The Mathematics of Juggling Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo Burkard Polster The Mathematics of Juggling With 114 Illustrations BurkardPolster DepartmentofMathematics MonashUniversity P.O.Box28M Victoria3800 Australia [email protected] MathematicsSubjectClassification(2000):05-02,05A05 LibraryofCongressCataloging-in-PublicationData Polster,Burkard. Themathematicsofjuggling/BurkardPolster. p.cm. Includesbibliographicalreferencesandindex. ISBN0-387-95513-5(alk.paper) 1.Sequences(Mathematics) 2.Juggling—Mathematics. I.Title QA292.P652002 515′.24—dc21 2002070555 ISBN0-387-95513-5 Printedonacid-freepaper. ©2003Springer-VerlagNewYork,Inc. All rights reserved. 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PrintedintheUnitedStatesofAmerica. 9 8 7 6 5 4 3 2 1 SPIN10881157 Typesetting:PagescreatedbytheauthorusingaSpringerLaTeX2emacropackage. www.springer-ny.com Springer-Verlag NewYork Berlin Heidelberg AmemberofBertelsmannSpringerScience+BusinessMediaGmbH To My Jugglable Anu Preface Recenthistorysawanunprecedented riseinamateurjuggling.Today,there arehundredsofjugglingclubsaroundtheworld,thereareanumberofspe- cialized juggling magazines, and every year thousands of jugglers gather at juggling conventions to practice, exchange ideas, take part in competi- tions,andhavefuntogether.Jugglingisparticularlypopularamongmath- ematicallymindedpeople such asmajorsincomputer science, engineering, mathematics, and physics. In fact, although exact numbers are hard to come by, an estimate that keeps popping up in this context is that up to forty percent of serious jugglers today have a mathematical background; see, for example, [118]. Given this strong interest in jugglingin the (hard) scientific community, it is not surprising that many mathematical aspects of jugglinghave been explored in depth, sophisticated computer programs have been written that simulatejuggling,and a number of jugglingrobots have even been built. First and foremost, the target audience of The Mathematics of Juggling consists of all mathematicallyminded jugglers. However, mathematics ed- ucators, readers of popular mathematical literature, and mathematicians interested in unusual applications of some of their favorite tools and tech- niques willalso be among the readers of this book. Also included in this book is a chapter dedicated to the mathematics of bell ringing and new connections between bell ringing and toss jug- gling. Furthermore, this chapter is the most comprehensive introduction to mathematical bell ringing availablein the literature and, consequently, many people interested in this ancient art will want to study at least this part of the book. viii Preface My aims in writing this book were the following: • Serious Mathematics. Many good books are availablethat are dedi- cated to teaching the different juggling tricks and skills; see, for ex- ample,[28],[40],and[94].Onthe other hand,manypopulararticles, serious papers,andevenPh.D.theses havebeen written dealingwith different aspects of the mathematics behind these tricks and skills. Themaingoalofthisbookistosummarize,connect, andexpandthe results dealing with mathematical aspects of juggling in the litera- ture. • Serious Juggling.Introducejugglerstonewsystematicwaysofthink- ing about jugglingand the way they acquire new juggling skills and tricks. First and foremost, this includes a description and analysis of a compact mathematical language for juggling patterns that has already led to the discovery of many new and attractive tricks. • Mathematics Education. Dueto the abstract nature of mathematics, mathematiciansoftenhaveahardtimecommunicatingtothegeneral publicthatmathematicsand,inparticular,puremathematicscanbe a lot of fun and very useful. Speaking from experience, I know that a talk/performance of mathematical jugglingis a perfect ice-breaker in this respect. I hope that this book willlead more scientists to use mathematical jugglingin outreach programs and to communicate to the general public that mathematics can be a lot of fun as well as useful in modeling and understanding just about every problem in real life. • Turn Jugglers into Mathematicians. Using juggling as a unifying theme, provide an intellectually stimulating introduction for mathe- matically wired jugglers to many beautiful results and techniques fromawiderangeofmathematicaldisciplinessuch ascombinatorics, graph theory, group theory, knot theory, classical mechanics, and number theory. • Turn Mathematicians into Jugglers.Makepeoplewhoareabletoap- preciate and enjoy the mathematics in this book also give the prac- tical side of jugglinga go. • Bell Ringing. As with juggling, the art of bell ringing has a strong following in the scientific community, and