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The Mathematics of Juggling

Burkard Polster
Publication Date: 
Number of Pages: 
[Reviewed by
Kevin R. Burger
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Given that the study of mathematics is at least 3,000 years old and the earliest known record of juggling was recorded by the Egyptians around 1800 B.C., it is interesting to note that no serious mathematical study of juggling occurred until Claude Shannon's work on his famous juggling theorems in the 1970's. Shannon proved the first mathematical theorems on juggling, but it was not until the mid 1980's that other mathematicians began to seriously study juggling patterns. Apparently three groups of people developed similar mathematical ideas on which this book is based independently: Bengt Magnusson and Bruce "Boppo" Tiemann at Caltech, Paul Klimek at UC Santa Cruz, and Adam Chalcraft, Mike Day, and Colin Wright at Cambridge (as with many discoveries, who was "first" may be in dispute here, but these are the discoverers according to Polster). Therefore, it is not surprising that since the 1980's a great deal of mathematics has been applied to the study of juggling, as both disciplines are essentially about patterns. The Mathematics of Juggling (hereafter referred to as TMOJ) by Burkard Polster is the first book of which I am aware that brings together much of this information into a single, unifying source, and it succeeds. Polster, according to the website at his institution is the Senior Logan Research Fellow in the School of Mathematical Sciences at Monash University, Australia, and as might be expected, he is also a juggler. In fact, according to the introduction of his book, around 40% of serious jugglers also have a mathematical background.

According to the author, the target audience of TMOJ is, "First and foremost ... all mathematically minded jugglers. However, mathematics educators, readers of popular mathematical literature, and mathematicians interested in unusual applications of some of their favorite tools and techniques will also be among the readers of this book." I concur with Polster; I think the readers who will most enjoy this book are jugglers with a mathematical background. Nonjuggling mathematicians will still find the material quite accessible because the requisite juggling knowledge is minimal and is presented as needed throughout the book. Polster does a good job of not assuming the reader has specific knowledge of juggling or, in Chapter Six, of change ringing (to be discussed later in this review). On the other hand, jugglers who are not mathematically inclined might find the mathematics intimidating — although one of Polster's stated aims is to "turn jugglers into mathematicians".

The mathematics in this book should be mostly accessible to the upper-level undergraduate student. The bulk of Chapters Two, Three, and Four uses combinatorial mathematics and a small amount of graph theory; Hamiltonian cycles are used in a later chapter. The more complex mathematical results involve the Möbius function μ(n), the affine Weyl group, the Euler totient function φ(n), Gaussian coefficients, Galois numbers, and Stirling numbers, although Polster provides footnotes explaining most of these in the appropriate place for those readers who might be unfamiliar with the terms. Chapter Five uses results from classical mechanics to model the physics of juggling, but this material should be readily accessible to the undergraduate physics or math student as well. Chapter Six, on change ringing, involves some group-theoretic concepts such as permutations, symmetric groups, Cayley graphs, and cosets. Chapter Seven includes a small section on juggling topological braids and proves that every finite braid can be "juggled". Polster does not provide mathematical proofs for every result in the book. In fact, the number of "formal" proofs is quite small. In other places, sketches of proofs are given, and some results are presented with just brief descriptions. The lack of formal proofs would not be a hindrance for mathematics educators as many of the results which are not proved would be good exercises for students.

The bulk of TMOJ examines three types of juggling patterns (written mathematically as sequences). Simple juggling sequences are juggling patterns in which at most one out of the b balls being juggled gets caught and thrown on every beat (trust me, as a very amateur juggler who has yet to master 4-balls, there is nothing simple about juggling more than three balls, but nonetheless, simple is what they call them). In a multiplex juggling sequence, more than one ball may be caught and thrown by a hand on a single beat. All simple sequences are also multiplex sequences. In theory, all simple and multiplex sequences can be juggled in one hand, although in practice, most, especially those involving more than three balls, cannot. The third type of juggling pattern is the multihand juggling sequence which involves juggling objects (i.e., props) using more than one hand. In practice, multihand juggling is performed by a two-handed juggler or by groups of jugglers tossing objects back and forth, the latter is called pass juggling by jugglers. A large chapter on the mathematics of change ringing is included, and relationships between juggling sequences and ringing sequences are explained.

