Pi Formula Is More Than Just Math

More than a hundred years ago, long before anyone imagined supercomputers or black hole simulations, there was one legendary Indian mathematician named Srinivasa Ramanujan who wrote down a set of formulas to calculate the digits of π (pi).

These equations, just 17 short expressions, were mysterious even to mathematicians of his time, as they produced incredibly accurate digits of pi using very few mathematical steps. Today, pi has been computed to over 200 trillion digits, using algorithms whose foundations trace back to Ramanujan’s ideas.

However, a new study by researchers at the Indian Institute of Science (IISc) has revealed something far more surprising. It suggests that Ramanujan’s old mathematical tricks are not just clever; they naturally appear in modern physics, popping up in models that describe turbulence, percolation, and even aspects of black holes.

"Ramanujan’s motivation might have been very mathematical, but without his knowledge, he was also studying black holes, turbulence, percolation, all sorts of things," Faizan Bhat, first author of the study and an ex-PhD candidate at IISc, said.

The study authors focused on a simple but deep question. Why do Ramanujan’s formulas work so beautifully? Mathematicians have admired the formulas for more than a century, and they form the basis for modern pi-computing methods such as the Chudnovsky algorithm, but their origin has always felt almost magical.

Instead of looking for a purely mathematical answer, the researchers tried something new. What if Ramanujan’s mathematics naturally arises from physical laws? In other words, could there be a real physical system where these formulas appear without being forced in?

To find out, they searched through different areas of high-energy theory. Their attention settled on conformal field theories (CFTs)—powerful frameworks used to describe systems that look the same no matter how much you zoom in. A well-known example is the critical point of water, where liquid and vapor become identical, and the system shows repeating behavior.

Within this large family, they looked at a special subset called logarithmic conformal field theories (LCFTs). LCFTs describe phenomena right at the edge of order and chaos, places where small changes ripple outward dramatically.

These include percolation (how things spread or seep through a network or material), turbulence onset (when smooth fluid flow suddenly breaks into chaotic eddies), and certain black hole descriptions, where spacetime behaves in exotic ways.

Using detailed mathematical examination, the researchers discovered that the starting structure of Ramanujan’s pi formulae, the part that sets up how the series expands, exactly matches the structure that appears in LCFTs.

Once they recognized the match, they used Ramanujan’s mathematical machinery to compute complicated quantities inside these physical theories. Calculating these values normally requires long, heavy computations, but Ramanujan’s approach—which was originally designed to compute digits of pi swiftly—allowed them to do it much faster and more efficiently.

This created a perfect mirror. Just as Ramanujan used a simple starting expression to generate many accurate digits of pi, the physicists used the same underlying structure to generate accurate physical predictions in LCFTs with surprisingly little effort.