beek lewbel - the science of juggling

To complete a delivery of muni-tions, a 148-pound man must tra-verse a high, creaking bridge that

can support only 150 pounds. The prob-lem is, he has three, one-pound cannon-balls and time for only one trip across.The solution to this old riddle is thatthe man juggled the cannonballs whilecrossing. In reality, juggling would nothave helped, for catching a tossed can-nonball would exert a force on thebridge that would exceed the weightlimit. The courier would in fact end upat the bottom of the gorge.

Though not practical in this case, jug-gling deÞnitely has uses beyond hobbyand entertainment. It is complex enoughto have interesting properties and sim-ple enough to allow the modeling ofthese properties. Thus, it provides acontext in which to examine other, morecomplex Þelds. Three in particular havebeneÞted.

One is the study of human movementand the coordination of the limbs. An-other is robotics and the constructionof juggling machines, which provides agood test bed for developing and ap-plying principles of real-time mechani-cal control. The third is mathematics;juggling patterns have surprising nu-merical properties.

Juggling is an ancient traditionÑthe

earliest known depiction is Egyptian, inthe 15th Beni Hassan tomb of an un-known prince from the Middle Kingdomperiod of about 1994 to 1781 B.C.; how-ever, the Þrst scientiÞc study we knowof did not appear until 1903. At thattime, Edgar James Swift published anarticle in the American Journal of Psy-

chology documenting the rate at whichsome students learned to toss two ballsin one hand. By the 1940s, early com-puters were being used to calculate thetrajectories of thrown objects, and theInternational Jugglers Association wasfounded. The 1950s and 1960s saw afew scattered applications, mostly suc-cessors to SwiftÕs work, which used jug-gling as a task to compare general meth-ods of learning sensorimotor skills.

Finally, in the 1970s, juggling beganto be studied on its own merits, as evi-denced by events at the MassachusettsInstitute of Technology. There, Claude E.Shannon created his juggling machinesand formulated his juggling theorem,which set forth the relation between theposition of the balls and the action ofthe hands. Seymour A. Papert and otherresearchers at Project MAC (which laterbecame M.I.T.Õs ArtiÞcial IntelligenceLaboratory) investigated how peoplemaster the art of juggling, and the M.I.T.juggling club, one of the oldest organi-

zations devoted to amateur jugglingstill in existence, was established. The1980s witnessed the rise of the mathe-matics of juggling, as several workersdeveloped a special kind of notation tosummarize juggling patterns [see box

on page 94].With three balls, most neophytes at-

tempt to juggle in the shower pattern(around in a circle), although the cas-cade patternÑin which the hands alter-nate throwing balls to each other, re-sulting in a Þgure eightÑis far easier. Itoften takes just hours or days to learnto juggle three balls. But learning timescan be weeks or months for four balls,months or years for Þve, as practition-ers reÞne their sense of touch and toss.

The world record for the greatestnumber of objects juggled (where eachobject is thrown and caught at leastonce, known as a ßash) is 12 rings, 11balls or eight clubs. The types of jug-gled objects, or props, aÝect the num-ber because they vary in correct orien-tation, in the diÛculty of holding andthrowing them and in the margin of er-ror to avoid collisions.

Limits on the Learning Curve

The obvious physical constraints thataÝect mastery and limit the number

of objects juggled arise from gravityÑmore speciÞcally, Newtonian mechanics.Each ball must be thrown suÛcientlyhigh to allow the juggler time to dealwith the other balls. The time that a ballspends in ßight is proportional to thesquare root of the height of the throw.The need for either speed or height in-creases rapidly with the number of ob-jects juggled.

