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Earlier this month, Indian Institute of Science, Bengaluru, professor Aninda Sinha and his former doctoral student Faizan Bhat linked the esoteric mathematics of Srinivasa Ramanujan with the principles underlying the physics of turbulent fluids and the expansion of the universe.
The bridge they laid was π (pi) — not the humble one but the transcendental one school students know to be the ratio of any circle’s circumference to its diameter.
Their paper appeared in Physical Review Letters.
Recipe for pi
While π is central to computing the volume and areas of objects, it is itself interminable and thus irrational. Its value is 3.14159265… There is no known pattern to the infinite avalanche of digits after the decimal point. Even today, professional mathematicians are developing formulae that rapidly and reliably predict this sequence.
For rough-ready use the ratio 22/7, first discovered by the Greek mathematician Archimedes 1500 years ago gives a series of numbers which is considered a coarse approximation to pi. There have been several improvements through the years, employing different branches of mathematics to compute pi, usually involving several terms and laborious substitutions.
More than a century ago Srinivasa Ramanujan, an accountant in Chennai and yet to be admitted to the pantheon of mathematical greats, discovered a set of astonishingly fast-converging formulas for 1/π. He discovered at least 17 distinct infinite series for 1/π. Each of them acts like a special “recipe”: add the first term, you get a rough value; add a second, it becomes dramatically more accurate; continue a bit more, and the approximation converges very quickly to π.
Some of these formulae are so efficient that they underpin the Chudnovsky algorithm, which scientists have used to compute π to over 200 trillion digits on modern supercomputers.
Like a rubber band
But Dr. Sinha wasn’t interested in merely adding to the pi. “We were interested in the maths behind Ramanujan’s thinking,” he said over the phone.
The trail began unexpectedly in string theory — a grand theory of theoretical physics that seeks to explain how all the fundamental particles of matter, electrons, neutrinos, quarks, gravitons, etc., could have emerged from the vibrations of invisible little coils of energy called ‘strings’.
Last year, Dr. Sinha and a collaborator were studying certain string-theoretic calculations and realised some of the existing answers in the literature were incomplete or incorrectly quoted.
“In the process of finding new representations of those string answers, we found a new formula for π,” he recalled. “In fact, an infinite number of new formulae.”
A string, Dr. Sinha explained, can be thought of like a rubber band: you can stretch it in many ways and its elasticity can take many values.
“If π is somehow hidden in the string answer, it should have an infinite number of different ways of looking at it. That is what we found.”
“That’s what pushed me to go back and look more carefully at Ramanujan’s formulae,” he continued. “Once I looked at the modern presentation, something jumped out. Because of my training, I immediately recognised structures I had seen before in conformal field theories.”
At a critical point
Conformal field theories (CFT) are the mathematical language of critical phenomena, those special points where systems are on the edge of change.
For instance, when water boils at 100°C and room pressure, you can clearly distinguish the liquid and vapour. But at a much higher temperature and pressure of 374°C and 221 atm, it reaches a critical point where that distinction vanishes: the fluid becomes ‘superfluid’ and is neither clearly liquid nor clearly gas, no matter how closely you zoom in.
“At the critical point, you cannot actually say which is liquid and which is vapour,” Dr. Sinha said. “That is the point where CFTs enter: they are used to explain what happens in this kind of critical phenomena.”
The Ramanujan equations, particularly the terms that are used, appeared to be analogous to those in certain kinds of CFTs. The mathematical engine Ramanujan intuitively deployed to find pi — involving modular equations, elliptic integrals and special functions — exactly matched the structure of correlation functions in the CFTs (specifically logarithmic CFTs).
As of now, their work doesn’t yet settle any grand conjecture in number theory or cosmology. Instead it stands as an intriguing bridge between two distant regions of thought: Ramanujan’s intuitive modular equations and modern CFT.
New line of inquiry
“[In] any piece of beautiful mathematics, you almost always find that there is a physical system which actually mirrors the mathematics,” Mr. Bhat in a press statement. “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 said, history is replete with examples of mathematical ideas developed in isolation, sometimes even as pure flights of fancy, eventually resonating with the physics of the real world decades later.
“Riemannian geometry (or the geometry of curved spaces) was being developed in the 19th century as pure mathematics. Much later, Einstein’s general theory of relativity showed that the geometry of spacetime itself is Riemannian (because of gravity’s impact on space-time). Today, we even use it with GPS,” Dr. Sinha said.
Napoleon Bonaparte’s mathematical advisor Joseph Fourier developed Fourier transforms as a mathematical tool to analyse heat flow. Today it underlines digital image and music compression.
For now, the Ramanujan-CFT connection has already spawned a new line of inquiry in Dr. Sinha’s group: the mathematical structure they identified appears again, he said, in models of an expanding universe.
On the mathematical side, the work hints that other transcendental numbers — of which π is just one example — could admit similarly efficient representations rooted in physics.
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