Our favorite mathematical constant, pi (π), describes the relationship between the circumference of a circle and its diameter, has taken on a new meaning.
The new representation emerged from the twists and turns of string theory and from the attempts of two mathematicians to better describe particle collisions.
“Our efforts initially were never intended to find a way to look at pi,” says Aninda Sinha of the Indian Institute of Science (IISc), who co-authored the paper with fellow IISc mathematician Arnab Priya Saha the new work.
“All we did was study the high-energy physics in quantum theory and try to develop a model with fewer and more precise parameters to understand how particles interact. We were excited when we got a new way to look at pi to look.”
Because it is a mathematical constant, the value of pi has not changed no matter how irrational the number; over time we have simply acquired more accurate representations of its precise value, reaching 105 trillion figures at last count.
This new work by Saha and Sinha posits a new series representation of pi, which they say provides an easier way to extract pi from calculations used to decipher the quantum scattering of high-energy particles spun around in particle accelerators.
In mathematics, the components of a parameter such as pi are arranged in a series in such a way that mathematicians can quickly arrive at the value of pi based on its constituent parts. It’s like following a recipe and adding each ingredient in the right amount and order, to prepare a tasty dish.
Only if you don’t have the recipe, you don’t know what ingredients a meal consists of or how much to add and when.
Finding the right number and combination of components to represent pi has puzzled researchers since the early 1970s, when they first tried to represent pi in this way, “but soon quit because it was too complicated,” Sinha explains.
Sinha’s group looked at something completely different: ways to mathematically represent subatomic particle interactions using as few and simple factors as possible.
Saha, a postdoctoral researcher in the group, tackled this so-called “optimization problem” by trying to describe these interactions – which release all kinds of strange and difficult-to-understand particles – based on different combinations of the particles’ masses. , vibrations and the wide spectrum of their erratic movements, among others.
What helped unlock the problem was a tool called a Feynman diagram, which represents the mathematical expressions describing the energy exchanged between two particles interacting and spreading.
Not only did this provide an efficient model of particle interactions that captured ‘all the major fibrous features up to a certain energy’, but it also produced a new formula for pi that is very similar to the very first series representation for pi in history, put forward by the Indian mathematician Sangamagrama Madhava in the 15th century.
The findings are purely theoretical at this stage, but could have some practical applications.
“One of the most exciting prospects of the new representations in this paper is to use appropriate adaptations of them to re-examine experimental data for hadron scattering,” Saha and Sinha write in their published paper.
“Our new representation will also be useful in connecting to celestial holography,” the pair add, referring to an intriguing but still hypothetical paradigm that seeks to reconcile quantum mechanics with general relativity through holographic projections of spacetime.
For the rest of us, we can take satisfaction in knowing that researchers can more accurately describe what exactly the famous irrational number is made of.
The research was published in Physical Assessment Letters.