Then there is human imperfection,leading to errors of both space and time.Juggling low leaves little room to avoidcollisions and hence requires catchingand throwing quickly, which can causemistakes. Throwing higher providesmore time to either avoid or correctmistakes but also ampliÞes any error.For throws of only a few meters, a devi-

92 SCIENTIFIC AMERICAN November 1995

The Science of JugglingStudying the ability to toss and catch balls and rings provides insight into human coordination, robotics and mathematics

by Peter J. Beek and Arthur Lewbel

EXPERT JUGGLING relies more on the sense of touch and less on sight than doesnovice juggling. Hence, a professional can manage to juggle common patternsblindfolded for several minutes, from a simple exchange of balls (opposite page ) tothe four-ball fountain (top left corner ) and the three-ball cascade (above ). Pho

togr

aphs

by

Ken

Reg

an C

amer

a 5;

pro

ps c

ourt

esy

of D

ubé

Jugg

ling

Equ

ipm

ent

Copyright 1995 Scientific American, Inc.

ation of just two or three degrees in thetoss can cause an error in the landinglocation of 30 centimeters or more.

The temporal constraints on jugglingare elegantly summarized by ShannonÕstheorem. It deÞnes relations that mustexist among the times that the handsare empty or full and the time each ballspends in the air. In other words, in-creasing the number of balls leaves lessroom to vary the speed of the juggle. Ifone were to juggle many balls to a cer-tain height, the theorem indicates thateven the smallest variation in tossingspeed would destroy the pattern.

How jugglers coordinate their limbsto move rhythmically and at the samefrequency within these constraints hasbecome a primary focus in the study ofhuman movement. Researchers haveborrowed concepts from the mathemat-ical theory of coupled oscillators [seeÒCoupled Oscillators and Biological Syn-chronization,Ó by Steven H. Strogatz andIan Stewart; SCIENTIFIC AMERICAN, De-cember 1993].

The key phenomenon in coupled os-cillation is synchronization: the tenden-cy of two limbs to move at the samefrequency. The particular type of coor-

dination displayed by juggling handsdepends on the juggling pattern. In thecascade, for instance, the crossing ofthe balls between the hands demandsthat one hand catches at the same ratethat the other hand throws. The handsalso take turns: one hand catches a ballafter the other has thrown one.

The fountain pattern, in contrast, canbe stably performed in two ways: bythrowing (and catching) the balls simul-taneously with both hands ( in sync) orby throwing a ball with one hand andcatching one with the other at the sametime (out of sync). Theoretically, onecan perform the fountain with diÝerentfrequencies for the two hands, but thatcoordination is diÛcult because of thetendency of the limbs to synchronize.

DeÞning the physical and temporalconstraints is one aspect of jugglinganalysis. A realistic model must alsoincorporate at least three other compli-cating factors. First, the oscillation of ajuggling hand is not uniform, becausethe hand is Þlled with a ball during partof its trajectory and empty during theremaining part. Second, the movementsof both hands are aÝected by the phys-


EARLIEST DEPICTION OF JUGGLING known shows skillful Egyptian women on the15th Beni Hassan tomb of an unknown prince from the Middle Kingdom period ofabout 1994 to 1781 B.C.

Red

raw

ings

from

Bild

atla

s zu

m S

port

im A

lten

Ägy

pten

, by

Wol

fgan

g D

ecke

r

One useful method that many jugglers rely on to summa-rize patterns is site-swap notation, an idea invented inde-

pendently around 1985 by Paul Klimek of the University ofCalifornia at Santa Cruz, Bruce Tiemann of the California Insti-tute of Technology and Michael Day of the University of Cam-bridge. Site swaps are a compact notation representing the or-der in which props are thrown and caught in each cycle of thejuggle, assuming throws happen on beats that are equallyspaced in time.

To see how it works, consider the basic three-ball cascade.The first ball is tossed at time periods 0, 3, 6.. . , the second attimes 1, 4, 7.. . , and the third ball at times 2, 5, 8.. . . Site-swapnotation uses the time between tosses to characterize the pat-tern. In the cascade, the time between throws of any ball isthree beats, so its site swap is 33333.. . , or just 3 for short.The notation for the three-ball shower (first ball 0, 5, 6, 11,12... , second ball 1, 2, 7, 8, 13... , third ball 3, 4, 9, 10, 15... . )consists of two digits, 51, where the 5 refers to the duration ofthe high toss and the 1 to the time needed to pass the ballfrom one hand to the other on the lower part of the arc. Otherthree-ball site swaps are 441, 45141, 531 and 504 (a 0 repre-sents a rest, where no catch or toss is made).

