The Theorem of Juggling and the Ultimate Machine

Unicycles, Flame-Throwing Trumpets, and the Box That Turns Itself Off

On the desktop of Claude Shannon's study inside his Winchester, Massachusetts home sat a small polished wooden casket fitted with a single brass toggle switch. Whenever a visitor flipped the switch to the ON position, an electric motor buzzed inside the box. The wooden lid creaked open, and a pale mechanical hand emerged. Slowly extending a single index finger, the hand pushed the brass toggle switch back to OFF, retreated under the lid, and shut itself in total silence.

The concept had originated during a playful conversation in the early 1950s with artificial intelligence pioneer Marvin Minsky, who sketched a machine whose sole purpose was to turn itself off. Shannon immediately built several working models in his shop. When science fiction writer Arthur C. Clarke saw the casket on Shannon's desk in 1952, Clarke wrote that there was something unspeakably sinister about a small box that did nothing except switch itself off.

After joining the Massachusetts Institute of Technology as a visiting professor in 1956 and becoming Donner Professor of Science in 1958, Shannon spent less time publishing formal academic papers and more time filling Entropy House with mechanical inventions. Those devices blurred the boundary between engineering and vaudeville. In his basement machine shop—lined with lathes, milling machines, thirty unicycles, and wall-to-wall pegboards of gears—he built a motorized pogo stick and a bicycle with an off-center rear hub that bounced like a galloping horse. He also constructed a brass trumpet that shot three-foot plumes of fire when blown and a motorized ski chairlift that carried passengers down to Mystic Lake.

Among all those pastimes, juggling held a special mathematical grip on his imagination. At Bell Labs in Murray Hill, colleagues had often jumped out of the way in the narrow hallways as Shannon rode past on a unicycle while juggling three balls in the air. In the 1970s, paying tribute to the deadpan vaudeville juggler and comedian W. C. Fields, Shannon constructed a life-sized mechanical mannequin of Fields out of Erector-set beams, cams, and electric motors. Instead of tossing balls upward—which requires rapid visual tracking and delicate finger timing—Shannon's automaton bounce-juggled three steel balls off a taut horizontal drumhead. Because a bouncing steel ball slows to zero vertical velocity at the exact apex of its arc, the grooved rocking hands of the W. C. Fields mannequin could scoop and release each ball gently at the top of its bounce with purely passive mechanical stability.

In his 1980 paper Scientific Aspects of Juggling, Shannon asked why human jugglers who mastered five or seven balls hit a sharp physical ceiling as they tried to add more props. By tracking a periodic juggling pattern over a long time interval, he derived the fundamental conservation law now known as Shannon's Juggling Theorem:

Here $H$ is the number of hands (typically two) and $N$ is the number of balls being juggled. The variable $F$ is the flight time a ball spends in the air between throws, $D$ is the dwell time a ball spends held in a hand, and $V$ is the vacant time a hand remains empty between releasing one ball and catching the next.

The proof follows directly from two ways of counting the total time balls spend touching hands. From the perspective of the $H$ hands, each hand repeats a cycle of length $V + D$ and holds a ball for a fraction $\frac{D}{V + D}$ of the time. From the perspective of the $N$ balls, each ball repeats a cycle of length $F + D$ and rests inside a hand for a fraction $\frac{D}{F + D}$ of the time. Setting the total hand-occupancy equal to the total ball-occupancy immediately yields $(F + D)H = (V + D)N$. Because flight time in Earth's gravity grows only with the square root of throw height ($F = \sqrt{8h/g}$), juggling twice as many balls at a fixed hand rhythm requires throwing the balls four times as high. When high-speed video cameras timed world-record jugglers tossing up to nine balls in the 1980s, every measured throw and catch obeyed Shannon's conservation equation linking flight time, dwell time, vacant hand time, hands, and balls.