a collision. We could do a 4. And if we do 4, we get to, this guy will now land after that. I'll get x dash x and another x.
OK? And at this point, we could fill in a 1. So I'm in 3. And I go 4, 5, 1, 5, 1, 5, 1. Now, instead of doing a 5 out of my right hand, those high crossing ones, I'm going to do a 4 out of my right hand. There it is. I'm back at 3.
So 4 to get in, and right hand 4 to get out, left 4 to get in. I'll do it all with the red ball. There will be the left 4 to get in, and right 4 to get out. Here it is. No, not that time. Right 4 to get out. There we are.
And note, here's 4, 4 1. You saw that one before. Here's 4, 4, 1, 3, 4, 4, 1, 3, 4, 4, 1, 3. You saw that one before. Where's 5, 2, 5, 1, 2? Well, we'd start here. And we do 5. And we do 2. That'll take me to there. And then we do 5, 1, 2. 5, 2, 5, 1, 2. So a juggling pattern is the same as a closed loop in this directed graph of juggling states.
And so you could ask questions then like, what does a juggling pattern look like if it doesn't repeat a state? If it repeats a state, that means you can break it up as two juggling patterns put together. And you get these impossible combinatorial problems.
A very interesting one, though, is, what does random juggling look like? You could say I want to have only n balls, and let's say b balls, length and pattern, then I'm going to get only finitely many juggling states. And choose b, many juggling states, and I could just do a random walk on there.
At any point, I do one of the throws available to me. Unless my hand is empty, and then I have to do a 0. So that has been analyzed, this random juggling, the stationary state of it. So the likelihood you'll find yourself in a given juggling state, if you're juggling randomly, has been calculated in this paper called "Juggling Probabilities." in the American Mathematical Monthly.
So this is the part of things that I was personally involved in, in the theoretical, coming up with this picture of juggling states to describe juggling patterns. And it's really handy for coming up with how to get from one trick to another. And we'll stop there.
SPEAKER 2: Thank you very much, Allen. This was very nice. This is part of our Math Awareness Month celebration. The Math Awareness Month is a national event. Each year, it has a theme. This year it's mathematics in sports, and we're declaring juggling a sport.
Here's your certificate of appreciation.
ALLEN KNUTSON: Thank you.
SPEAKER 2: Another means by which we celebrate Math Awareness Month is by having a T-shirt design contest with high-schoolers at Ithaca High School. The students design T-shirts in recognition of the theme for Math Awareness Month each year. And then people at Ithaca High School vote on the T-shirt design, send their favorite designs here.
And then we go put all of that together and come up with a winner. And this is this year's winning design. So I'd like to present you with a Math Awareness Month T-shirt.
ALLEN KNUTSON: Thank you.
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Allen Knutson, Cornell professor of mathematics and former world record-holding juggler, gives a public demonstration and lecture on how juggling patterns can be represented the mathematically.
The mathematics of juggling was codified about 25 years ago, when three different groups of jugglers came up with a notational system, called "siteswaps," which made it easy to record patterns, note similarities and find new juggling patterns--even by computer.