Earlier this month, Indian Institute of Science, Bengaluru Professor Aninda Sinha and his former PhD student Faizan Bhat linked the esoteric mathematics of Srinivasa Ramanujan with the principles underlying the physics of turbulent fluids and the expansion of the universe.
The bridge they laid was π (pi)—not the humble but transcendent one known to schoolchildren as the ratio of the circumference of any circle to its diameter.
Their paper appeared in Physical Review Letters.
Recipe for pi
While π is central to calculating the volume and areas of objects, it is itself infinite and therefore irrational. Its value is 3.14159265… There is no known pattern for the endless avalanche of digits after the decimal point. Even today, professional mathematicians are developing formulas that quickly and reliably predict this sequence.
For rough use, the ratio 22/7, first discovered by the Greek mathematician Archimedes 1,500 years ago, gives a series of numbers that is considered a rough approximation to pi. Over the years there have been several refinements employed by various branches of mathematics to calculate pi, usually involving several terms and laborious substitutions.
More than a century ago, Srinivasa Ramanujan, an accountant in Chennai and not yet admitted to the pantheon of mathematical greats, discovered a set of surprisingly rapidly converging formulas for 1/π. He discovered at least 17 distinct infinite series for 1/π. Each of them works as a special “recipe”: add the first term, you get an approximate value; add a second, it becomes dramatically more accurate; continue a little further and the approximation converges to π very quickly.
Some of these formulas are so powerful that they support Chudnovsky’s algorithm, which scientists have used to calculate π to more than 200 trillion digits on modern supercomputers.
Like a rubber band
But Dr. Sinha was not interested in merely adding to pi. “We were interested in the mathematics behind Ramanujan’s thinking,” he said by phone.
The trail began unexpectedly in string theory—the grand theory of theoretical physics that seeks to explain how all the fundamental particles of matter, electrons, neutrinos, quarks, gravitons, etc., could have arisen from the vibrations of invisible tiny coils of energy called “strings.”
Last year Dr. Sinha and his collaborator studied certain string theory calculations and found that some existing answers in the literature were incomplete or misquoted.
“In the process of finding new representations of these string responses, we found a new formula for π,” he recalled. “In fact, an infinite number of new formulas.
The string, explained Dr. Sinha can be thought of as a rubber band: you can stretch it in many ways and its elasticity can take on many values.
“If π is somehow hidden in the string answer, there should be an infinite number of different ways to look at it. That’s what we found.”
“That made me go back and look more closely at Ramanujan’s formulas,” he continued. “As soon as I looked at the modern presentation, something jumped out. Thanks to my training, I immediately recognized structures that I had previously seen in conformal field theories.”
At a critical point
Conformal field theory (CFT) is the mathematical language of critical phenomena, those special points where systems are on the verge of change.
For example, when water boils at 100°C and room pressure, you can clearly distinguish between liquid and vapor. But at a much higher temperature and pressure of 374°C and 221 atm, it reaches a critical point where this difference disappears: the fluid becomes a “superfluid” and is neither clearly liquid nor clearly gaseous, no matter how close you zoom.
“At the critical point, you can’t really tell what’s liquid and what’s vapor,” said Dr. Sinha. “That’s the point where CFTs come in: they’re used to explain what’s going on in these kinds of critical phenomena.”
Ramanujan’s equations, especially the terms used, appeared to be analogous to those in certain types of CFT. The mathematical engine Ramanujan intuitively deployed to find pi—involving modular equations, elliptic integrals, and special functions—exactly matched the structure of correlation functions in CFTs (specifically, logarithmic CFTs).
Their work has not yet resolved any major conjectures in number theory or cosmology. Instead, it stands as an interesting bridge between two distant fields of thought: Ramanujan’s intuitive modular equations and modern CFT.
New line of inquiry
“(In) any beautiful mathematics, you will almost always find that there is a physical system that actually mirrors the mathematics,” Mr. Bhat said in a press statement. “Ramanujan’s motivation may have been very mathematical, but unbeknownst to him, he was also studying black holes, turbulence, seeps, all kinds of things.”
That said, history is full of examples of mathematical ideas developed in isolation, sometimes even as purely fanciful, ultimately resonating with real-world physics decades later.
“Riemannian geometry (or the geometry of curved spaces) was developed in the 19th century as pure mathematics. Much later, Einstein’s general theory of relativity showed that the geometry of spacetime itself is Riemannian (due to the effect of gravity on spacetime). Today we even use it with GPS,” said Dr. Sinha.
Napoleon Bonaparte’s mathematical advisor Joseph Fourier developed Fourier transforms as a mathematical tool for heat flow analysis. Today it underlines the digital compression of images and music.
The Ramanujan-CFT connection for the time being in the group of Dr. Sinhy sparked a new line of inquiry: the mathematical structure they identified reappears, he said, in models of the expanding universe.
Mathematically, the work suggests that other transcendental numbers—of which π is just one example—could admit of similarly efficient representations rooted in physics.
jacob.koshy@thehindu.co.in
Published – 11 Dec 2025 08:30 IST
