The research team, led by Professor Aninda Sinha and his former PhD student Faizan Bhat, has discovered a startling connection between mathematical formulas for calculating $pi$ developed a century ago by the legendary Indian mathematician Srinivasa Ramanujan and some of today’s most pivotal ideas in fundamental physics. These profound connections manifest in theoretical descriptions of seemingly disparate fields, including percolation phenomena, the chaotic dynamics of fluid turbulence, and even certain peculiar features associated with black holes, the cosmic entities from which not even light can escape. This finding implies that Ramanujan, working in relative isolation in early 20th century India, may have intuitively grasped mathematical structures that are now considered foundational to our understanding of the universe at its most extreme scales.

The story begins in 1914, a pivotal year for Ramanujan. Shortly before his departure from Madras for Cambridge, he published a seminal paper that introduced 17 distinct formulas for computing $pi$. These expressions were not merely elegant mathematical curiosities; they were remarkably efficient, enabling the calculation of $pi$ with unprecedented speed compared to the techniques available at the time. What is particularly astonishing is that these formulas, despite containing a surprisingly small number of mathematical terms, yielded an impressive number of accurate digits of $pi$. Their impact has been enduring, shaping the trajectory of mathematical and computational approaches to calculating $pi$. Indeed, Ramanujan’s methods became the bedrock upon which modern algorithms for $pi$ computation were built, including those employed by today’s most advanced machines. Professor Sinha elaborates on this point, stating, "Scientists have computed pi up to 200 trillion digits using an algorithm called the Chudnovsky algorithm. These algorithms are actually based on Ramanujan’s work." This direct lineage underscores the lasting legacy of Ramanujan’s mathematical genius.

However, for Sinha and Bhat, the fascination extended beyond mere computational prowess. They were driven by a deeper, more philosophical question: why should such extraordinarily powerful formulas for calculating $pi$ exist in the first place? Rather than viewing them solely as abstract mathematical constructs, the team embarked on a quest to find a physical explanation for their existence, seeking to ground these elegant equations in the tangible reality of the physical world. "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?" This question led them down a path of rigorous investigation, seeking to bridge the perceived gap between pure mathematics and applied physics.

Their meticulous investigation led them to a broad and sophisticated family of theories known as conformal field theories (CFTs). More specifically, their attention was drawn to a subset of these theories called logarithmic conformal field theories (LCFTs). These theoretical frameworks are adept at describing physical systems that exhibit scale invariance symmetry. This symmetry implies that the system appears fundamentally the same regardless of the scale at which it is observed, much like the intricate, self-repeating patterns found in fractals. A relatable physical example of scale invariance can be observed at the critical point of water. At a precise temperature and pressure, liquid water and water vapor become indistinguishable, and at this juncture, water exhibits scale invariance symmetry. The behavior of water at this critical point can be elegantly captured using the mathematical tools of conformal field theory. The IISc researchers realized that similar critical behaviors, characterized by scale invariance, arise in a variety of important physical phenomena. These include the process of percolation, which describes how substances, like water or oil, spread through porous materials; the complex and often chaotic onset of turbulence in fluids; and even in certain theoretical descriptions of black holes, where gravity warps spacetime to an extreme degree. Crucially, these phenomena are precisely the kinds of systems that fall within the domain of logarithmic conformal field theories.

The breakthrough moment for the IISc team came when they discovered that the underlying mathematical framework that forms the very heart of Ramanujan’s remarkable pi formulas also appears within the equations that govern these logarithmic conformal field theories. This shared mathematical structure provided them with a powerful new tool. By exploiting this common ground, they were able to compute key quantities within these complex physical theories with significantly greater efficiency. This advancement holds immense promise for improving scientists’ understanding of intricate and notoriously difficult-to-model processes such as fluid turbulence and percolation. The researchers noted that this approach bears a striking resemblance to Ramanujan’s own genius: starting from a compact, elegant mathematical expression and rapidly arriving at highly precise results.

Faizan Bhat eloquently captures this parallel, stating, "[In] any piece of beautiful mathematics, you almost always find that there is a physical system which actually mirrors the mathematics. Ramanujan’s motivation might have been very mathematical, but without his knowledge, he was also studying black holes, turbulence, percolation, all sorts of things." This suggests that Ramanujan, through his profound mathematical intuition, was inadvertently tapping into the fundamental physical principles that govern the universe. His formulas, conceived for the abstract purpose of calculating $pi$, possessed an inherent structure that mirrored the behavior of complex physical systems.

The implications of these findings are far-reaching. They reveal that Ramanujan’s formulas, conceived over a century ago, offer a previously unrecognized and significant advantage in making modern high-energy physics calculations faster and more manageable. Beyond their immediate practical value in computational physics, the researchers emphasize that this work underscores the extraordinary and often serendipitous reach of Ramanujan’s ideas. The discovery highlights how profound mathematical insights, born from pure intellectual curiosity, can resonate with and illuminate our understanding of the physical universe in ways that were unimaginable at the time of their conception.

Professor Sinha concludes with a sense of awe at the enduring power of Ramanujan’s legacy: "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." This sentiment encapsulates the profound wonder and intellectual thrill of discovering that the elegant mathematics of $pi$, as revealed by Ramanujan, continues to unlock the deepest secrets of the cosmos, a testament to the interconnectedness of abstract thought and the physical reality we inhabit. The century-old formulas are not just mathematical marvels; they are nascent windows into the fundamental workings of the universe itself.