A particular simple juggling pattern (e.g. the 3-ball cascade or 4-ball fountain) can be represented in what jugglers call site swap notation, but in TMOJ these are called juggling sequences, as they are just mathematical sequences of integers. The 3-ball cascade for example is represented by the sequence 333..., which is abbreviated to simply 3, and the 4-ball fountain is the sequence 444..., similarly written as just 4. The numbers in such a sequence indicate how long a particular ball stays in the air between being tossed and being caught. The higher the number, the higher the ball is thrown and the longer before it is caught. One of the first mathematical questions addressed in Chapter Two after explaining juggling sequences and juggling diagrams is the average theorem, which states that the number of balls necessary to juggle a juggling sequence equals its average [p. 15]. For example the average of the sequence 3 is 3 so we know that sequence represents a 3-ball juggling pattern. It is interesting to wonder, then, if given a particular sequence such as 632 whether this sequence represents a simple juggleable pattern. The average test [p. 16] tells us it is not because the average of this sequence is 3.667 which is not an integer. Lest the reader be deluded into believing that he or she has found an easy way to determine new juggling sequences, Polster explains that not every sequence in which the average is an integer is a simple juggleable pattern, e.g., 321 is not because it would require the juggler to catch three balls on the third beat (the average being an integer is a necessary but not sufficient condition).

In a multiplex juggling sequence more than one ball may be caught and thrown on a particular beat, so a different notation than for simple juggling sequences is required. Multiplex sequences are represented by "finite sequences of nonempty (ordered) sets of nonnegative integers" [p. 65]. For example one particular multiplex juggling sequence is {2} {2} {7,2} {5,4} {2,1} which is normally written in the shorthand notation 22[72][54][21].

In a multihand juggling sequence, we are dealing with more than one hand, so the notation must be extended to represent these additional hands. The notation described in TMOJ, called multihand notation or MHN, was developed by Ed Carstens in a 1991 paper (available here). MHN extends the multiplex notation to an h-hand juggling matrix of period p, where the period is the number of catches and throws made before the pattern begins repeating. The juggling matrix is a p x h matrix where each entry is a multiplex juggling sequence with additional subscripts denoting which hands are doing the throwing and catching.

I hope this short introduction to juggling patterns and sequences has whetted your appetite, because there is a substantial amount of mathematics to be explored here. To give you a feel for the contents of the book, I will provide a description of each chapter. TMOJ contains seven chapters, one appendix, a large number of references (160) to juggling and the mathematics of juggling, and eight black-and-white photographs of the author in various juggling poses.

Chapter One "Juggling: An Introduction" is a six page introduction to juggling and the history of juggling. It begins with a definition of juggling and discusses the many forms of juggling, but emphasizes that throughout TMOJ, in general, references to the term "juggling" will mean "toss juggling". Toss juggling is the type of juggling familiar to most nonjugglers — balls, rings or clubs being the most common props; bowling balls, meat cleavers, and chainsaws being less common. Mathematically, the analysis of a particular juggling pattern is not dependent on the prop being juggled, so the reader can just assume the juggler is juggling balls. At the end of Chapter One, Polster recommends that the reader download one of the many juggling programs which can be used to animate the juggling patterns discussed in the book. These programs are essential for helping the reader to visualize the sequences under discussion. I found Polster's recommendation of JuggleAnim to be a good choice, although it should be noted that the author of JuggleAnim has released a newer, more advanced juggling animator called Juggling Lab (JuggleAnim can be found here and Juggling Lab here). Both of these programs are written in Java and consequently will run in a standard Java-enabled web browser.

Chapter Two "Simple Juggling" is where the real fun begins. Here we are introduced to simple juggling sequences and juggling diagrams which are visual representations of juggling sequences. The chapter begins by stating three important properties of simple juggling sequences: (J1) The balls are juggled to a constant beat; that is, throws occur at discrete equally spaced moments in time; (J2) Patterns are periodic; and (J3) At most one ball gets caught and thrown on every beat, and if one is caught the same ball is thrown. Implicit in these properties is that the juggler does not hold balls in his or her hand — a ball is caught and thrown immediately — although in practice this is impossible (jugglers call the amount of time a ball is held the dwell time). Later, in Chapters Four and Five, Polster discusses some mathematical juggling results which assume a nonzero dwell time, however, for many of the results in Chapters Two through Four, these three properties J1-J3 simplify the mathematical analysis.

As discussed earlier, not all finite sequences of integers represent juggleable patterns. The average test [p. 16] gives us a practical way to determine if a finite sequence of integers is not a juggleable sequence, and the juggling diagram can reveal if the sequence is juggleable, but the permutation test [p. 22] provides a practical, algorithmic way to determine if a particular sequence is juggleable. Here we find that 6424 is a juggleable sequence (with how many balls?), but 6244 is not.