The easiest way to unpack a site swap to discover how theballs are actually tossed is to draw a diagram of semicircles ona numbered time line. The even-numbered points on the linecorrespond to throws from the right hand, the odd-numberedpoints to throws from the left.

As an example, consider the pattern 531. Write the numbers5, 3 and 1 a few times in a row, each digit under the next pointin the number line starting at point 0 [see top illustration atright ]. The number below point 0 is 5, so starting there, drawa semicircle five units in diameter to point 5, representing athrow that is high enough to spend five time units (beats) inthe air. The number below point 5 is a 1, so draw a semicircle

of diameter 1 from point 5 to 6. Point 6 has a 5 under it, so thenext semicircle is from point 6 to 11. You have now traced outthe path in time of the first ball, which happens to be the sameas the first ball in the three-ball shower pattern 51 describedabove. Repeat the process starting at times 1 and 2, respec-tively, to trace out the path of the remaining two balls. The re-sult is that the first and third balls both move in shower pat-terns but in opposite directions, and the second ball weavesbetween the two showers in a cascade rhythm. Leaving outthis middle ball results in the neat and simple two-ball siteswap 501.

Not all sequences of numbers can be translated into legiti-mate juggling patterns. For example, the sequence 21 leads toboth balls landing simultaneously in the same hand (althoughmore complicated variants of site-swap notation permit morethan one ball to be caught or thrown at the same time, a featjugglers call multiplexing).

Site-swap notation has led to the invention of some patternsthat are gaining popularity because they look good in perfor-mances, such as 441, or because they are helpful in masteringother routines, such as the four-ball pattern 5551 as a preludeto learning to cascade five balls. Several computer programsexist that can animate arbitrary site swaps and identify legiti-mate ones. Such software enables jugglers to see what a pat-tern looks like before attempting it or allows them simply togaze at humanly impossible tricks.

The strings of numbers that result in legitimate patterns haveunexpected mathematical properties. For instance, the num-ber of balls needed for a particular pattern equals the numer-ical average of the numbers in the site-swap sequence. Thus,the pattern 45141 requires (4 + 5+1+4 + 1)/5, or three balls.The number of legitimate site swaps that are n digits long usingb (or fewer) balls is exactly b raised to the n power. Despiteits simplicity, the formula was surprisingly difficult to prove.

The Mathematics of Juggling


ical demands of accurate throwing andcatching. Third, the timing between thehands is based on a combination of vi-sion, feel and memory.

These three factors render jugglingpatterns intrinsically variable: howeversolid a run, no two throws and no twocatches are exactly the same. Analyzingthis changeability provides useful cluesabout the general strategy of jugglers toproduce a solid pattern that minimizesbreakdown.

Variables associated with throwing(angle of release, release velocity, loca-tion of throws, height of throws) arethose most tightly controlled: jugglersattempt to throw the balls as consistent-ly as possible, the timing of which mustobey ShannonÕs theorem. Given a height,a crucial measure of the rate of jugglingis the so-called dwell ratio. It is deÞnedas the fraction of time that a hand holdson to a ball between two catches (orthrows). In general, if the dwell ratio islarge, the probability of collisions in theair will be small. That is because thehand cradles the ball for a relativelylong time and hence has the opportuni-ty to throw accurately. If the dwell ratio

is small, the number of balls in the airaveraged over time is large, which is fa-vorable for making corrections, becausethe hands have more time to repositionthemselves.

