The iconic constant we commonly know as pi (π), an irrational number famously approximated as 3.14 and extending infinitely without any repeating pattern, first enters most of our lives during elementary school mathematics lessons, typically when exploring the geometry of circles. In recent decades, the relentless march of computational power has propelled this familiar mathematical entity far beyond the confines of the classroom. Today, sophisticated supercomputers are capable of calculating pi to an astonishing depth, reaching trillions of decimal places. However, a groundbreaking discovery has revealed an unexpected and profound connection: mathematical formulas for computing pi, conceived by the legendary Indian mathematician Srinivasa Ramanujan over a century ago, are now being found to hold deep resonance with some of the most pressing and complex ideas in contemporary fundamental physics. These remarkable connections are not confined to abstract theory; they are manifesting in the theoretical descriptions of phenomena as diverse as percolation, the chaotic dance of fluid turbulence, and even certain enigmatic characteristics of black holes.

The story of Ramanujan’s extraordinary contribution to the calculation of pi is as fascinating as the recent physics revelations. In 1914, a pivotal year shortly before his departure from Madras to pursue his academic endeavors at Cambridge, the prodigious Srinivasa Ramanujan published a seminal paper that presented a collection of 17 distinct formulas for calculating the value of pi. These formulas were not merely novel; they were strikingly efficient, offering a pathway to compute pi with a speed and accuracy that far surpassed any existing techniques of that era. Astonishingly, despite their relatively concise mathematical structure, containing only a small number of terms, these formulas were capable of generating an impressive number of accurate digits of pi. The enduring legacy of these expressions is undeniable. Ramanujan’s innovative methods have become foundational to the modern mathematical and computational approaches employed for calculating pi, forming the bedrock upon which even the most advanced algorithms used by today’s supercomputers are built. As Aninda Sinha, a distinguished Professor at the Centre for High Energy Physics (CHEP) at the Indian Institute of Science (IISc) and the senior author of the recent study, elaborates, "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."

For Professor Sinha and Faizan Bhat, the lead author of the study and a former PhD student at IISc, the significance of Ramanujan’s formulas extended beyond their remarkable computational efficiency. They were driven by a deeper, more fundamental question: why did such powerful and elegant mathematical structures for calculating pi even exist in the first place? Eschewing the view of these formulas as purely abstract mathematical curiosities, the research team embarked on a quest to uncover an explanation rooted in the physical world. "We wanted to see whether the starting point of his formulae fit naturally into some physics," Professor Sinha explains. "In other words, is there a physical world where Ramanujan’s mathematics appears on its own?"

This profound inquiry led their investigation into a broad and intricate family of theoretical frameworks known as conformal field theories (CFTs). More specifically, their focus narrowed to a particularly intriguing subset called logarithmic conformal field theories (LCFTs). These theories are designed to describe physical systems that exhibit a property known as scale invariance symmetry. This means that the system appears fundamentally the same, regardless of the scale at which it is observed, much like the self-similar nature of fractal patterns. A tangible and relatable example of scale invariance in the physical world can be observed at the critical point of water. At a very precise temperature and pressure, the distinction between liquid water and water vapor vanishes, and the substance enters a state of critical behavior. At this unique point, water exhibits scale invariance symmetry, and its complex behavior can be elegantly captured and understood using the principles of conformal field theory. Similar critical phenomena, characterized by scale invariance, also emerge in diverse physical processes such as percolation (the study of how substances spread through porous materials), the complex onset of turbulence in fluid dynamics, and in certain theoretical models used to describe the enigmatic nature of black holes. These diverse phenomena, from the microscopic to the cosmic, all fall within the conceptual domain of logarithmic conformal field theories.

The pivotal breakthrough for Sinha and Bhat’s research team came with the discovery that the underlying mathematical framework that forms the very essence of Ramanujan’s celebrated pi formulas also finds a striking parallel within the complex equations that govern these logarithmic conformal field theories. By ingeniously exploiting this shared mathematical structure, the researchers found themselves equipped to compute key quantities within these intricate theories with unprecedented efficiency. Such advancements hold the potential to significantly enhance scientists’ understanding of highly complex and often intractable processes, including the intricate dynamics of fluid turbulence and the multifaceted nature of percolation phenomena. This approach remarkably mirrors Ramanujan’s own ingenious method of deriving precise results for pi by starting from a compact, elegant mathematical expression and rapidly arriving at highly accurate values. "In any piece of beautiful mathematics, you almost always find that there is a physical system which actually mirrors the mathematics," observes 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 implications of these findings are far-reaching and underscore the enduring power of intellectual inquiry across disciplines and time. The research unequivocally demonstrates that Ramanujan’s formulas, meticulously crafted more than a century ago with no knowledge of modern physics, possess a previously unrecognized advantage: they offer a potent tool for accelerating and simplifying complex calculations in contemporary high-energy physics. Beyond their immediate practical utility in scientific computation, the researchers emphasize that this work serves as a profound testament to the extraordinary breadth and depth of Ramanujan’s visionary 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," Professor Sinha concludes, highlighting the remarkable prescience embedded within Ramanujan’s mathematical legacy. The universe, it seems, continues to whisper its secrets through the elegant language of mathematics, and Ramanujan’s century-old formulas are proving to be an invaluable key to deciphering them.