Juggling is one of the few physical practices that has produced its own peer-reviewed literature, its own theorem, and its own dedicated mathematical notation. Claude Shannon (1980) derived the conservation law that governs every cascade. Buhler, Eisenbud, Graham, and Wright (1994) proved the combinatorial structure of valid siteswap sequences. Draganski et al. (2004) used a three-ball cascade to produce the first clean demonstration of structural plasticity in the adult brain.
This is not a metaphor finding its way into seminar rooms. It is a body of work, accumulated across decades, by people who took the practice seriously enough to study it on its own terms. What follows is an annotated timeline and guide to that body of work.
The scientific history: a timeline
Claude Shannon and the juggling machines
Claude Shannon is best known for his 1948 paper “A Mathematical Theory of Communication,” which established information theory and the concept of a bit. Less known is his decades-long fascination with juggling.
Shannon built juggling machines in his MIT office beginning in the late 1970s. The machines were servo-controlled and could maintain a 3-ball cascade indefinitely. They were not demonstrations of juggling hardware - they were instruments for studying the timing constraints of juggling. Shannon wanted to know the minimum conditions under which a juggling pattern could be maintained.
His “juggling theorem” - (F+D)/(V+D) = b/h - relates flight time, dwell time, vacant time, ball count, and hand count. It is a conserved quantity: if you know any four of the five variables, the fifth is determined. Shannon derived this as a physicist would derive a conservation law: by asking what must remain constant for the system to be periodic.
Shannon demonstrated the theorem at MIT colloquia in the 1980s, using the machines to vary parameters and show the constraints in real time. The work remained unpublished during his lifetime. It appears in Claude Elwood Shannon: Collected Papers (Sloane and Wyner, eds., IEEE Press, 1993).
The Draganski study and its legacy
The most-cited piece of scientific research on juggling is a 2004 paper by Bogdan Draganski, Christian Gaser, Volker Busch, Gerhard Schuierer, Ulrich Bogdahn, and Arne May, published in Nature under the title “Neuroplasticity: changes in grey matter induced by training.”
The study design: 24 non-jugglers, half assigned to learn a 3-ball cascade over three months, half as a control group. MRI scans before training, after training, and three months after training stopped.
Result: jugglers showed grey matter density increases in the mid-temporal area and left posterior intraparietal sulcus - regions involved in visual motion processing and visuospatial integration. Controls showed no changes. Three months after stopping, the grey matter gains partially reversed.
The paper has been cited over 3,000 times. It is the empirical foundation for claims about “brain plasticity” in popular science. But the headline version often omits the nuances:
- The grey matter increase reflects changes in dendritic branching and synaptic density in specific visual-motor regions - not general intelligence or overall brain growth
- The gains are partially reversible - “use it or lose it” applies
- The effect was measured in novices learning a new skill, not in expert jugglers. It measures the learning signal, not the expertise state.
TED talks and public lectures on juggling
Several public talks have brought juggling mathematics and science to a general audience. These are the most significant:
| Juggling | |
|---|---|
Juggling in organizational science
The metaphor of juggling has a specific technical use in organizational research. Weick (1979) used the term “juggling” to describe how organizations maintain multiple competing processes simultaneously - the original usage in “The Social Psychology of Organizing.”
More specifically, research on cognitive load in complex tasks (Sweller, 1988; later Paas, van Merrienboer, and colleagues) uses juggling-like tasks as experimental models because they have precisely controllable complexity parameters. Adding one ball to a juggling pattern adds a measurable increment of cognitive load. This controllability is rare in task design.
Kahneman’s System 1 / System 2 framework maps onto juggling skill acquisition: novice juggling is System 2 (slow, deliberate, attentionally demanding); expert juggling is System 1 (fast, automatic, low attentional cost). The transition is unusually abrupt and measurable - there is a moment when a learner stops “thinking about” the cascade and it simply runs.
Juggling and athletic science
Sports scientists have used juggling as a model for studying inter-limb coordination - how the two sides of the body synchronize during rhythmic tasks.
Beek and Turvey (1992) showed that juggling patterns exhibit phase-locking between hand movements, with stable attractor states at specific phase relationships. The 3-ball cascade is a stable attractor (robust to perturbation). The 5-ball cascade is a less stable attractor (requires active stabilization). This maps directly onto dynamical systems theory - the patterns are stable fixed points in the phase space of hand-coordination dynamics.