The permutation test also provides a nice way of enumerating all possible b-ball juggling sequences of period p [p. 24]. It turns out for example, that for 3-balls and a period of 3 there are 37 possible juggling sequences. One particular juggling sequence may be transformed into another juggling sequence by performing site swaps and cyclic shifts, e.g., 126, 612, and 261 are essentially the same sequence by cyclic shifts. Of the 37 possible 3-ball juggling sequences of period 3, there are only 13 up to cyclic shifts.

The reader may naturally ask for b balls and period p how many unique juggling sequences are there? Well, Polster explains in the context of juggling cards how to generate all possible b-ball juggling sequences of period p with height h, and spends considerable time answering this question. The answer is determined with help from the Möbius function.

Juggling states are introduced in Section 2.8, and here graph theory is used to illustrate how for b balls and a maximum throw height of h there exists a unique state graph. As an example, the 3-ball state graph with maximum height five has ten vertices (h choose b in general). This graph can be used to enumerate juggling sequences and also to determine if and how a transition may be made from one particular b-ball sequence to another. A juggling sequence corresponds to a closed path in the state graph that begins and ends at the same state and includes at least one vertex and one edge [p. 45]. We also find that a new juggling sequence may be constructed by concatenating two existing juggling sequences under appropriate conditions.

Next, considerable effort is spent discussing prime juggling sequences with b balls and height h. A prime juggling sequence is generated by a loop in the state graph in which no state is visited more than once. It is shown that every juggling sequence can be decomposed into a number of prime juggling sequences. A prime loop is maximal if "its length is maximal among the lengths of prime loops in the state graph" [p. 52]. Given b and h, two outstanding questions are: (1) what is the length MP(b,h) of a maximal prime loop, and (2) how many (maximal) prime loops are there? Polster tells us that the answers to those two questions are not known in general, but he derives an upper bound for MP(b,h) that is close to the actual value in the known cases.

There is much more mathematics in this chapter, but in the interest of brevity, I will stop here simply by noting that Polster also discusses the flattening algorithm, vertical shifts, the inverse of a juggling sequence, a procedure for constructing some b-ball juggling sequences of period p from a p x p matrix, the converse of the average theorem, scramblable juggling sequences, magic juggling sequences, weights of juggling sequences, ground-state and excited-state sequences, complements of state graphs, and transition matrices.

Chapter Three "Multiplex Juggling" Multiplex juggling sequences are sequences in which more than one ball may be caught and thrown on a beat, and much of the material in Chapter Two serves as a prerequisite to this chapter as most of the properties and results relating to simple juggling sequences also pertain to multiplex juggling sequences. Consequently, a thorough understanding of the material in Chapter Two will facilitate understanding of Chapter Three.

Polster first gives the definition of a multiplex juggling sequence (essentially one that does not adhere to property J3 of Chapter Two) and describes multiplex juggling diagrams, which are quite similar to simple juggling diagrams. The next few sections are multiplex counterparts to simple juggling sequences, discussing such topics as the average theorem for multiplex juggling sequences, the permutation test for multiplex juggling sequences, the number of distinct multiplex juggling sequences (using multiplex juggling cards), weights of multiplex juggling sequences (the results of which depend on Gaussian coefficients and Galois numbers), multiplex state graphs, prime multiplex juggling sequences, and maximal prime loops.

Concerning maximal prime loops, it was shown in Chapter Two for simple juggling sequences that a good upper bound on MP(b,h) could be found. However, for multiplex juggling sequences, MM(b,h), the length of a maximal prime loop in the multiplex state graph, a good upper bound is not known.

The chapter concludes with a discussion of operations which can be used to turn existing multiplex juggling sequences into new multiplex juggling sequences. Some of these operations also function on simple juggling sequences (such as concatenation, cyclic shifts, vertical shifts, site swaps), but some are specific to multiplex juggling sequences (such as permutations of integers within the square brackets of the sequence).

Chapter Four "Multihand Juggling" begins by introducing Carstens' multihand notation, which is a matrix whose entries are multiplex juggling sequences with additional subscripts, and showing multihand juggling diagrams as in the manner of simple and multiplex juggling sequences. Polster notes on pages 87 and 94 that both simple and multiplex juggling sequences can be represented by multihand juggling matrices, and such matrices "capture the mathematical essence of just about any juggling pattern that you will ever come across in practice, and they are what juggling animators use as their principal input" [p. 87].

As we saw in the transition from the simple juggling sequences of Chapter Two to the multiplex juggling sequences of Chapter Three, many of the mathematical results for the multiplex juggling sequences of Chapter Three extend to the multihand juggling sequences of this chapter, and the resulting coverage of most of those topics (the average theorem, the permutation test, multihand state graphs, operations involving juggling matrices) is less extensive than in the previous chapters.