Novice jugglers will opt for largerdwell ratios to emphasize accurate toss-es. More proÞcient jugglers tend towardsmaller values, especially when juggling

three balls, because of a greater ßexibil-ity to shift the pattern. Measurementsby one of us (Beek) demonstrate thatthe ratio attained in cascade jugglingchanges roughly between 0.5 and 0.8,with ratios close to 3/4, 2/3 and 5/8 mostcommonly observed. That is, in a three-ball cascade, the balls may spend up totwice as much time in the air as they

SCIENTIFIC AMERICAN November 1995 95

JOH

NN

Y J

OH

NS

ON

Site-swap theory does not come close to describing com-pletely all possible juggling feats, because it is concerned onlywith the order in which balls are thrown and caught. It ignoresall other aspects of juggling, such as the location and style of

throws and catches. Many of the most popular juggling tricksto learn, such as throwing balls from under the leg or behindthe back, are done as part of a regular cascade and so have thesame site-swap notation.

JUGGLING THEOREM proposed by Claude E. Shannon of the Massachusetts Insti-tute of Technology is schematically represented for the three-ball cascade. The ex-act equation is (F +D )H = (V +D )N, where F is the time a ball spends in the air, Dis the time a ball spends in a hand, V is the time a hand is vacant, N is the numberof balls juggled, and H is the number of hands. The theorem is proved by follow-ing one complete cycle of the juggle from the point of view of the hand and of theball and then equating the two.

HAND PERSPECTIVE

BALL PERSPECTIVE

TIME

HAND IS EMPTYHAND HOLDS A BALL

BALL IN A HANDBALL IN FLIGHT

FIRST BALL THIRD BALLSECOND BALL

RIGHT HAND LEFT HAND

CA

RE

Y B

ALL

AR

D

TIME

THROW

THROW 5 THROW 3 THROW 1 THROW 5

THROW 3 THROW 1 THROW 5 THROW 3

5

0

3

1

1

2

5

3

3

4

1

5

5

6

3

7

1

8

5

9

3

10

1

11

5

12

3

13

1

14

0 1 2 3TIME

4 5 6 7TIME

The 531 in action

Unpacking the site swap 531


do in the hands. Such a range suggestsjugglers strike a balance between theconßicting demands of stability andßexibility, correcting for external per-turbations and errors. Moreover, thetendency toward dwell ratios that aresimple fractions subtly illustrates a hu-man tendency to seek rhythmic solu-tions to physical tasks.

Juggling more than three balls leavesless room to vary the ratio, because theballs have to be thrown higher and,hence, more accurately. This fact great-ly limits what jugglers can do. Jugglingthree objects allows ample opportunityfor modiÞcation, adaptation, tricks andgimmicks. At the other extreme, thereare few ways to juggle nine objects.

Keep Your Eye OÝ the Ball

Modeling the movement patterns ofjuggling as such, however, says lit-

tle about the necessary hand-eye coor-dination. Jugglers must have informa-

tion about the motions of both thehands and the balls. There are few con-texts in which the coaching advice toÒKeep your eye on the ballÓ makes aslittle sense as it does in juggling. Atten-tion must shift from one ball to thenext, so that a juggler sees only a partof each ballÕs ßight.

Which part is most informative andvisually attended to? ÒLook at the high-est pointÓ and ÒThrow the next ballwhen the previous one reaches the topÓare common teaching instructions. As agraduate student at M.I.T. in 1974, How-ard A. Austin investigated how large aregion around the zenith had to be seenby practitioners of intermediate skillfor them to be able to sustain juggling.He placed between the hands and theeyes of the juggler a fanlike screen thathad a wedge-shaped notch cut out of it.Successful catches of a ball occurredeven when as little as one inch of thetop of the ball ßight was visible. Thatroughly corresponds to a viewing time

of 50 milliseconds, implying that brießyglimpsing the zeniths of the ball ßightswas suÛcient to maintain a juggle.

In 1994 Tony A. M. van Santvoord ofthe Free University in Amsterdam ex-amined the connection between handmovements and ball viewing in moredetail. He had intermediate-level jug-glers perform a three-ball cascade whilewearing liquid-crystal glasses, whichopened and closed at preset intervalsand thus permitted only intermittentsightings of the balls. From the relationbetween the motion of the balls in theair and the rhythm deÞned by the glass-es, one could deduce the location of theballs when the glasses were open, thepreferences for any segments of theßight paths during viewing and the de-gree of coordination between the handmovements and visual information.

