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Doctoral Student Juggles Math and … Juggling!

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Working to make his passion an Olympic sport, Jonah Botvinick-Greenhouse knows how to keep many balls in the air (literally)

This story was condensed from a feature in the Cornell Chronicle.

By James Dean

Outside Gates Hall as spring classes ended, Jonah Botvinick-Greenhouse juggled a flurry of white balls—five, if you could keep track—hands occasionally crisscrossing, at times juggling entirely behind his back or above his head.

Suddenly launching several balls skyward, the doctoral student in applied math pirouetted twice before calmly receiving the plummeting projectiles and lofting them back to lower altitudes.

“Wow,” says a passing student, taking out a phone to record the scene. “Cool,” remarks another. “That’s dope,” agrees a third, pausing mid-jog. Others wander by, too distracted to notice one of the world’s best jugglers at work.

In late June, Botvinick-Greenhouse—a two-time winner of the World Juggling Federation’s Advanced Overall Championship—competed in Paris for the WJF’s first Ultimate Overall Championship. He finished a very close second, only about a 10th of a point behind the winner.

The Paris Olympic Committee had invited the WJF to host the competition and an annual convention as part of pre-Summer Games festivities—a platform the WJF is using to build a case for juggling as an Olympic sport.

ESPN will broadcast the event—which featured routines with balls, rings, and club—on August 2.

“His patterns, his control, his symmetry, and his accuracy I think are unparalleled,” federation president Jason Garfield says of Botvinick-Greenhouse.

“Jonah’s juggling is very aesthetically pleasing. It’s rare, and I would say comforting, but also inspiring—and almost not believable, because it looks like it was programmed, like it’s a simulation.”

His patterns, his control, his symmetry, and his accuracy I think are unparalleled.
Jason Garfield, president, World Juggling Federation

Perhaps that’s appropriate for someone whose research in the Center for Applied Mathematics focuses on dynamical systems, and who as an undergraduate co-authored an article called “Juggling Dynamics” in Physics Today.

Botvinick-Greenhouse, who earned math and physics degrees at Amherst, not only takes pleasure in the act of juggling, but in the math underlying its patterns and the numerical notation system used to describe them, called “siteswap.”

“It turns out there are a lot of exciting connections between mathematics and juggling, so many people who happen to like math will take to juggling pretty easily,” he says. “It’s kind of the go-to language for how jugglers talk to each other and communicate about the tricks that they’re doing.”

Jonah Botvinick-Greenhouse, math PhD student, writes out "siteswap" notation in Malott Hall. — Writing out “siteswap” notation in Malott Hall.

"Siteswap" is a mathematical abstraction developed in the mid-1980s that counts the number of beats until an object is manipulated again—a number roughly corresponding to its throw height—and the order in which objects are swapped between hands.

Jugglers can analyze siteswaps to determine if a pattern can be juggled.

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For example, since averaging a siteswap’s digits reveals the number of objects being juggled, the average cannot be a fraction.

Allen Knutson, a math professor who once held a world record—68 catches of 12 balls with a partner—says jugglers can read siteswaps to confirm two essential conditions: that no two objects will land at the same time, and that an object will always be available to throw. Validate one condition and the other follows—a mathematical statement, says Knutson.

“If you boil something down sufficiently, then it will become mathematics,” he says. “Things sufficiently well understood can’t help but become mathematics.”

If you boil something down sufficiently, then it will become mathematics.
Professor Allen Knutson

Bennett Santora ’26, an information science major and former International Jugglers’ Association junior gold medalist, wrote a college admissions essay about siteswap.

“It made this technical progression of juggling possible,” says Santora, who can juggle seven balls consistently and has juggled as many as nine. “When everything was just kind of vaudeville tricks, it wasn’t obvious what the next hardest thing in the progression was. Now it’s just a question of, can you do it?”

For Botvinick-Greenhouse, more than most, the answer is yes.

Introduced to juggling by a friend when he was 10, he adopted YouTube as his mentor.

Jonah Botvinick Greenhouse, math PhD student, juggles with the Cornell Juggling Club in the Physical Science Building.. — Practicing with the Cornell Juggling Club in the Physical Sciences Building.

Hours of daily practice ensued, and he won a WJF junior title at age 12, just two years before claiming his first advanced overall championship—the youngest to do so. His progression is documented on YouTube and Instagram.

“I’m not surprised that I’m the kind of person who enjoyed learning to juggle four or five balls when I was 10 and 11, and now I’m doing math research,” says Botvinick-Greenhouse, also an advanced cellist who plays in a campus quartet.

“It seems like a similar thing going on in the brain that enjoys that type of progression. You’re not going to find the solution to a research question right away, and it’s going to take a lot of persistence and failing over and over again, but you enjoy the process of trying to work up to it.”

I’m not surprised that I’m the kind of person who enjoyed learning to juggle four or five balls when I was 10 and 11, and now I’m doing math research.

As in figure skating or gymnastics, whose scoring systems the WJF’s mimics, sport jugglers must balance ambitions to perform very difficult, high-scoring moves with the risk of errors that would earn deductions—ranging from collisions and drops to asymmetrical patterns, moving one’s feet too much, bad posture, or other poor form.

“There’s less crossover than you would hope between the theory and practice of it,” he says.

“It’s fun to think about mathematically and physically, but at the end of the day, the only thing that can help me improve and do good routines is practice. Just look up, and juggle.”

Top: Botvinick-Greenhouse juggles outside Milstein Hall. (All photos and video by Jason Koski / Cornell University.)

Published June 26, 2024; updated July 3, 2024