Youth Incorporated Magazine

In recent times, there have been significant advancements across various scientific fields. One notable research development that captured headlines in June 2024 involved Professor Aninda Sinha from the Indian Institute of Science (IISC) in Bangalore, India, along with Arnab Saha have discovered a new series representation for the irrational number pi during their research on how string theory may be utilized to explain certain physical events. Professor Aninda Sinha, a distinguished theoretical physicist at the Center for High Energy Physics, IISC, played a pivotal role in this research breakthrough.

Why Is Pie Essential, And How Is It Used?
The Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for π. The first calculation of π was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world. Archimedes was the first person to calculate an accurate estimate for pi, which we’ve since discovered is equal to about 3.14159.

Engineers use Pi to calculate the stress and strain on materials, design bridges and buildings, and develop new technologies for building and construction, quantum physics, communications, music theory, medical procedures, air travel, and space flight. Pi is an important number used in many different areas of physics, astronomy, and mathematics.

Hence, we reached out to professor Aninda Sinha on this achievement and to get to know more about this research that led to the discovery of pi.

1. Can you explain in simple terms the new series representation of pi that your research has uncovered?

Ans: In school we learn that pi is approximated by 22/7 but in reality, it is a number which cannot be written as the ratio of two integers. Mathematicians have been interested in writing pi as an infinite sum of terms made by ratios of integers. The earliest known such series is what is called the Madhava-Leibniz series. This series converges quite slowly. Many representations of pi exist in the literature which converge to pi much faster. Our goal was not to come up with a competing representation. Rather pi was a by-product of a different, physics-motivated question we were asking. The new series representation that we landed up finding had a parameter in it (think of it as an ingredient in a recipe that is beneficial in the intermediate stages of cooking but leaves no trace in the final product! Maybe a catalyst in a chemical reaction could also be an analogy). In a certain limit, where this parameter is large, the terms approximate those in the Madhava-Leibniz series. This parameter gives a family of new representations, which converge much faster than the Madhava-Leibniz series, which is a member in this family. Each term has a physics interpretation and the different limits of this parameter in our representation also have very interesting physics interpretations. For instance, for a range of this parameter, one can meaningfully interpret each term as corresponding to a specific Feynman diagram in a high energy physics process described by string theory.

2. What motivated you to use string theory in relation to calculating pi? Were there specific challenges or gaps in existing methods that prompted this research?

Ans: We were trying to write a specific string calculation in the language of Feynman diagrams. Feynman diagrams are the bread and butter of high energy particle physicists. The best understood microscopic description of nature called the standard model of particle physics also relies on using Feynman diagrams. Previous known attempts in string theory for the same calculation (specifically string field theory) do not lead to convenient formulas as roughly speaking, they need many more ingredients than what one would like to use. We used some new techniques that I have developed with other collaborators over the last several years (and also a crucial input by a physicist by the name of Chaoming Song of Univ. of Miami; in spite of being outside the community of researchers working in similar areas, he found our earlier work fascinating and built upon it). Finally, we talked with experts on this topic like Ashoka Sen and gained confidence that the question we were trying to address was an interesting one!

3. What impact will this discovery have on theoretical physics as a whole? OR How will your research team use this newfound understanding of pi in future projects?

Ans: The long term impact is difficult to predict. The techniques we developed will enable us to investigate an optimisation problem with far fewer ingredients than one may have thought possible. In follow-up work, we have been investigating this line of research. As I mentioned, our original goal was not to find a formula for pi. The new series representations of pi and its other cousins called Zeta functions were by-products of our investigations.

4. How do you envision this discovery impacting educational curricula or public understanding of mathematics and physics?

Ans: One thing that is fascinating is that our findings suggest that classical Mathematical functions (classical in the sense that they have been known and studied for a very long time) that have been known for many centuries (the key one in our research was the Euler-Beta function which was invented by the Swiss genius Euler around 1730) can still have useful new representations. The new one that we found was motivated by physics considerations. It is unlikely that mathematicians would have been looking for similar results without the underlying physics considerations. This suggests the importance of having a solid grounding in traditional mathematics and physics since their interplay can still lead to novel, interesting results. It is also important that universities encourage intermingling of different areas of science, since it is very likely that answers to inter-related questions will feed into different areas that now belong to distinct departments.

5. What advice would you give to young physicists or mathematicians who are interested in pursuing similar research paths?

Ans: It is important to have a breadth of knowledge (which can only be gained initially by learning from masters (here I refer to acclaimed experts) and then later on by reading widely) and not just information that is now readily available on google, but an in-depth understanding of how things work. Also in today’s day and age, it is important to learn coding since even in mathematics and theoretical physics, computer checks form a vital ingredient. Finally, it is very important to be patient. Contrary to the impressions we form with the present overdose of electronic content, science is not a race but a marathon.