Where Math Meets Art: New Frontiers in the Mathematics of Juggling

Dominic Klyve – professor of mathematics at Central Washington University – recently published a paper titled “A Zeta Function of Juggling Sequences” in the Journal of Combinatorics and Number Theory. When we found out about this publication, we all got very excited and made high-pitched squealing noises, then we emailed him about writing a piece for us at eJuggle. Previously being an IJA member, himself, Dominic enthusiastically agreed to write a little something for us! He and I spoke back and forth a bit about how mathematical an article this could be before a majority of readers started having flashbacks to their torturous high school math years, and decided that it might be nice to include a piece about Dr. Klyve’s personal experiences with juggling (sans math). As a result, I got to have a series of lovely phone and email conversations with him! The following “interview” is a compilation of the most interesting and fun information we exchanged during these conversations.

Okay, so just to get me in the right mind for this whole question-asking thing: where did you go for your undergraduate education? Grad school?

I went to Hamline University in St. Paul Minnesota as an undergrad. You are from New Hampshire, right?

Yes, I am!

I went to Dartmouth, up in Hanover, for grad school!

No way – small world!! Okay, so what was your early experience with juggling? Did you pick it up at Hamline?

I was in junior high school when I first started juggling; my little sister got a copy of Juggling for the Complete Klutz, so she locked herself in her room and started learning. I was annoyed that she had a skill that I did not have, so I sort of “borrowed” her book, and started learning in secret. She ended up only maintaining interest for a few weeks, but I stuck with it. When I started, youtube wasn’t around, so I learned strictly through reading about juggling – juggling artists were more graphical than kinetic, to me.

Were you interested in siteswap back then?

No – I was first exposed to the concept of mathematics and juggling when I was 17. I read an article in Discover magazine, I think (or maybe it was Juggler’s World?) about some work Ron Graham had done with the mathematics of juggling. I remember thinking that this guy – who is interested in both math and juggling – this guy is cool. I was aware of the concept of siteswap at that point, but I didn’t really make an effort to learn it until my first year of grad school.

What made you look into it in grad school?

I read Ron Graham’s article Juggling Drops and Descents at a friend’s suggestion. We were both pretty excited about it, and we even made some pretty tenuous plans to present the information at club, but I don’t think that ever actually happened!

Have you gotten to any juggling festivals, since you started?

Not very many…I went to MONDO in…I don’t know – maybe 1996? That’s all, though.

You know, St. Louis Jugglefest 2012 is coming up in October…you could bring the whole family…

<laughter>

Speaking of – you have kids (I can hear them in the background)!

<laughs> Yes! We have a two-year-old girl and a five-year-old boy. You learn to do things one-handed, when you become a dad. (This was said as Dominic was cleaning something up on his daughter’s arms, I believe.)

How do your kids feel about juggling? They are probably a bit young to have started, but what do I know?

My son started mimicking my juggling at around one year, actually! He picks up balls and things and will wave them around and tell people that he juggles! It’s very cute! My daughter has started to show more of an interest, too. I keep juggling balls in my office, and when she tells people about the time she spends there, she tells them she juggles. This, to her, seems to mean taking the balls into the hallway and throwing them around! Usually, though, she seems more interested in watching.

She seems more like me! What about your wife? What is her opinion on your work in juggling and math?

She has long since stopped trying to understand the mathematics of what I do! She teaches science, so it’s not that her field is very far-removed from what I do. I suppose she likes my juggling!

What is the reaction that most mathematicians give, when you explain your research? Non-mathematicians?

Most mathematicians seem vaguely aware of a relationship between math and juggling. They tend to accept it as legitimate research. People outside of the field are generally more surprised – even non-mathematicians at CWU tend to show shock! <laughs>

Speaking of CWU – is there a juggling club that exists on campus? Are you at all involved?

Not to my knowledge, no. However, since this piece was published and students started hearing about it, I have been approached by a handful of them: “You juggle?! We should hang out, sometime!” So, there might be some rather informal gathering in the works, and I may be involved!

Are you at all interested in performing as a juggler, or is it more of a hobby to you?

