In 1914, Indian mathematician Srinivasa Ramanujan published a short paper detailing several unusual formulas for calculating the value of the number π. Using only a few mathematical terms, the formulas generated far more digits of pi (π) than any existing methods.
Ramanujan’s work later became central to modern algorithms for computing pi. For physicists at the Indian Institute of Science (IISc), the formulas raised a different question. In a recent study published in Physical Review Letters, researchers explored what makes these compact expressions work so well. That question eventually led them to an unexpected place.
Looking Beyond Computation
For decades, researchers have primarily treated Ramanujan’s formulas as tools for efficient calculation. Powerful computers can now use similar methods to compute pi to trillions of digits.
“Scientists have computed pi up to 200 trillion digits using an algorithm called the Chudnovsky algorithm,” said Aninda Sinha, a physicist at IISc and senior author of the new study. “These algorithms are actually based on Ramanujan’s work.”
Sinha and his collaborator, Faizan Bhat, chose to explore these formulas from a different angle. Instead of asking how the formulas worked, they asked why such formulas existed in the first place.
“We wanted to see whether the starting point of his formulae fit naturally into some physics,” Sinha said. “In other words, is there a physical world where Ramanujan’s mathematics appears on its own?”
A Familiar Pattern in Unfamiliar Places
The researchers turned to theoretical physics, where certain mathematical patterns appear repeatedly across very different systems. In particular, they focused on a class of models known as conformal field theories.
These theories describe systems that look the same across different scales. They are often observed in physics at critical points where small changes can ripple across all scales simultaneously. For example, the temperature and pressure of water can reach a critical point where liquid and vapor become indistinguishable. This could also apply to percolation or the early stages of fluid turbulence. Some theoretical descriptions of black holes also rely on related formulas.
Within this broader category lies a more specialized group called logarithmic conformal field theories. These models are mathematically complex to work with, but they do appear in several real physical contexts. While examining these theories, the researchers noticed something familiar.
An Overlapping Mathematical Structure
Logarithmic conformal field theories and Ramanujan’s pi formulas both rely on closely related mathematical structures. That overlap enabled the use of Ramanujan-style techniques to compute quantities in the physics models. Calculations that normally require lengthy, complex steps can be approached more directly. The strategy echoed Ramanujan’s original approach of extracting precise results from a compact expression.
“In any piece of beautiful mathematics, you almost always find that there is a physical system which actually mirrors the mathematics,” Bhat said. “Ramanujan’s motivation might have been very mathematical, but without his knowledge, he was also studying black holes, turbulence, percolation — all sorts of things.”
Modern Applications for Old Mathematics
The study does not claim that Ramanujan predicted practical applications of these concepts in modern physics. Instead, it demonstrates that mathematical concepts developed in one field can later become useful in another.
“We were simply fascinated by the way a genius working in early 20th-century India, with almost no contact with modern physics, anticipated structures that are now central to our understanding of the universe,” Sinha said.
More than a century later, Ramanujan’s formulas are still being rediscovered, not just as historical curiosities, but also as tools for navigating complex concepts in modern physics.
Austin Burgess is a writer and researcher with a background in sales, marketing, and data analytics. He holds a Master of Business Administration, a Bachelor of Science in Business Administration, and a Data Analytics certification. His work combines analytical training with a focus on emerging science, aerospace, and astronomical research.