Claude Shannon: Mathematician, Engineer, Genius…and Juggler?

By Rob Goodman and Jimmy Soni, co-authors

A MIND AT PLAY: How Claude Shannon Invented the Information Age

(Credit: EECS Michigan Poster)

 “Do you mind if I hang you upside down by your legs?”

From any other professor, this question might have elicited concern. But from Claude Shannon, it was par for the course. Shannon had in mind an elaborate experiment: combining two forms of juggling—bounce and toss—by suspending a juggler by his feet.

Toss juggling is the airborne form of the art most familiar to us. Bounce juggling, on the other hand, consists in keeping objects in motion by hitting them against the ground in a pattern, a motion akin to the beating of hand drums. As generations of jugglers have discovered in the early stages of mastering their craft, bouncing items against the ground requires far less energy than tossing them in the air; in a bounce juggle, the balls arrive at the hand at the very top of their arc, the slowest speed they’ll have during the entire sequence. But even though the bounce juggler has the advantage of catching a ball at the nadir of its speed, conventional juggling of the tossing-and-catching kind is a more fluid motion, one that comes more naturally to us and affords the juggler more control than the percussive effort of bounce juggling.

Shannon wondered: Was it possible to marry the physics of these two styles? Could you capture, in one motion, the fluidity of toss juggling and the efficiency of bounce juggling? In practical terms: If you were dangling by your feet, could you toss balls in the air, let gravity do the work of bringing the balls down to earth, and then catch them again? Both the inquiry and the method were vintage Shannon: whimsical, indifferent to practicalities, and originating in an activity that typical professors might have dubbed unserious, but which Shannon, a tenured member of the MIT faculty, found amusing enough to merit scholarly time and attention.

That’s how Arthur Lewbel, an MIT student, found himself suspended by his feet in the middle of Shannon’s living room. The balls went up…and dropped unceremoniously to the ground. “As a physical experiment, it was a complete failure,” remembered Lewbel. Some physical limits even perfect math can’t crack; and in this case, even the great Claude Shannon couldn’t overcome the most obvious problem with the experiment’s design: How well does anyone do anything upside down?


Lewbel had become accustomed to inquiries of the may-I-hang-you-upside-down variety. He was the founder of the MIT Juggling Club and first met Shannon when the famous information theorist casually dropped by the club’s meeting unannounced. Shannon was there for the same reason that parents the world over find themselves in rooms not necessarily of their choosing: his daughter, Peggy, wanted him to go. She’d read about the club in the Boston Globe, and while it probably required minimal arm-twisting to get her unicycling, tinkering father to attend, the initial interest in the Juggling Club was Peggy’s.

He just showed up, and didn’t tell anyone who he was. There were a bunch of jugglers standing outside practicing and he just walked up and said, ‘Can I measure your juggling?’” remembered Lewbel. “That was the first thing he said to us, and it was something no one had ever asked us before.” Lewbel and the other jugglers agreed to be measured, and Shannon and Lewbel developed a fast friendship.

Drop-ins from star faculty of Shannon’s stature weren’t unusual. As Lewbel tells it, “One nice thing about juggling at MIT is that you never know who will show up. For example, one day Doc Edgerton, inventor of the strobe light, stopped by the juggling club and asked if he could photograph some of us juggling under strobe lights.” What was unusual was a return visit. But return Shannon did, repeatedly, even hosting the club at his Winchester home when they needed space for pizza-and-movie night. “The Juggling Club and the jugglers enchanted us,” remembered Peggy Shannon.

(Credit: Courtesy of the Shannon family)

Shannon had dabbled in juggling for decades. As a boy, he had imagined himself a fairground performer. At Bell Labs, the stories of his achievements in information theory were almost always accompanied by tales of his juggling while riding a unicycle in the Labs’ narrow halls. At home in Winchester, there were ample objects stashed in the playroom to toss and catch. By that point, Shannon had developed his talents as an amateur juggler far past the point of mere amusement: he was said to be able to juggle four balls, which, as anyone who has tried to juggle knows, is a worthy achievement.

Ronald Graham, a fellow mathematician-juggler, attributed some of his success to a trick borrowed from Galileo. “When Galileo wanted to slow gravity down, he just tilted a table” and let a ball roll from one end to the other, said Graham. “Imagine a big table, and then as you tilt the table, you get closer to 1 g.” By sliding pucks up a tilted air hockey table, Shannon was able to study their patterns, and refine his juggling technique, in a kind of slow motion. The pucks’ paths “were not parabolas, just pointy, and you could practice doing that.”