Juggling is just for fun – I’m not good enough to be serious about it! I have done small performances, but nothing very serious.

Okay, Dominic! See you in St. Louis!!

<more laughter>

 

The following article is a piece written by Dominic briefly outlining some relevant history of mathematics in juggling, and describing new work published in the same field by Klyve, Erik Tou, and Carsten Elsner. Enjoy!

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Many eJuggle readers are probably vaguely familiar with the fact that some people study the mathematics of juggling. Fewer, I expect, know exactly what this means. The purpose of this article is to give jugglers a sense of the sorts of questions people look at when they work on the mathematics of juggling, and to give readers some insight into brand new work (hot off the press!) in this field, just published in the second half of 2012.

I should also note that I’ll assume the reader is already familiar with basic siteswap notation. If you’re not, this is a good time to change this! The IJA just teamed up with Mike Moore on a video tutorials project. In less than ten minutes, you can watch the introduction to vanilla siteswap, at which point you’ll know all you need to know to understand what I’m about to write about.

So what is it that mathematicians study? One good example of a math-juggling idea is the “average theorem.” It states that you can determine the number of balls being juggled in any siteswap pattern just by averaging the values in the siteswap. So if you see a new siteswap (say, 424413), you can quickly determine how many balls you’ll need to try it. If you got “3” for this example, you’re off to a good start!

Another well-known result is that the number of siteswap sequences with a fixed length (call it n) using any number of balls up to some number (call it b) is (b + 1)ⁿ.

Mathematics also tells us that jugglers will never run out of things to do. Mathematically, we can prove that there are infinitely many siteswaps. We also know that there are infinitely many primitive siteswaps. (A primitive siteswap is one that can’t be broken down into smaller ones. Our earlier example is not primitive, since 424413 is a combination of 42, 441, and 3). Interestingly, no one yet knows if there are infinitely prime siteswaps – these are siteswaps with no repeated states. See [1] for the mathematical details.

Finally, mathematicians have developed a systematic way to write out all possible siteswaps. Mathematicians sometimes say that before they did this, no one ever juggled the 771 pattern. I don’t know whether this is true, but if it is, it’s a great story!

All of this is old information, though, and we’d like to move on to something new. The newest object in the mathematics-of-juggling world is the juggling zeta function, and this is what I want to explain.

The juggling zeta function first appeared in a mathematics journal in the fall of 2012, in a paper co-authored by me, Erik Tou, and Carsten Elsner [2]. I’ll tell you the highlights, and if you want more details, you can read the original paper!

First, we shall need a little bit of background. Readers are warned that there is some actual mathematics (at about the junior high school level) coming!

Zeta functions

Zeta functions are some of the most valuable tools in mathematics – they take an infinite set of numbers, and roll them up in one neat little package to make them easier to study. This usually involves four steps:

1)      Choose an infinite set of numbers.

2)      Raise them all to a power (we usually call this “s”).

3)      Turn these powers into fractions.

4)      Add them up, and see what we get.

Let’s look at this with a famous example in the history of mathematics. The problem, in the early 18^th century, was called the “Basel Problem.” Here, we note that the basic idea of the problem was to determine the zeta function for the positive integers with s=2. (In math-speak, the goal is to find ζ(2) – the funny looking symbol is the Greek letter “zeta”). How does this work? Let’s follow our four steps:

1)      Choose an infinite set of numbers. We’ll make it easy, and choose

1, 2, 3, 4, 5, 6, 7, 8, …

2)      Raise them to a power. We’ve chosen the power 2, which means we square the numbers:

1, 4, 9, 16, 25, 36, 49, 64, …

3)      Turn these powers into fractions. The math term for this is “take reciprocals:”

1, 1/4, 1/9, 1/16, 1/25, 1/36, 1/49, 1/64, ….

4)      Add them up. This part is easy:

1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + 1/49 + 1/64 + ….

After many years of searching for the answer, a mathematician named Leonhard Euler found the precise value of the final sum. He surprised the mathematical world by finding that the sum above, which we have called ζ(2), equals precisely π²/6! The fact that pi, which usually concerns round things, showed up in a problem about squares, which are emphatically not round at all, is one of the things that made mathematicians so excited about zeta functions in the first place.