Section 4.6 in this chapter is where we are introduced to Claude Shannon's groundbreaking work on juggling and his three juggling theorems. Polster first notes that in this section, "unlike earlier in this chapter, throws are not necessarily required to occur at distinct equally spaced moments in time (at least not to start with), and juggling is no longer performed in hot-potato style", i.e., we are not assuming properties J1, J2, and J3 as in previous chapters. He describes uniform juggling as satisfying the synchronicity principles at all times: (1) the dwell time of a ball d is constant; (2) the flight time of a ball f is constant; i.e., the time the ball is in the air between being thrown and caught again is f; and (3) the vacant time of a hand v is constant; i.e., the time that a hand spends between throwing and catching again is v.

Shannon's first theorem states that in uniform juggling with b balls, h hands, dwell time d, flight time f, and vacant time v, we have (f + d) / (v + d) = b / h, and Polster provides a proof. From this theorem we note: (1) if we hold b, d, and h constant, and increase f then v must also increase — which is logical, if balls are spending more time in the air, then they are spending less time in our hands; (2) if we hold f, d, and v constant and increase b then h must also increase — from this we conclude that to juggle more balls without changing f, d, or v requires more hands; (3) if we hold d, v, and h constant and increase b, then f must increase — hence, increasing the number of balls we are juggling without changing the dwell time, vacant time, or number of hands, means the balls must spend more time in the air, i.e., they must be thrown higher. Polster goes on to describe Shannon's second and third theorems. Section 4.7 presents some work by J. Song and Y. Yam which correlate Shannon's theorems with the mathematics of juggling sequences.

The remainder of Chapter Four discusses what happens if we switch balls with hands in a juggling sequence (leading to a duality principle for juggling sequences), the mathematics of juggling labeled or colored balls, and the decomposition of simple juggling sequences into smaller juggling sequences in terms of orbits of balls in the juggling diagram.

Chapter Five "Practical Juggling" By this time, Polster has covered a substantial amount of simple, multiplex, and multihand juggling sequences, discussing the mathematics of many sequences that are in practice either impossible to juggle or simply uninteresting from a performance perspective. In the first section of Chapter Five he provides a description of several of his favorite practical juggling sequences and gives insight to how these can be used in performances or lectures on the mathematics of juggling.

Section 5.2 discusses practical techniques for making juggling easer, such as simulating juggling in zero gravity on a billiard table, juggling on an elliptical-shaped billiard table with hands at the foci (this would be a fun research project for a physics or mathematics student), bounce juggling, and robot juggling (an active research area in robotics). It is interesting to note that Shannon also built the first known juggling machine. A fascinating, short QuickTime movie of Shannon discussing his juggling machine can be found here.

Section 5.3 might have been titled "The Physics of Juggling" because this section describes a paper by Bengt Magnusson and Bruce Tiemann (two of the inventors of site swap notation) that describes "some of the basic physical laws that govern the actual juggling of the basic b-ball juggling sequence with two hands in a cascade or fountain pattern" [p. 129]. The physics in this section is certainly accessible to the undergraduate, as it only involves classical mechanics and projectile motion. Table 5.1 on page 131 provides some insight into the difficulty of juggling many objects. To juggle eight balls in the basic 8-ball pattern requires the juggler to throw each ball around 3.8 m high, 9-balls requires throws of 4.9 m, ten balls 6.2 m, and eleven balls 7.7 m. The world record for flashing balls is 12 which requires throws of around 9.2 m in the air (flashing is a juggling term meaning the juggler throws and catches each object once). Not only is that difficult, but as the juggler throws objects higher, the accuracy of each throw must improve. Polster states, "This means that Sergei Ignatov, who juggles 11 rings at this height [6 m], has to aim every ring to within 0.6 degrees just to be able to catch it at the far end of the trajectory. Of course the aim has to be even better to avoid any collisions" [p. 131]. The remainder of Section 5.3 discusses why clubs and balls tend to line up in certain patterns as they are being juggled.

Chapter Six "Jingling, or Ringing the Changes" Discusses change ringing and the "links between change ringing and the kind of numbers juggling developed in the previous chapters." Perhaps I've lived a sheltered life, but I had never heard of change ringing, although it seems to be an activity most prevalent in the U.K. and Ireland. Polster states that of the 6,000 churches and secular buildings that house change ringing bells, only about 100 are located outside those two countries. I can assure you there were none in my small, rural Kansas hometown.

