For most people, the irrational number π (pi) is a familiar companion from school mathematics—introduced as the ratio of a circle’s circumference to its diameter and approximated as 3.14. Beyond the classroom, pi has taken on a life of its own. With the advent of modern computing, scientists have calculated pi to trillions of decimal places, pushing the limits of numerical precision and computational power. Yet beneath this seemingly technical pursuit lies a deeper and more surprising story—one that connects a century-old mathematical breakthrough to some of the most profound ideas in modern physics.
At the heart of this story is Srinivasa Ramanujan, one of India’s greatest mathematical geniuses. In 1914, shortly before leaving Madras for Cambridge, Ramanujan published a remarkable paper presenting 17 highly efficient formulas for computing pi. These expressions were revolutionary. Using only a small number of terms, they produced an extraordinary number of correct digits, far outperforming existing methods of the time. Today, these formulas are not merely historical curiosities; they form the foundation of modern algorithms used to compute pi to extreme precision.
Ramanujan’s Legacy in Computing Pi
Modern supercomputers rely on algorithms that trace their intellectual ancestry directly to Ramanujan. One prominent example is the Chudnovsky algorithm, which has enabled scientists to compute pi to over 200 trillion digits. According to Aninda Sinha, Professor at the Centre for High Energy Physics (CHEP) at the Indian Institute of Science (IISc), these modern methods are deeply rooted in Ramanujan’s work. His formulas remain among the fastest-converging series ever discovered for pi, demonstrating a level of insight that continues to astonish mathematicians.
But for Sinha and his collaborator Faizan Bhat, a former IISc PhD student, the real intrigue lay not in computational speed alone. Instead, they asked a more fundamental question: Why do Ramanujan’s formulas exist at all? What deeper structure makes such compact and powerful expressions possible? Rather than viewing them as isolated mathematical tricks, the researchers wondered whether these formulas might naturally arise within a physical theory.
Seeking Physics Behind Pure Mathematics
This line of inquiry led the team into the realm of theoretical physics, specifically to conformal field theories (CFTs). These theories describe systems that exhibit scale invariance symmetry—meaning the system looks the same regardless of the scale at which it is observed. Fractals offer an intuitive visual analogy, where zooming in reveals repeating patterns.
Scale invariance is not just a mathematical abstraction; it appears in real physical systems at critical points. A classic example is water at its critical temperature and pressure, where the distinction between liquid and vapor vanishes. At this point, fluctuations occur at all length scales, and the system can be described using conformal field theory.
Among the many variants of CFTs, Sinha and Bhat focused on a special class known as logarithmic conformal field theories (LCFTs). These theories play a role in describing complex phenomena such as percolation (how substances spread through porous materials), the onset of turbulence in fluids, and even certain theoretical aspects of black holes. Despite their importance, LCFTs are notoriously difficult to analyze due to the mathematical complexity involved.
Where Pi Meets the Physics of Extremes
In their research, the IISc team discovered that the mathematical structures underlying Ramanujan’s pi formulas also appear naturally within LCFTs. This was a striking result. The same framework that allows pi to be computed with extraordinary efficiency turns out to govern the equations describing scale-invariant physical systems.
By exploiting this shared structure, the researchers were able to calculate key quantities in logarithmic conformal field theories more efficiently than before. This approach mirrors Ramanujan’s own strategy: starting from a compact, elegant mathematical expression and extracting precise results with remarkable speed. In practical terms, this could help physicists better understand phenomena like turbulence, where calculations often become overwhelmingly complex.
Faizan Bhat reflects on the elegance of this connection, noting that beautiful mathematics often finds a mirror in the physical world. Although Ramanujan’s motivations were purely mathematical, his formulas unknowingly captured structures that now appear central to modern physics. In this sense, his work anticipated ideas that would only fully emerge decades later.
Ramanujan, Scale Invariance, and the Universe
The implications of this research extend far beyond pi. Logarithmic conformal field theories also appear in theoretical models of black holes, particularly in contexts where gravity, quantum mechanics, and thermodynamics intersect. The discovery that Ramanujan’s mathematics can simplify calculations in these theories suggests a profound unity between abstract mathematics and the laws governing the universe.
For Aninda Sinha, this unity is a source of continual fascination. Ramanujan worked in early 20th-century India with limited access to the scientific developments that would later shape modern physics. Yet his intuition led him to mathematical structures that now play a central role in our understanding of nature. This underscores a recurring theme in the history of science: ideas developed in one context can find powerful applications in entirely different domains, sometimes generations later.
A Century-Old Insight with Modern Impact
The IISc study highlights how Ramanujan’s formulas, created more than a hundred years ago, offer new tools for tackling contemporary problems in high-energy physics. Beyond improving computational efficiency, the work deepens our appreciation of how mathematics and physics inform one another. What began as an effort to understand an ancient numerical constant has opened new pathways into the study of turbulence, percolation, and black holes.
Ultimately, this research serves as a tribute to Ramanujan’s extraordinary legacy. His work reminds us that mathematical beauty is not confined to chalkboards and textbooks—it resonates through the fabric of the physical world. As modern scientists continue to uncover connections between his ideas and the universe’s most extreme phenomena, Ramanujan’s influence proves as infinite and enduring as pi itself.
Source: Indian Institute of Science (IISc)
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