Part of juggling’s appeal to Shannon might have been the fact that it didn’t come easily. For all his mathematical and mechanical gifts, “it was something he simply could not master, making it all the more tantalizing,” wrote Jon Gertner. “Shannon would often lament that he had small hands, and thus had great difficulty making the jump from four balls to five—a demarcation, some might argue, between a good juggler and a great juggler.” Here, at least, Shannon was destined to be merely good.


To some, juggling lacks the nobility of mathematical pastimes like chess or music. And yet the tradition of mathematician-jugglers is an ancient and distinguished one. As best as we can tell, that tradition began in the tenth century CE in an open-air market in Baghdad. It was there that Abu Sahl al- Quhi, later one of the great Muslim astronomers, got his start in life juggling. A few years later, Al-Quhi became a kind of court mathematician for the local emir, who, fascinated by planetary motion, built an observatory in the garden of his palace and put Al-Quhi in charge. The appointment bore some fine mathematical fruit: Al-Quhi invented an adjustable geometrical compass, likely the world’s first, and led the revival among Muslim geometers of the study of the Greek thinkers Archimedes and Apollonius.

From juggling in a market to measuring the courses of planets: what they had in common, what drew Al-Quhi and so many of future number-crunching jugglers, was the patterns of parabolas and arcs, equations played out over open space. As Graham observed, “mathematics is often described as the science of patterns. Juggling can be thought of as the art of controlling patterns in time and space.” So it’s no surprise that generations of mathematicians could be found on university quads, tossing things in the air and catching them. Burkard Polster, author of The Mathematics of Juggling, writes, “next time you see some jugglers practicing in a park, ask them whether they like mathematics. Chances are, they do. . . . Most younger mathematicians, physicists, computer scientists, engineers, etc. will at least have given juggling three balls a go at some point in their lives.”

So what drew Shannon to the study of juggling? “He liked peculiar motions…I think what he liked about juggling was that it was a peculiar physical motion,” noted Lewbel. It was in the early 1970s that these peculiarities finally tugged enough to provoke him to write a mathematical paper on the topic.

Juggling, observed Lewbel, “is complex enough to have interesting properties and simple enough to allow the modeling of these properties.” But for all of its mathematical richness, when Shannon first began his work on the topic, he was starting from scratch: the field had no body of written work.

The first important scientific work on the topic was in the field of psychology. In 1903, Edgar James Swift published a paper in the American Journal of Psychology that studied the time it took to learn how to juggle, as an examination of the most effective ways to teach neurosensory skills. The nature of juggling itself seems to have been something of an afterthought. The insight Swift was after wasn’t “How does a juggler learn his craft?” so much as “How can a human being learn any craft?” Following in his footsteps, psychologists continued to use juggling as a research tool into the mid-twentieth century. But where psychologists had found juggling useful in their research, mathematicians had been reluctant to use a favorite pastime as a source of data and experiments. Until Shannon arrived on the scene, no papers had explored the math of juggling.

How could that be? How had millennia’s worth of mathematicians tried their hands at juggling but published no mathematical results on the topic? In some ways, it’s not hard to understand. Mathematics was then, as now, a fiercely competitive discipline, and while card games, puzzles, juggling, and other such entertainments may have been amusing mathematical hobbies, no serious, ambitious mathematician would have mistaken a circus routine for a topic deserving sustained research or publication. No one, that is, until Claude Shannon. Unmoved by material concerns, freed of the need to burnish his reputation, and driven by curiosity for curiosity’s sake, he could throw himself headlong into the study of juggling without any of the misgivings his colleagues might feel about doing the same.


In the context of Shannon’s other work, the juggling paper is unremarkable. It didn’t inaugurate a new field of study, and it didn’t bring him international acclaim. Shannon neither published it nor entirely finished it. Though Shannon was perhaps the first scientist to study juggling with mathematical rigor, the paper’s striking feature isn’t its originality or the quality of its mathematics, but rather what it reveals about its author’s wide-ranging reading and research. If information theory, genetics, and switching proved the depth of Shannon’s thinking, juggling displayed his dexterity. It is a testament, as well, to  Shannon’s belief that just about anything could be the object of serious mathematical analysis.

