Most people first encounter the irrational number π (pi) -- commonly approximated as 3.14 and extending infinitely without repeating -- during school lessons about circles. In recent decades, advances in computing have pushed this familiar constant far beyond the classroom, with powerful supercomputers now calculating pi to trillions of decimal places.
Researchers have now uncovered an unexpected twist. Physicists at the Centre for High Energy Physics (CHEP), Indian Institute of Science (IISc) report that mathematical formulas developed a century ago to compute pi are closely linked to some of today's most important ideas in fundamental physics. These connections appear in theoretical descriptions of percolation, fluid turbulence, and even certain features of black holes.
Ramanujan's Remarkable Pi Formulae
In 1914, shortly before leaving Madras for Cambridge, renowned Indian mathematician Srinivasa Ramanujan published a paper presenting 17 different formulas for calculating pi. These expressions were strikingly efficient, allowing pi to be computed much faster than existing techniques of the time. Despite containing only a small number of mathematical terms, the formulas produced an impressive number of accurate digits.
Their impact has endured. Ramanujan's methods became foundational to modern mathematical and computational approaches for calculating pi, including those used by today's most advanced machines. "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 study. "These algorithms are actually based on Ramanujan's work."
For Sinha and Faizan Bhat, the study's first author and a former IISc PhD student, the mystery went beyond computational efficiency. They asked why such powerful formulas should exist in the first place. Rather than treating them as purely abstract results, the team searched for an explanation rooted in physics.
"We wanted to see whether the starting point of his formulae fit naturally into some physics," says Sinha. "In other words, is there a physical world where Ramanujan's mathematics appears on its own?"
Their investigation led them to a broad family of theories known as conformal field theories, and more specifically to logarithmic conformal field theories. These theories describe systems that exhibit scale invariance symmetry -- meaning they look the same regardless of how closely they are examined, similar to fractals.
A familiar physical example appears at the critical point of water, defined by a precise temperature and pressure at which liquid water and water vapor become indistinguishable. At this point, water displays scale invariance symmetry, and its behavior can be captured using conformal field theory. Similar critical behavior arises in percolation (how substances spread through a material), during the onset of turbulence in fluids, and in certain theoretical treatments of black holes. These phenomena fall within the domain of logarithmic conformal field theories.
Using Ramanujan's Structure to Solve Physics Problems
The researchers discovered that the mathematical framework at the heart of Ramanujan's pi formulas also appears in the equations underlying these logarithmic conformal field theories. By exploiting this shared structure, they were able to compute key quantities within the theories more efficiently. Such calculations could ultimately improve scientists' understanding of complex processes like turbulence and percolation.
The approach mirrors Ramanujan's own method of starting from a compact mathematical expression and rapidly arriving at precise results for 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 findings reveal that Ramanujan's formulas, developed more than 100 years ago, offer a previously hidden advantage for making modern high-energy physics calculations faster and more manageable. Beyond their practical value, the researchers say the work highlights the extraordinary reach of Ramanujan's ideas.
"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.