Nonetheless, Polster does an excellent job in the introductory section of describing the history and terminology of change ringing. A ring is a set of b bells arranged in descending order of pitch 1, 2, ..., b, and a change is the ringing of those b bells in some order, i.e., a change is a permutation on the ordered set {1, 2, ..., b}. An extent on b-bells is a ringing sequence containing all possible changes; there are b! + 1 such changes in an extent (one is added because the first change is repeated at the end of the extent).

Next Polster describes how to replace a ring of b bells by b bell-balls, that is, juggling balls with bells attached. He then goes on to discuss the relationship between changes and juggling sequences and proves that ringing sequences can be "juggled".

Section 6.3 is an introduction to the mathematics of bell ringing, and proceeds to discuss a mathematical notation for denoting changes. The transitions from one change to the next are considered as elements of the symmetric group Sb. Next we see how existing ringing sequences can be transformed into a new ringing sequence by operations such as inverse, reverse, cyclic shift, vertical shifts, and elevation.

In Section 6.5 a graphical representation of extents is given in terms of Cayley graphs, and produces the result that extents are oriented Hamiltonian cycles in such a graph.

In the next section Polster notes, "it is interesting to note that 200 years before groups were formally introduced, composers of ringing sequences implicitly used group-theoretic concepts, such as the decomposition of groups into cosets of other groups to compose and prove their compositions." [p. 168], and then goes on to discuss this topic.

The final section of this chapter discusses how computers are used today to compose and analyze ringing sequences. He discusses how, "recently, computers have also been used to solve a century-old problem in change ringing." The problem would be difficult to describe in this review, but it basically concerned whether it was possible to ring a certain extent. The proof that this is possible was obtained in 1994 with the help of a computer.

Chapter Seven "Juggling Loose Ends" is a collection of juggling-related topics that did not fit neatly into any of the other chapters. The first section, with the provocative title "Does God Juggle", discusses the n-body problem and how recently a new solution to the 3-body problem has been found in which three bodies orbit each other in 3-ball cascade pattern. Polster mentions the probability of such a system existing in the Universe is extremely small but nonzero.

Next, Polster discusses juggling braids and discusses the relationships to topology. The main result of this section is that every finite braid can be juggled. He then discusses spinning balls in the hand (what jugglers call contact juggling) and some mathematical results.

The next section entitled "Useless Juggling" discusses how words, rational, and irrational numbers can be juggled, and considers antiballs (a ball that travels backward in time when thrown, as opposed to a regular ball that travels forward in time). Ball/antiball juggling does not lead to many interesting results, but if we consider antithrows (a throw of a regular ball back in time) some interesting results are obtained, and Polster links these to causal diagrams used by jugglers as another way of denoting juggling patterns.

Chapter Seven concludes with some short stories and riddles about juggling, and a section on the famous juggler-mathematicians Ronald Graham and Claude Shannon.

Appendix "Stereograms of Hamiltonian Cycles" presents 18 stereogram pictures of Hamiltonian cycles representing change ringing extents in the 4-bell Cayley graph. I love these things, although certain people have visual anomalies which means they cannot view them. I once worked with a colleague who lacked depth perception, and he could not view stereograms to save his life. It was hilarious to watch him try though.

I did not find a lot of errors in this book, and the ones I did were relatively minor. On page 5, Paul Klimek's name is spelled with an a rather than e — my research has led me to believe e is correct. On p. 190 when discussing juggling words he states, "A4B is a short form of 10, 1, 11". I think that should read "10, 4, 11". Although not an error, according to the Juggling Information Service Committee on Numbers Juggling, the latest record for flashing rings is 13 by Albert Lucas in 2002. Polster's Table 5.2 on page 132 states the official record is 12 and the unofficial record is 14. Most likely Lucas set the new record after TMOJ had gone to print, so this is not an error.

I think this book is essential reading for any mathematical juggler, and although many may be familiar with much of the material in TMOJ from articles published elsewhere, this is the first book that books pulls together much of this material into one source. Mathematics and physics educators will find many interesting examples for problems that are accessible to undergraduates, and computer science instructors could devise challenging programming assignments involving the mathematics of juggling.

Kevin R. Burger is an Assistant Professor of Computer Science at Rockhurst University in Kansas City, Missouri. His primary scholarly interests are in undergraduate computer science education, teaching programming, discrete mathematics, and algorithms. Outside of work, he enjoys throwing pottery on the wheel, lifting weights at the gym, hiking, listening to music, and drinking coffee at the Broadway Café. He is a master of the 3-ball cascade and completely incompetent at every other juggling pattern.

Juggling-an Introduction * Simple Juggling * Multiplex Juggling * Multihand Juggling * Practical Juggling * Jingling, or, Ringing the Changes * Juggling Loose Ends * Appendix A: Stereograms of Hamiltonian Cycles * References * Index