Shannon opens the paper with a dialogue from Lord Valentine’s Castle, Robert Silverberg’s science fiction novel set on the distant planet of Majipoor. It’s a chronicle of the adventures of an itinerant juggler named Valentine, who, as it happens, is actually a king whose throne and title have been taken from him:

Do you think juggling’s a mere trick?” the little man asked, sounding wounded. “An amusement for the gapers? A means of
picking up a crown or two at a provincial carnival? It is all those things, yes, but first it is a way of life, friend, a creed, a species of worship.”

And a kind of poetry,” said Carabella.

Sleet nodded. “Yes, that too. And a mathematics. It teaches calmness, control, balance, a sense of the placement of things and the underlying structure of motion. There is a silent music to it. Above all there is a discipline. Do I sound pretentious?”

For Shannon, it was important that people reading this paper “try not to forget the poetry, the comedy and the music of juggling for the Carabellas and Margaritas future and present.” We can sense something of Shannon’s self-consciousness in the next sentence, when he interrupts this train of thought and cribs Sleet to ask the reader, “Does this sound pretentious?”

If it did, Shannon knew it. It might explain why his next paragraphs seek to soften the lofty-sounding beginnings of the paper by grounding the reader in the history of juggling. In the span of roughly two pages, he travels over 4,000 years and covers a
considerable range of popular and cultural nods to juggling. The paper’s historical tour opens in early Egypt, circa 1900
BCE, in tombs with juggling scenes etched into the walls, four women each tossing three balls apiece. From there it’s off to the
Polynesian island of Tonga, with the sailor- adventurer Captain James Cook and scientist Georg Forster. The year was 1774, and Forster observed, in
A Voyage Round the World, that the Tongans had a flair for keeping multiple objects suspended in the air in sequence. Shannon quotes Forster’s observation of one girl who, “lively and easy in all her actions, played with five gourds, of the size of small apples, perfectly globular. She threw them up into the air one after another continually, and never failed to catch them all with great dexterity, at least for a quarter of an hour.”

From there it was back to dry land and 400 BCE, to Xenophon’s The Banquet and an audience with Socrates, who, upon seeing a young woman juggle twelve hoops in the air, is moved to observe, “This girl’s feat, gentlemen, is only one of many proofs that woman’s nature is really not a whit inferior to man’s, except in its lack of judgment and physical strength. So if any one of you has a wife, let him confidently set about teaching her whatever he would like to have her know.” For Shannon, Socrates’s comment is interesting on two levels. For one thing, if the girl in this scene did in fact juggle twelve hoops, she would hold the world record for the most objects juggled at a single time. On this fact Shannon is willing to give Xenophon and Socrates the benefit of the doubt: “Who could ask for better witnesses than the great philosopher Socrates and the famous historian Xenophon? Surely they could both count to twelve and were careful observers.”

But that’s Shannon’s only concession; Socrates’s chauvinism doesn’t sit well. Mustering as much antagonism as we might imagine Shannon capable of, he dismisses Socrates’s blinkered view of a woman’s capacities. “It is amusing to note how Socrates, departing from his famous method of teaching by question, makes a definite statement and immediately suffers from foot-in-mouth disease. Had he but put his period nine words before the end he could have been the prescient prophet of the women’s equality movement.” Later in the paper, Shannon makes the case for female jugglers more explicit. Two he singles out for special mention: Lottie Brunn, the “world’s fastest female juggler” and a fixture on the 1920s European theater circuit; and Trixie Firschke, the “first lady of juggling,” a German child star born to a Budapest circus family.

So, beginning in ancient Egypt and passing through the medieval minstrel’s hybrid of “juggling, magic, and comedy,” Shannon ends in the world of twentieth-century variety shows. Their leading lights— including W. C. Fields—inspired a generation of girls and boys, including the young Claude Shannon, to terrify their parents with talk of running away to join the circus.

The history lesson concluded, it was on to a more serious inquiry: how to understand the psyche of a juggler and the practice of juggling? Specifically, how does one make sense of a practice that requires precision but is also the stuff of comedy? A gymnast’s fault elicits pity, a kind of disappointment shared by performer and audience; a juggler, failing to catch a club, is just as likely to be met with laughter. How do jugglers deal with this?

Jugglers are surely the most vulnerable of all entertainers,” Shannon writes, walking up again to the edge of autobiography. Indeed, most serious jugglers are forced to develop a range of mind games and public feints to deal with the anguish of “the missed catch or the dropped club.” Their coping strategies vary with skill level: lesser jugglers paper over their failures with comedy and props; the experts make their failures appear as intentional as their successes.

