Physicists have uncovered a hidden connection between a famous 350-year-old mathematical formula for *pi,* everyone’s favorite irrational number, and quantum mechanics. At least one mathematician has pronounced the discovery “a cunning piece of magic.”

The English mathematician John Wallis published his formula for calculating *pi* as the product of an infinite series of ratios in 1655. In a paper published this week in the *Journal of Mathematical Physics,* University of Rochester physicists announced they had discovered the same formula popping out of their calculations of a hydrogen atom’s energy levels.

Wallis isn’t well known today outside of academic circles, but he rubbed elbows with some of the the greatest names in science in his era. Initially he intended to become a doctor when he started university at the tender age of 13, but he was far more interested in mathematics, and showed a knack for cryptography in particular. It began as just a hobby, but years later, he applied his skills deciphering coded Royalist dispatches on behalf of their political rivals, the Parliamentarians. (The two parties were in the midst of a civil war at the time.) Eventually he became part of the group of scientists who founded the Royal Society of London. There, his love of math blossomed into a bona fide academic pursuit.

Among his peculiar skills: he could perform complicated mental calculations in his head — something he did frequently, given his tendency toward insomnia. One such feat was recorded in the Society’s *Philosophical Transactions* in 1685: Wallis had calculated the square root of a 53-digits (27 digits in the square root) one sleepless night, and recorded it from memory the next morning.

So, yeah, the guy could do the math. In 1656, Wallis published his most famous work, *Arithmetica infinitorum*, containing his classic formula for *pi*. (No less a luminary than Christopher Huygens remained highly skeptical until Wallis walked him through it to show his work.)

“The value of *pi* has taken on a mythical status, in part, because it’s impossible to write it down with 100% accuracy,” Rochester physicist Tamar Friedmann, lead author of the new paper, told Science 2.0. “It cannot even be accurately expressed as a ratio of integers and is, instead, best represented as a formula.”

Friedmann and her co-author, Carl Hagen, weren’t actually looking for anything remotely *pi*-related. “It just sort of fell in our laps,” Hagen said in a press release. He was just trying to teach his students a particular technique to approximate the energy states of quantum systems — in this case, the hydrogen atom.

But when he set about solving the problem himself, he noticed something odd about the error bars. It was around 15% for a hydrogen atom’s lowest energy state (the ground state), 10% for the first excited state (which occurs when the atom gets an infusion of energy that bumps the electron up to the next energy level), and then kept getting smaller with each successive higher energy level. That’s the opposite of what this particular technique is supposed to produce: the best approximations are usually at the ground state.

Intrigued, Hagen enlisted Friedmann’s help, and they found themselves going back to Niels Bohr’s model of the hydrogen atom from the earliest days of quantum mechanics, depicting the electron orbits as perfectly circular. “At the lower energy orbits, the path of the electron is fuzzy and spread out,” Hagen explained. “At more excited states, the orbits become more sharply defined and the uncertainty... decreases.”

Apparently it took a mere 24 hours for the journal to accept their paper, which must be some kind of record. “The special thing is that it brings out a beautiful connection between physics and math,” said Friedmann. “I find it fascinating that a purely mathematical formula from the 17th century characterizes a physical system that was discovered 300 years later.”

**Reference**:

Friedmann, Tamar, and Hagen, C.R. (2015) “Quantum mechanical derivation of the Wallis formula for pi,” *Journal of Mathematical Physics* 56: 112101.

*Top image: Still from Irrational Numbers: Pi and Pies, a ClickView original Mathematics series. Bottom image: Pages from Wallis’s Arithmetica Infinitorum, digitized by Google.*