the appeal of bell ringing andjugglingtothescientificmindseems tobeverysimilar.However, not many jugglers seem to know about bell ringing and, vice versa, not many bell ringers also practice juggling. I hope that including and linking the mathematical aspects of these seemingly unrelated pastimes willenrich and complement the overallmathematicalmenu offered in this book and introduce and awaken the interest of many Preface ix readerswhoareonlyawareofjugglingorbellringingtotherespective other activity. Thisbookisdesigned tobeaccessibleat different levelsofmathematical sophistication. Anybody who is not put off from picking up this book in the first place by the word “mathematics” in the title should be able to get something out of it. At the same time, full proofs, or at least detailed sketches of proofs, havebeen included for allthose people who are ableto appreciate “serious” mathematics. Description of Contents In the following,I summarize the different chapters and the material cov- ered in this book. The first chapter consists of a brief introduction to jugglingand its his- tory. In particular, it contains a detailed account of the development of mathematical juggling starting around 1985. This also includes brief re- views of a number of freely availablecomputer jugglingprograms that are basedonthemathematicallanguageforjugglingpatterns developedinthis book. These programsillustratevividlythepower ofthe mathematicalap- proach to juggling by conjuring up virtual jugglers that can juggle any conceivable number of props in infinitely many ways and all this without drops. I highlyrecommend that youuse one or two of these programs side by side with this book. I also take the opportunity to acknowledge the important, yet not very well-known, contributions of many people to the developmentofmathematicaljugglingviatheirarticlespostedtothenews- group rec.juggling, their development of computer juggling programs, and so forth. InChapter2,westripawayalltheflourishesandcontortionsinvolvedin aperiodicjugglingpatterninwhichthrowsoccuratdiscreteequallyspaced moments in time and in which at most one ball is caught and thrown at any given time. What we are left with can be described by a (simple) jug- gling sequence; that is, a special finite sequence of nonnegative numbers that records thenumber ofbeats theindividualthrows inthe pattern stay in the air. We investigate tests based on averages, juggling diagrams, and permutations that allow us to recognize juggling sequences and the num- ber of balls that are used in juggling a given juggling sequence. We prove that any finite sequence of nonnegative numbers, the sum of whose ele- ments is divisible by the length of the sequence, can be rearranged into a juggling sequence. Using juggling cards, we generate all b-ball juggling sequences, derive a formula for the number of b-ball juggling sequences of period p,and describe some close relationships between jugglingsequences and elements ofcertain affine Weyl groups. Weintroduce state graphs and transition matrices based on which all b-ball jugglingsequences of a given maximal throw height can be generated. Furthermore, we derive various results about state graphs such as upper and lower bounds for the lengths x Preface ofprimeloopsofmaximallengths,thefactthatthestates visitedbyajug- gling sequence determine its throws, and the fact that the complement of theb-ballstate graphofheighthisthe(h−b)-ballstategraphofheighth. In Chapter 3, we consider multiplex juggling sequences, a first general- ization ofthesimplejugglingsequences considered inChapter 2. Unlikein simple jugglingsequences, in a multiplex jugglingsequence several throws can be made simultaneously on every beat. We find that many of the re- sults that we derived for simple juggling sequences have counterparts for multiplex jugglingsequences. Also included in this chapter are several ap- plications that illustrate how to derive, entirely within the framework of multiplex juggling sequences, combinatorial identities involving Gaussian coefficients, Stirling numbers of the second kind, and their q-analogues. Finally, we include a summary of operations that enable us to transform simple and multiplex jugglingsequences into other jugglingsequences. All juggling sequences in Chapters 2 and 3 can, at least in theory, be juggledusing onehandonly.In Chapter 4,weconsider multihandjuggling inwhichseveralmultiplexthrowscanbemadesimultaneouslybyanumber of different hands. Multihand juggling patterns are described by juggling matrices. Again, we find that virtually all basic results for simple juggling sequences have counterparts for juggling matrices. This includes the dif- ferent ways to calculate the number of balls required to juggle