On some occasions, subjects modi-Þed their juggling to match the fre-quency of the opening and closing ofthe glasses. In that case, the balls be-

came visible immediately af-ter reaching their zeniths.The experiments also sug-gest that seeing the balls be-comes less important aftertraining. In general, noviceand intermediate jugglersrely predominantly on theireyes. Expert performers de-pend more on the sensationscoming from the contact be-tween the hands and theballs. Indeed, in his 1890 The

Principles of Psychology, Wil-liam James observed thatthe juggler Jean-Eug�ne Ro-bert-Houdin could practicejuggling four balls whilereading a book. Many skilledjugglers can perform blind-folded for several minutes.

A plausible hypothesis isthat viewing the moving ball


KA

RL

GU

DE

ROBOTIC ARM can bat two balls in a fountain pattern indeÞnitely. A camera records theßights of the balls, and a special juggling algorithm, which can correct for errors and changethe cadence of the juggle, controls the robotÕs motion. Daniel E. Koditschek and Alfred A.Rizzi of the University of Michigan built the apparatus.

DA

VID

KO

ET

HE

R

THREE-BALL CASCADE THREE-BALL SHOWER FOUR-BALL FOUNTAIN (“IN SYNC”)

Common Juggling Patterns


gradually calibrates the sense of touchin the course of learning. An expert im-mediately detects a slight deviation inthe desired angle of release or in theenergy imparted to the ball, whereas anovice has to see the eÝect of mistakesin the ßight trajectories. As a conse-quence, the corrections made by an ex-pert are often handled with little dis-turbance to the integrity of the pattern.Fixes made by less proÞcient jugglers,in contrast, often disrupt the overallstability of the performance.

Robots That Juggle

Insights into human juggling have ledresearchers to try to duplicate the

feat with robots. Such machines wouldserve as a basis for more sophisticatedautomatons. Indeed, juggling has manyof the same aspects as ordinary life,such as driving an automobile on a busystreet, catching a ßy ball on a windy dayor walking about in a cluttered room.All these tasks require accurately antic-ipating events about to unfold so as toorganize current actions.

Shannon pioneered juggling robotics,constructing a bounce-juggling machinein the 1970s from an Erector set. In it,small steel balls are bounced oÝ a tight-ly stretched drum, making a satisfyingÒthunkÓ with each hit. Bounce jugglingis easier to accomplish than is toss jug-gling because the balls are grabbed atthe top of their trajectories, when theyare moving the slowest.

In ShannonÕs machine, the arms areÞxed relative to each other. The unitmoves in a simple rocking motion, eachside making a catch when it rocks downand a toss when it rocks up. Throwingerrors are corrected by having short,grooved tracks in place of hands. Caughtnear the zenith of their ßight, balls landin the track; the downswing of the armrolls the ball to the back of the track,thus imparting suÛcient energy to theball for making a throw. ShannonÕs orig-inal construction handled three balls,

although Christopher G. Atkeson andStefan K. Schaal of the Georgia Instituteof Technology have since constructed aÞve-ball machine along the same lines.

Although the bounce-juggling robotsare Þendishly clever, a robot that cantoss-juggle a three-ball cascade and ac-tively correct mistakes has yet to bebuilt. Some progress, however, has beenachieved. Machines that can catch, batand paddle balls into the air have beencrafted. Engineers have also built robotsthat juggle in two dimensions. In the

1980s Marc D. Donner of the IBM Thom-as J. Watson Research Center used a tilt-ed, frictionless plane, similar to an air-hockey table. It was equipped with twothrowing mechanisms moving on tracksalong the lower edge of the table.

In 1989 Martin B�hler of Yale Univer-sity and Daniel E. Koditschek, now atthe University of Michigan, took thework a step further. Instead of a launch-ing device running on a track, they useda single rotating bar padded with a bil-liard cushion to bat the pucks upwardon the plane. To control the bar so as toachieve a periodic juggle, the research-ers relied on the help of the so-calledmirror algorithm.

