Ramanujan’s 100-Year-Old Pi Formula Is Still Unlocking Secrets of the Universe

Uncover the surprising connection between Ramanujan's pi formulas and the universe. Learn how his century-old math helps explain turbulence, black holes, and more.

Most people first learn about the number π (pi) in school, usually when studying circles. It is often written as 3.14, but this is just an approximation. In reality, pi is an irrational number, meaning its decimal digits go on forever without repeating. For a long time, this made pi mostly a curiosity in mathematics. However, in recent decades, powerful computers have taken pi far beyond the classroom, calculating it to trillions of decimal places.

Now, scientists have uncovered a surprising new connection. Physicists at the Indian Institute of Science (IISc) in Bengaluru have shown that formulas for pi written over 100 years ago are closely related to some of today’s most important ideas in physics. These links appear in theories that describe how fluids become turbulent, how substances spread through materials, and even in certain mathematical descriptions of black holes. The research was published in Physical Review Letters.

At the heart of this discovery is Srinivasa Ramanujan, one of the most remarkable mathematicians of the early 20th century.

Ramanujan’s Extraordinary Formulas

In 1914, just before leaving Madras (now Chennai) to study at Cambridge, Ramanujan published a paper containing 17 different formulas for calculating pi. These formulas were astonishing. With just a few mathematical steps, they could produce highly accurate values of pi much faster than methods known at the time.

Over the years, mathematicians came to understand that Ramanujan’s ideas were not just clever tricks. In fact, modern computers rely on methods inspired by Ramanujan’s work. One such method, known as the Chudnovsky algorithm, has been used to calculate pi to over 200 trillion digits. As Professor Aninda Sinha from IISc explains, these powerful modern algorithms are built directly on Ramanujan’s original ideas.

Asking a Deeper Question

For Sinha and his colleague Faizan Bhat, the real puzzle went beyond speed or efficiency. They were interested in a more fundamental question: why do such powerful formulas exist at all?

Instead of seeing Ramanujan’s formulas as purely abstract mathematics, they wondered whether these expressions might naturally arise from the physical world. In simple terms, they asked whether the mathematics behind pi could also describe real physical systems.

Where Pi Meets the Physics of Patterns

Their search led them to a class of theories known as conformal field theories. These theories are used by physicists to describe systems that look the same no matter how much you zoom in or out. This property is called “scale invariance.” A familiar example is a fractal, where the same pattern appears at many different sizes.

One everyday example of scale invariance occurs at the critical point of water. At a very specific temperature and pressure, liquid water and water vapour become indistinguishable. At this moment, water shows scale-invariant behaviour, and physicists can describe it using conformal field theory.

Similar behaviour appears in many complex systems. These include percolation, which describes how liquids or gases spread through materials, the onset of turbulence in fluids, and even certain theoretical models of black holes. Many of these systems are described using a special type of theory called logarithmic conformal field theory.

Using Old Mathematics in New Physics

The IISc researchers discovered something remarkable. The same mathematical structure that lies at the heart of Ramanujan’s pi formulas also appears in the equations used in these physical theories.

By recognising this shared structure, the team found new ways to calculate important quantities within these theories more efficiently. This could help scientists better understand difficult problems such as turbulence and percolation, which is important in fields ranging from geology to materials science.

As Bhat puts it, beautiful mathematics often mirrors the real world. Although Ramanujan was focused purely on mathematics, his ideas unknowingly touched on phenomena such as black holes, turbulence and spreading processes.

A Century-Old Legacy

This work shows that Ramanujan’s formulas are more than historical achievements. Even after 100 years, they continue to offer new tools for modern physics and fresh insights into how the universe works.

For Sinha, the discovery is deeply inspiring. Ramanujan, working in early 20th-century India with little exposure to modern physics, somehow anticipated mathematical structures that are now central to our understanding of nature.