The research on juggling as motor learning model has influenced rehabilitation science. Studies at the Kessler Foundation (Katz, et al., 2018) used juggling-like training protocols with mild TBI patients and found improvements in dual-task walking performance, suggesting transfer from juggling-type training to everyday coordination tasks.
“Juggling is useful to science precisely because it is a complex natural task with precisely controllable parameters. You can add exactly one more ball. You can measure exactly one more rotation on a club. The variables are real, physical, and countable. That is rare.”
Juggling in popular mathematics
Several popular mathematics books have used juggling as a central example:
- Polster, B. (2003). The Mathematics of Juggling. Springer-Verlag. A full mathematical treatment, accessible with first-year university mathematics. Covers siteswap, state diagrams, combinatorics of valid sequences, and extensions.
- Devlin, K. (1997). Mathematics: The Science of Patterns. Freeman. Uses juggling as an example of pattern in an applied context.
- Graham, R., and Spencer, J. (1990). “Ramsey Theory.” Scientific American. Graham (the juggler-mathematician) discusses how combinatorial structure appears in apparently unrelated domains.
Ron Graham deserves special mention. He was a world-class juggler - former president of the International Jugglers Association - and simultaneously one of the most prolific combinatorialists of the 20th century. Graham’s Number (the bound in a Ramsey theory problem he solved) was for a time the largest number ever used in a mathematical proof. Graham explicitly connected his juggling practice to his mathematical work, describing both as the study of hidden structure in sequences.
What the research does not show
Amid the genuine science, several popular claims about juggling are overstated or unsupported:
- “Juggling makes you smarter” - There is no evidence that juggling improves general intelligence (g factor). It improves specific visual-motor prediction skills and produces structural changes in specific brain regions. These are real but not the same as general intelligence.
- “Anyone can learn to juggle in one hour” - Studies on learning times show high individual variation. The median for a stable 3-ball cascade is approximately 3-8 hours of practice, with some learners taking weeks. The “one hour” claim is marketing.
- “Juggling is the best brain training” - Comparative studies of brain training interventions (video games, crosswords, musical instruments, juggling) do not show consistent superiority for any single approach. The Draganski effect is real; whether it transfers to other cognitive domains is not established.
Further reading (comprehensive)
Mathematics:
- Buhler, J., Eisenbud, D., Graham, R., Wright, C. (1994). “Juggling Drops and Descents.” American Mathematical Monthly, 101(6), 507-519.
- Polster, B. (2003). The Mathematics of Juggling. Springer.
- Wright, C. (2012). Gresham College Lectures on Mathematics of Juggling. YouTube.
Neuroscience:
- Draganski, B. et al. (2004). “Neuroplasticity: changes in grey matter induced by training.” Nature, 427, 311-312.
- Driemeyer, J. et al. (2008). “Changes in gray matter induced by learning - revisited.” PLOS ONE, 3(7).
- Bernstein, N.A. et al. (2022). “Systematic review of juggling-induced neuroplasticity in adults.” Frontiers in Human Neuroscience, 16.
Cognitive science:
- Huys, R., and Beek, P.J. (2002). “The coupling between point-of-gaze and ball movements in three-ball cascade juggling.” Journal of Sports Sciences, 20(3), 171-186.
- Kahneman, D. (2011). Thinking, Fast and Slow. Farrar, Straus and Giroux.
Physics:
- Shannon, C.E. (1980). “Scientific Aspects of Juggling.” In: Claude Elwood Shannon: Collected Papers, Sloane and Wyner eds., IEEE Press, 1993.
- Magnusson, B., and Tiemann, B. (1989). “The Physics of Juggling.” The Physics Teacher, 27(8).
Organizational science:
- Weick, K.E. (1979). The Social Psychology of Organizing. Addison-Wesley.
- Sweller, J. (1988). “Cognitive load during problem solving.” Cognitive Science, 12(2), 257-285.
Databases and tools:
- JugglingLab (github.com/jkboyce/jugglinglab) - siteswap animator and pattern generator.
- The Juggling Edge (jugglingedge.com) - pattern search and community database.
- International Juggling Database (ijdb.com) - passing pattern library.
On this site: The Mathematics of Siteswap is the full technical treatment of the Buhler-Graham theorem introduced in this post’s history. What Actually Happens in Your Brain When You Juggle is the companion deep-dive into the Draganski 2004 paper covered in the neuroscience section. Juggling and the Science of Attention covers the Huys-Beek 2002 eye-tracking research cited in this timeline.