This concept essentially combines

two ideas. The Þrst is to translate (orÒmirrorÓ) the continuous trajectory ofthe puck into an on-line reference tra-jectory for controlling the motion of therobot (via a carefully chosen nonlinearfunction). The advantage of this mirroralgorithm is that it avoids the need tohave perfect information about the stateof the puck at impact, which is diÛcultto obtain in reality. The second idea, tostabilize the vertical motion of the puck,analyzes the energy of the ball to see ifit matches the ideal energy from a per-fect throw. Thus, the program registersthe position of the puck, calculates thereference mirror trajectory, as well asthe puckÕs actual and desired energy,and works out when and how hard thepuck must be hit. With an extended ver-sion of the mirror algorithm, the robotcan also perform a kind of two-dimen-sional, two-puck, one-hand juggle. Itbats the pucks straight up, alternatelywith the left and the right part of thepivoting bar, in two separate columns.

Watching the mirror algorithm in ac-tion is spooky. If you perturb one of thepucks, the robot arm will make somejerky movements that look completelyunnatural to human jugglers but resultin a magically rapid return to a smoothjuggle. The mirror algorithm cleverlycontrols batting, but it does not extendto the more diÛcult problem of jug-gling with controlled catches.

In addition to bouncing and batting,robots have managed a host of otherjuggling-related activities, including tap-ping sticks back and forth, hopping,balancing, tossing and catching balls ina funnel-shaped hand and playing amodiÞed form of Ping-Pong. Despitethese advances, no robot can juggle in away that seems even passingly human.But the science of juggling is a relative-ly new study, and the pace of improve-ment over the past two decades hasbeen remarkable. It may not be toolong before we ask, How did the robotcross the creaking bridge holding threecannonballs?

SCIENTIFIC AMERICAN November 1995 97

The Authors

PETER J. BEEK and ARTHUR LEWBEL can jugglethree and eight balls, respectively. Beek is a move-ment scientist at the Faculty of Human Movement atthe Free University in Amsterdam. His research in-terests include rhythmic interlimb coordination, per-ception-action coupling and motor development. Hehas written several articles on juggling, tapping andthe swinging of pendulums. Lewbel, who received hisPh.D. from the Massachusetts Institute of Technolo-gyÕs Sloan School of Management, is now professorof economics at Brandeis University. He founded theM.I.T. juggling club in 1975 and writes the columnÒThe Academic JugglerÓ for JugglerÕs World magazine.

Further Reading

TEMPORAL PATTERNING IN CASCADE JUGGLING. Peter J. Beek and Michael T. Turvey inJournal of Experimental Psychology, Vol. 18, No. 4, pages 934Ð947; November 1992.

SCIENTIFIC ASPECTS OF JUGGLING. In Claude Elwood Shannon: Collected Papers. Edit-ed by N.J.A. Sloane and A. D. Wyner. IEEE Press, 1993.

JUGGLING DROPS AND DESCENTS. J. Buhler, D. Eisenbud, R. Graham and C. Wright inAmerican Mathematical Monthly, Vol. 101, No. 6, pages 507Ð519; JuneÐJuly 1994.

PHASING AND THE PICKUP OF OPTICAL INFORMATION IN CASCADE JUGGLING. Tonyvan Santvoord and P. J. Beek in Ecological Psychology, Vol. 6, No. 4, pages 239Ð263;Fall 1994.

JUGGLING BY NUMBERS. Ian Stewart in New Scientist, No. 1969, pages 34Ð38; March18, 1995.

JUGGLING INFORMATION SERVICE. On the World Wide Web at http://www.hal.com/services/juggle/

PROFESSIONAL JUGGLER Tony Duncan,whose skilled hands appear in the open-ing photographs, also handles pins.

KE

N R

EG

AN

Cam

era

5


beek lewbel - the science of juggling

Documents