Pi (π) — the mathematical constant that links circles to everything from pendulums to river bends — continues to surprise. In early 2024, physicists Arnab Priya Saha and Aninda Sinha of the Indian Institute of Science, while working on string theory, stumbled upon a brand-new formula for pi.
Their work, published in Physical Review Letters, adds to humanity’s long history of uncovering the mysteries of pi. String theory, which posits that the universe’s fundamental building blocks are vibrating strings rather than particles, led Saha and Sinha to investigate how these strings interact. Their goal was to develop a unifying theory of fundamental forces. Along the way, they found new mathematical relationships, including one that provides an infinite series to calculate pi.
For centuries, mathematicians have tried to find efficient methods for calculating pi, with one of the earliest approaches dating back to Archimedes. His method of drawing polygons inside and outside of circles narrowed down pi’s value, but was impractical for precise calculation. In the 15th century, Indian scholar Madhava discovered a simpler series that allowed pi to be determined with increasing accuracy by summing fractions with odd-numbered denominators. Saha and Sinha’s discovery built on Madhava’s work, revealing that his series was just a special case of a much larger equation. Their formula introduces a parameter, λ, which can take an infinite range of values, each leading to pi.
For instance, when λ is infinite, their formula reduces to Madhava’s original series. What makes their formula so striking is its efficiency.
While Madhava’s series requires billions of terms to approximate pi to 10 decimal places, Saha and Sinha’s formula for a small λ requires only 30 terms to reach the same precision. Although their formula for pi was an unexpected by-product of their string theory research, it adds a fascinating chapter to the long history of pi. “Our motivation wasn’t to find a new formula for pi,” Sinha explained in an interview.
“Pi was just a by-product.” In fact, their research could have even broader implications, potentially shedding light on the Riemann zeta function — one of the greatest unsolved mysteries in mathematics.
For now, though, their new formula for pi stands as a remarkable mathematical discovery, born from the search for the building blocks of the universe.
