When Tadashi Tokieda’s eldest son was just 2 1/2 years old, the pair discovered a new physical phenomenon while splashing in the cool water at the Stanford West Apartments pool.

Tokieda held his son in one arm and a lightweight ball about 20 centimeters in diameter, decorated with Mickey Mouse’s face, in the other. They began to experiment.

When the Stanford University mathematics professor pulled the ball as far down as his arm could reach in the deep end and then released it, the ball floated up and gently bobbed at the surface. When he pulled the ball down only slightly below the surface of the water, the ball behaved similarly.

Kurt Hickman

The younger Tokieda then had another idea. “My son told me, in his imperfect language, that I should release it from somewhere halfway, not too deep, not too shallow,” Tokieda said.

When Tokieda did so, something unexpected happened: the ball shot up in the air, clear off the surface of the water. He later reasoned that the height to which the ball rose up in the air above the water was a function of how deep he released from. In mathematical terms, it had a sharp maximum. Tokieda was intrigued. “What is going on? It’s very, very strange.”

There isn’t a name for the phenomenon that Tokieda and his son studied in their neighborhood pool. But Tokieda observed the ball acted similarly to how dolphins and porpoises jump from the water into the air. In the biomechanics of zoology, people call it porpoising.

“I’m emphasizing that even an inanimate object can do this provided you release it from a suitable depth – not too deep, not too shallow,” he said, noting matter-of-factly that he’s since figured out all the mechanics and mathematics about why releasing from the intermediate depth does the trick.

One week after playing in the pool with his son, he was tutoring the concept in his advanced graduate course on fluid flows.

Whether Tokieda is in the pool, folding paper, or playing with a Slinky, he is constantly looking for the strange and surprising in our every day. Then he begins to calculate scientific and mathematical ideas to explain his observations. Prior to teaching math at Stanford, he was a painter in Japan, studied ancient languages as a classical philologist in France, and explored mathematics and physics at Princeton.

Tokieda is best known for his entertaining public outreach on behalf of physics and mathematics with his trove of toys: spinning tops, coffee mugs, and folded squares of paper. The simple toys work to illustrate complex concepts, like the Mickey Mouse ball in the pool, and his online lessons have millions of views. He says he finds joy in producing fresh material that appeals to everyone from Nobel laureates to school children.

Andrew Brodhead
This is a special brand of magic that has a traditional name. It’s called science. If you try it, it works too.”
Tadashi Tokieda Professor of Mathematics

“What I really like is to have those small human beings who have been given to us, to whom we owe the future, and play with them and discover with them,” Tokieda said in English, his seventh language. “No further agenda. I’m not using them as tools in the advance of my glory in adult society. That, I cannot care less about. If I cared, I would destroy the whole thing. Every time it works, I’m very appreciative. I’m grateful. If it doesn’t work, well, I’ll do it another time and I’ll try to do it better.”

Tokieda’s lectures have often been billed as magic shows. But magicians who pull rabbits out of top hats guard their magic tricks, so as not to spoil the secrets of their show. Tokieda describes his magic as completely open and reproducible. “This is a special brand of magic that has a traditional name. It’s called science,” he said. “If you try it, it works too. There’s no distance between me and the audience. All I’m doing is discovering and sharing physical phenomena, natural phenomena, as mathematics is a part of nature. I’m just introducing spectators to nature, and nature to spectators.”

A scientific discovery can follow a long, painstaking stage of hard work. Afterward, Tokieda says one can feel a sense of relief, or some pleasure at the prospect of publishing a paper in a high-impact journal, advancing one’s career, and earning a bigger salary. Sometimes, he admits, one finally arrives at the answer to a math problem and feels completely fed up. “In classrooms, when you’re going through a very complicated argument and finally you see QED at the end of a mathematical proof, people have already turned into skeletons and they can’t take it anymore,” he jokes.

But on very few occasions, people have a different reaction. When Tokieda and his son saw the ball pop out of the water into the mid-summer sky, they responded with joy. “When you reach the end of a scientific argument and you have just proved something or some truth comes out, it is such a wonder that you start laughing,” he said.

Andrew Brodhead

With his tricks and toys, Tokieda helps the general public appreciate, and sometimes even laugh at, how mathematics is an essential connection to the rest of our universe. Physics, for example, completely collapses without mathematics. But even he admits that explaining what mathematicians do is challenging. “I’m hoping that people will catch a glimpse and imagine that there’s something out there.”

Last summer, Tokieda and his family spent three months in Paris and he gave a public lecture at the Institut Henri Poincaré, “the nerve center of French mathematics,” he said. “It was an absolutely super awful technical lecture, no prisoners taken. The who’s who of the French mathematical society, and I hit them with the hard stuff.”

His family, including his now 6-year-old son, sat in the back of a large auditorium watching Papa perform. Throughout the lecture, the boy couldn’t stop laughing. And the mathematicians around him started laughing too. “Papa, when you gave that lecture in Paris, it was so funny,” the boy told his father later.

“But that’s part of the story of my life,” Tokieda said. “When I give, especially public lectures, people don’t always laugh when they’re supposed to, but they sometimes laugh when they’re not supposed to. So I break even on average.”