IISc physicists discovered that Ramanujan’s classic π-formulas arise naturally in modern theories describing critical phenomena and black holes. The connection suggests his early mathematics may have foreshadowed key structures in today’s high-energy physics. Credit: Stock

A new study reveals that Srinivasa Ramanujan’s century-old formulas for calculating pi unexpectedly emerge within modern theories of critical phenomena, turbulence, and black holes.

In school, many of us first encounter the irrational number π (pi) – rounded off as 3.14, with an infinite number of decimal digits – when we learn how a circle’s circumference relates to its diameter. Since then, computing power has advanced enormously, and modern supercomputers can now determine trillions of digits of this constant.

Researchers at the Centre for High Energy Physics (CHEP), Indian Institute of Science (IISc) have now shown that some of the purely mathematical formulas created a century ago to calculate pi are closely linked to present-day fundamental physics. These old formulas reappear in theoretical models used to study percolation, turbulence, and certain aspects of black holes.

The trail leads back to 1914. Just before leaving Madras for Cambridge, the renowned Indian mathematician Srinivasa Ramanujan published a paper that introduced 17 formulas for calculating pi. These formulas were exceptionally efficient, allowing pi to be computed more quickly than with other methods available at the time. Although they contained only a small number of mathematical terms, they still produced many correct decimal places of pi. Over the years, they became so important that they now underpin modern computational and mathematical techniques for evaluating pi, including the methods used on today’s supercomputers.

“Scientists have computed pi up to 200 trillion digits using an algorithm called the Chudnovsky algorithm,” says Aninda Sinha, Professor at CHEP and senior author of the new study. “These algorithms are actually based on Ramanujan’s work.”

A Deeper Question: Why Do These Formulas Exist?

The question that Sinha and Faizan Bhat, first author and former IISc PhD student, asked was: Why should such astonishing formulas exist at all? In their work, they looked for a physics-based answer. “We wanted to see whether the starting point of his formulas fit naturally into some physics,” says Sinha. “In other words, is there a physical world where Ramanujan’s mathematics appears on its own?”

They found that Ramanujan’s formulas naturally come up within a broad class of theories called conformal field theories, specifically within logarithmic conformal field theories. Conformal field theories describe systems with scale invariance symmetry – essentially systems that look identical no matter how deep you zoom in, like fractals. In a physical context, this can be seen at the critical point of water, a special temperature and pressure at which both liquid and vapor forms of water become indistinguishable from each other.

At this point, water shows scale invariance symmetry and its properties can be described using conformal field theory. Critical behavior also comes up in percolation (how things spread through a medium), at the onset of turbulence in fluids, and certain descriptions of black holes – phenomena that can be explained by the more specific logarithmic conformal field theories.

Ramanujan’s Mathematics Reappears in Physics

The researchers found that the mathematical structure underlying the starting point of Ramanujan’s formulas also comes up in the mathematics underlying these logarithmic conformal field theories. Using this connection, they could efficiently calculate certain quantities in these theories – ones that could potentially help them understand phenomena like turbulence or percolation better. This is similar to Ramanujan going from the starting point of his formulas and efficiently deriving pi. “[In] any piece of beautiful mathematics, you almost always find that there is a physical system which actually mirrors the mathematics,” says Bhat. “Ramanujan’s motivation might have been very mathematical, but without his knowledge, he was also studying black holes, turbulence, percolation, all sorts of things.”

The study shows that Ramanujan’s century-old formulas have a hitherto hidden application in making current high-energy physics calculations faster and more tractable. Even without this, however, Sinha and Bhat say they were just baffled by the beauty of Ramanujan’s mathematics. “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,” says Sinha.

Reference: “Ramanujan’s 1/π Series and Conformal Field Theories” by Faizan Bhat and Aninda Sinha, 2 December 2025, Physical Review Letters. DOI: 10.1103/c38g-fd2v

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