a juggling matrix, an algebraic test that allows one to check whether or not a given matrixoftheappropriateformisajugglingmatrix,andresultsaboutstate graphs ofjugglingmatrices. Wealsofindthat mostofthe usual operations introduced for jugglingsequences have counterparts for jugglingmatrices. However, we also encounter some new operations that allow one to “con- tract” any juggling matrix into a juggling sequence and to systematically construct all jugglingmatrices from jugglingsequences. Wehaveacloserlookatanumberofdistinguishedsubclasses ofjuggling matrices. This includes the simple juggling matrices in which every hand handles at most one ball at a time and the distributed juggling matrices in which, as the name suggests, on every beat at most one of the hands does something. Cyclic juggling matrices are special distributed juggling matrices. In a cyclicjugglingmatrix,the different hands taketurns throw- ing the balls in cyclical order. We are particularly interested in cascades andfountains;thatis,jugglingpatterns described bysimplecyclicjuggling matrices in which all nonzero throws are to the same height and one such throw is made on every beat. Also included in this chapter are Shannon’s fundamental results about uniform multihand jugglingpatterns. In a uniform jugglingpattern, there is notnecessarily anunderlyingbeat onwhich allthethrows occur. Inthis respect, these patterns do not have to be “uniform” at all. What makes them uniform is the fact that the dwell time that any ball is held by a hand is constant, the flight time that any ball spends in the air between being thrown and being caught is constant, and the vacant time that any Preface xi hand spends empty between throwingand catching is alsoconstant. Shan- non’s celebrated first theorem relates the dwell,flight, and vacant times of a uniform juggling pattern with the numbers of balls and hands used in juggling the pattern. His second and third theorems deal with the essen- tially different ways of uniformly juggling a given number of balls with a given number ofhands. Westate and prove Shannon’s theorems and some corollaries and extensions of these results. In particular, we derive gener- alizations of Shannon’s results in the framework of simple cyclic juggling matrices. Anotherinterestingideathatfirstpopsupintheinvestigationofuniform juggling patterns is that we never just juggle balls. What we really do is juggleballsandhands.Infact,itfollowsimmediatelyfromthedefinitionof uniform jugglingthat ifwe interpret ballsas hands and hands as balls,we arrive at a new uniform jugglingpattern. This leads to a dualityprinciple for uniformjugglingpatterns. Wealsoinvestigatejugglingballsand hands in the framework of simple distributed jugglingmatrices. In the followingsection, we show that jugglingsimple jugglingmatrices with labeled balls still yields periodic patterns. In the final section of this chapter, we consider natural decompositions of simple juggling sequences that arise when these juggling sequences are interpreted as simple cyclic juggling matrices. This includes a nice appli- cation of the cycle representations of the permutations that are associated with simplejugglingsequences. Chapter 5 is dedicated to practical aspects of mathematical juggling. First, we list and describe a number of jugglingsequences that are partic- ularly attractive from a purely visual point of view or useful for teaching purposes. This is followed by a discussion of ways to make juggling easier by taking away gravity in variousways or bouncing balls and descriptions of how these simple ways of juggling have been used in constructing jug- glingrobots. Followingthis,we modelthe 2-hand jugglingof cascades and fountains to deduce how high and how accurately these patterns have to be juggled. We derive a simple model for club throws, which we then use, in conjunction with previous results, to explain variouslining-upeffects in juggling 2-hand fountains and cascades. Finally, we summarize the main advantages of the mathematical language for juggling patterns developed in the previous chapters. Chapter6isanintroductiontomathematicalbellringinganditsconnec- tionswithjuggling.Bellringersringthechanges.Mathematicallyspeaking, changes arepermutations of the bells that are being rung, and ringing the changes is really ringing special sequences of permutations. We call these sequences “ringing sequences.” It turns out that ringing sequences can be juggled if we use balls instead of bells and that the simple juggling se- quences thatcorrespond toringingsequences areparticularlywell-behaved juggling sequences. Furthermore, the sequence of transitions that corre- sponds to a ringing sequence on b bells can be interpreted in terms of site