Of course, there’s no special reason that we should set s = 2. Any value of s would work: we could take s = 3, or s = 4.567, or even numbers like s = ½ + 14.1347i. (Do you remember i from high school math class? It hardly matters either way for this story.) What’s important is that choosing the value of s determines the final sum. Mathematicians say that the final value is a function of s.

Back to Juggling

The real question, of course, is: What does this have to do with juggling? We’d like to be able to do the same thing with siteswaps that we did with numbers – following steps 1-4 – and make a juggling zeta function. There’s just one problem. It wouldn’t make any sense. No one has ever found a way to find any meaning in, say, 1/(441).

Happily for our mathematics, there is a solution. All we had to do is to find a way to turn each siteswap into a single number.

In fact, the most important decision when trying to build a new zeta function is how to turn the objects in which we are interested (in this case, siteswap sequences) into numbers. We call this creating a norm. This is crucial – our zeta function will only work if we find a good way to do this. Moreover, there are rules about how this must work:

1)      All the numbers have to be positive.

2)      If we put together two small siteswap sequences, the number for the big one should be the numbers for the two small ones multiplied together.

Because we would like our function to have some meaning for jugglers, we added a third rule which is not mathematically necessary:

3)      The number assigned to each siteswap sequence should correspond, at least roughly, to how difficult it is to juggle.

The challenge is to capture as much information about the siteswap as possible, knowing we’ll have to leave out some of it. We tried various ideas involving the number of different heights to which the balls are thrown, or using the difference in the heights of consecutive throws, but we couldn’t make these work out well mathematically. In the end, we decided that we would associate each siteswap to the number bⁿ, where b is the number of balls, and n is the length of the siteswap. Juggling more balls is harder, of course, and generally long patterns are harder than short ones. The following table shows some of the values assigned to various siteswaps readers may know:

Siteswap	Balls	Length	Norm (bⁿ)
(3)	3	1	3
(51)	3	2	9
(441)	3	3	27
(531)	3	3	27
(333423)	3	6	729

 

This table shows quite well the strengths and weaknesses of our norm. It does succeed in capturing some information about each sequence, but it doesn’t really measure difficulty. (Is 333423 really harder than 441? Probably not.) I would here put out a call to jugglers – if you can find a better way to represent each siteswap as a number that follows rules 1) and 2) above, please let me know! There may be more interesting mathematics hidden behind your new method.

But never mind the new, better way that you may come up with – our way works quite well. Given the (infinite) list of all juggling sequences, we can still build a new function: a new zeta function. The steps now become:

1)      Choose an infinite set of siteswaps. In this case, we choose all of them (for three balls):

(3), (51), (441), (612), and so on….

1b) Turn them each into a number. This is the new step! And we use the table above

3, 9, 27, 27, …

Now the rest is easy!

2)      Raise them to a power (call it s).

3)      Turn these powers into fractions.

4)      Add them up.

That’s all there is to it. You now know how to define the juggling zeta function. All you have to do is choose your favorite value of s and follow the steps.

The rest of our work involves mathematically technical questions. For example, can we find a way to get the equation to make sense even when s is negative? Can we precisely find all the values of s which make the equation equal zero? (answer: yes!) And can we generalize the equation to several balls? (Interestingly, the mathematics gets easier for siteswaps involving more than 16 balls. At this point, even I will admit that the work is probably not practical!)

So there you have it. The mathematics inspired by juggling turns out to be terrifically interesting, and it’s a growing field. The field of mathematics owes juggling a debt for the many interesting problems it has suggested; I hope that someday mathematics can repay the favor.

 

References

[1] Fan Chung and Ron Graham. Primitive Juggling Sequences. The American Mathematical Monthly, March 2008. pp. 185—194.

[2] Carsten Elsner, Dominic Klyve, and Erik Tou. A Zeta Function for Juggling Sequences. The Journal of Combinatorics and Number Theory, Volume 4, Issue 1 (2012). Article 4. Also available in reprint form from the author’s web page at https://sites.google.com/site/profklyve/research.