Ramanujan’s Genius π Formulas From a Century Ago Might Help Explain the Deepest Secrets of the Universe

Ramanujan’s insights into pi are now guiding scientists toward a deeper understanding of how the universe works.

In 1914, Srinivasa Ramanujan arrived at Cambridge with a notebook filled with 17 extraordinary infinite series for 1/π. They were strikingly efficient, producing accurate digits of the world’s most famous irrational number much faster than any technique of the time. For over a century, mathematicians viewed these formulas as a pinnacle of number theory, but they lacked a “physical” explanation for why they worked so well.

Now, researchers at the Indian Institute of Science (IISc) have uncovered a hidden bridge between Ramanujan’s “recipes” for pi and the cutting-edge physics used to describe black holes and turbulent fluids. The study suggests that Ramanujan was intuitively using the same mathematical engine that governs how matter behaves at the edge of a total transformation.

Srinivasa Ramanujan’s life reads like fiction. Born into poverty in southern India and largely cut off from formal education, he taught himself mathematics by obsessively working through whatever books he could find, then pushing far beyond them. His notebooks filled with equations that seemed to arrive whole, without derivation, as if tuned in from some distant frequency.

In his early twenties, he began mailing these results to British mathematicians, most of whom ignored them. One didn’t. British Mathematician G.H. Hardy recognized that the strange, unproved formulas on the page could not be mere accidents. They were too original, too coherent. It was Hardy who brought Ramanujan to Cambridge, where the young mathematician produced a torrent of results before illness forced him back to India, where he died in 1920 at just 32. Ramanujan’s story was turned into the 2015 film The Man Who Knew Infinity.

Seventeen Formulas That Should Not Have Worked

In 1914, Srinivasa Ramanujan published 17 formulas for calculating 1/π. Each looked almost magical.

Add just a few terms, and π snaps into focus with uncanny speed. Compared with older methods — such as the long, laborious series dating back to Archimedes — Ramanujan’s expressions converged almost explosively.

Today, they still power the fastest π calculations on Earth. “Scientists have computed pi up to 200 trillion digits using an algorithm called the Chudnovsky algorithm,” says Aninda Sinha, a physicist at IISc and senior author of the new study. “These algorithms are actually based on Ramanujan’s work.”

Yet Ramanujan offered little explanation for why his formulas worked so well.

For more than a century, mathematicians treated them as isolated marvels. Sinha and his collaborator Faizan Bhat wanted to know whether that mystery pointed somewhere deeper.

“We wanted to see whether the starting point of his formulae fit naturally into some physics,” Sinha says. “In other words, is there a physical world where Ramanujan’s mathematics appears on its own?”

The Physics of Things on the Brink

Their search led to a class of theories with a long name and strange implications: logarithmic conformal field theories.

Physicists use conformal field theories to describe systems at critical points, such as the moments when matter teeters between phases and ordinary notions of scale break down.

Water provides a classic example. When water is heated to exactly 374°C under 221 atmospheres of pressure, at this precise moment, the distinction between liquid and vapor vanishes into a “superfluid” state. Zoom in or out, and the system looks statistically the same.

“At the critical point, you cannot actually say which is liquid and which is vapor,” Sinha told The Hindu. “That is the point where CFTs [Conformal field theories] enter: they are used to explain what happens in this kind of critical phenomena.”

Logarithmic versions of these theories describe even stranger systems: percolation (how fluids spread through porous materials), dense polymers, certain quantum Hall states, and the onset of turbulence. They also appear in theoretical descriptions of black holes.

What Bhat and Sinha found is that the mathematical backbone of Ramanujan’s π formulas — the same structure that makes them converge so quickly — also appears inside the equations that define these physical theories.

Pi Hiding in Plain Sight

At the heart of Ramanujan’s work with π series lies a mathematical identity known as the Legendre relation. On its own, it looks abstract and apparently has nothing to do with physics.

But when the IISc team rewrote it using the language of conformal field theory, something unexpected happened. The symbols began to line up with physical quantities: correlation functions, scaling dimensions, and operators that encode how systems fluctuate at criticality.

In their analysis, Ramanujan’s mysterious parameters map directly onto the properties of twist operators — mathematical objects that track how systems behave when boundaries or symmetries are disrupted.

The result is a new computational shortcut.

By borrowing Ramanujan’s strategy — compressing complex behavior into compact expressions — the researchers constructed faster ways to calculate key quantities in logarithmic conformal field theories. In some cases, calculations that normally require summing many contributions collapse to just one.

“Remarkably,” the authors write, the entire answer can emerge from what physicists call the “identity operator” alone.

That kind of simplification hints at a universal property of these theories, something fundamental hiding beneath the equations.

Black Holes, Turbulence, and a Rubber Band

The connection extends even further.

In the study’s appendix, the authors show that the same mathematical structure appears in models of black holes described using holography — a framework where gravity in higher dimensions maps onto quantum physics in lower ones.

In this setting, Ramanujan’s formulas correspond to how disturbances propagate between a black hole’s horizon and the edge of spacetime. The same equations also describe how polymers stretch, how fluids turn turbulent, and how clusters form in percolating materials.

Bhat sees a familiar pattern.

“[In] any piece of beautiful mathematics, you almost always find that there is a physical system which actually mirrors the mathematics,” he says. “Ramanujan’s motivation might have been very mathematical, but without his knowledge, he was also studying black holes, turbulence, percolation, all sorts of things.”

Sinha offers a metaphor drawn from string theory. A string, he explains, behaves like a rubber band: stretch it different ways, and it reveals different properties. Pi, hidden inside those equations, shows up through infinitely many mathematical perspectives.

When Pure Math Waits for Physics

This is not the first time mathematics has anticipated physics by decades.

Riemannian geometry began as a 19th-century abstraction, but Einstein later revealed that spacetime itself obeys its rules. Fourier transforms emerged from studies of heat flow and now underpin digital images, music, and data compression.

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Ramanujan’s work fits this pattern uncannily well.

Working largely in isolation in early 20th-century India, with little exposure to modern physics, he stumbled onto structures that now sit at the center of quantum field theory and cosmology.

“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 says.

Beyond Pi

The researchers are careful not to oversell their results. This work does not solve open problems in number theory or unlock a theory of everything.

But it opens a door.

The same approach could reveal fast-converging formulas for other irrational numbers. It could streamline calculations in theories that model turbulence and critical behavior.

The IISc researchers are already looking toward the next horizon. The same mathematical structure they identified in Ramanujan’s pi series has reappeared in their models of an expanding universe.

It turns out that when we calculate the circumference of a circle, we might be using the same rules that govern the very fabric of the cosmos.

The new findings appeared in the Physical Review Letters.