Researchers from the Indian Institute of Science (IISc) and the University of Calgary have come across a new way to represent the mathematical constant π. The new formula for expressing the irrational number was found unintentionally while the researchers were working on applying quantum field theory (QFT) principles to string theory amplitudes.

A happy accident

The mathematical constant π has been calculated in numerous ways over the centuries. For instance, π is defined as the ratio of the circumference of a circle to its diameter. However, the constant can also be expressed as integrals, continued fractions, and — perhaps most elegantly — using infinite series.

One of the most important π infinite series that math students learn early on is the Gregory-Leibniz and Nilakantha series. This new approach, however, is rooted in the intersection of QFT and string theory, two fundamental frameworks in modern physics.

Researchers Arnab Priya Saha and Aninda Sinha stumbled upon this novel mathematical series by accident, while exploring string theory’s applications in high-energy physics. Even more remarkably, the researchers found that the series closely mirrors the representation of pi proposed by 15th-century mathematician and astronomer Sangamagrama Madhava.

“Our efforts, initially, were never to find a way to look at pi. All we were doing was studying high-energy physics in quantum theory and trying to develop a model with fewer and more accurate parameters to understand how particles interact. We were excited when we got a new way to look at pi,” Sinha says.

A new and fast ‘recipe’ for pi

Traditional series representations of π can require millions of terms to achieve high precision. The new representation developed by Saha and Sinha converges much more rapidly. For instance, setting a parameter to 41.5 allows convergence to 15 decimal places with only 40 terms. Meanwhile, the traditional series may need 50 million terms for similar precision.

The new representation is derived from the Euler-Beta function and tree-level string theory amplitudes. By tweaking parameters and utilizing the crossing symmetric dispersion relation, the researchers obtained a series that converges quickly to π.

In mathematics, a series breaks down a complex parameter like pi into simpler components. This new series offers a rapid approach to approximate pi, which is super important for calculations in high-energy particle physics. The formula they presented is as follows:

“Physicists (and mathematicians) have missed this so far since they did not have the right tools, which were only found through work we have been doing with collaborators over the last three years or so,” Sinha explains. “In the early 1970s, scientists briefly examined this line of research but quickly abandoned it since it was too complicated.”

The rapid convergence of the new series can significantly reduce the computational effort required to calculate π to high precision. This can be particularly useful in fields where π is used extensively, such as numerical simulations, cryptographic algorithms, and computational geometry.

The study showcases how ideas from high-energy physics can lead to practical advancements in pure mathematics. It opens up new avenues for exploring other mathematical constants and functions using similar techniques. Researchers may apply these principles to develop new representations for constants like e or the natural logarithm, further enhancing our computational toolkit.

The findings appeared in the Physical Review Letters.