But this vulnerability is precisely why, Shannon notes, juggling’s ranks can be roughly divided into two camps: performance jugglers and technical jugglers. Technicians are in a numbers game, an arms race of objects juggled. The more objects in the air, the bigger the bragging rights. Shannon references one of the world’s great technicians, Enrico Rastelli, about whom Vanity Fair would say in eulogy: “In his twenty years’ devotion to his craft this son of Italy elevated it, for probably the first time, to what was unmistakably an art.” Rastelli, Shannon noted, was able to keep ten balls in the air simultaneously. Shannon also remarks that Rastelli “could do a one-armed handstand while juggling three balls in the other hand and rotating a cylinder with his feet.”

(Credit: Eugene Daub. This was one of the early mock-ups of the Shannon statue)

Rastelli and his breed of technical juggler held the most interest for Shannon, and mathematicians since. Call it seriousness of purpose, or the possibility of organizing by numbers and implicit for- mulas the quest to manage an ever-increasing number of objects. For the mathematician, performance juggling, enjoyable though it might be to watch, possesses none of these qualities. The joy of the crowd, the thrill of the motion, the comedy of the effort—all of these are amusing but ultimately uninteresting to a mind trained in math. The paper’s journey begins here: in the challenge of increasing the number of objects juggled while still maintaining precision, the intersection of mathematics and movement.


It shouldn’t come as any particular surprise that Shannon, whose love of juggling was surpassed only by his love of music, opens the mathematical section of the paper with a reference to jazz—in particular, to the drummer Gene Krupa, who said that “the cross rhythm of 3 against 2 is one of the most seductive known.” For Shannon, the pattern of three against two is a useful analogue for an introduction to the mathematics of juggling. It’s the pattern by which most people first learn to juggle: three balls in two hands.

Pick apart the motions of a juggler and what emerges are a series of predictable parabolas. One ball tossed in the air produces one arc; multiple balls, multiple arcs. All that’s left is to combine them into a consistent pattern, set to a rhythm. This was how Shannon approached the problem of juggling—not only as an exercise in coordination, but as an algebraic formula. His juggling theorem stated the following:

(F + D) H = (V + D) N

F = how long a ball stays in the air

D = how long a ball is held in a hand

H = number of hands

V = how long a hand is empty

N = number of balls being juggled

Shannon’s theorem tracks time continuously. As Lewbel put it, “The way the juggler achieves the rhythm in Shannon’s theorem is by trading off time in a continuous way; the more time one ball spends in the air relative to the time it spends in your hand, the more time you have to deal with the other balls, and so the more balls you can juggle. Shannon’s theorem makes this trade off in times precise.” (He also pointed out the irony, given the rest of Shannon’s digital innovations, that the juggling theorem’s measurement of continuous time makes it analog.) Each side of the equation tracks a different part of the act of juggling: the left side tracks the pattern of the balls, and the right side tracks the pattern of the hands. Because, as Lewbel put it, “the amount of time balls spend being juggled is the same as the time the hands spend juggling them,” the equality is maintained.


Shannon’s work on juggling might have ended here; already he had lent the study of juggling considerable legitimacy, and given a generation of mathematician-jugglers the ability to combine their two passions without fear of embarrassment. But a paper was, in this case, insufficient. In 1983, as he had so often before, Shannon brought the work from the world of theory into the realm of mechanics: he set out to build his own juggling robot.

It all started when Betty brought home a little four-inch clown, doing a five ball shower, from the cake decorating store ($1.98). I was both amused and bemused—amused as a long-time amateur juggler who even as a boy wished to run away and join the circus, but be- mused by the unlikely shower pattern and the plastic connections between the balls,” Shannon wrote.

The cake-store clown only appeared to juggle—but Shannon’s robot actually did. Assembled from his Erector Set, the finished product was able to handle three balls. The balls bounce off a tom-tom drum, and the robot moves its paddle arms in a rocking motion, “each side making a catch when it rocks down and a toss when it rocks up.” Though Shannon never completed the bounce-juggling robot’s counterpart—a robot that could authentically toss juggle—he still built his own set of clowns that put on a convincing imitation.

And there was one way, he noted with pride, in which they outclassed any human: “The greatest numbers jugglers of all time cannot sustain their record patterns for more than a few minutes, but my little clowns juggle all night and never drop a prop!”



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