Our favorite mathematical constant, pi (Ï€), describing the ratio between the circumference of a circle and its diameter, took on a new meaning.
The new representation was born out of twists in string theory and two attempts by mathematicians to better describe particle collisions.
“Our effort initially was never to find a way to look at pi,” says Aninda Sinha of the Indian Institute of Science (IISc), who co-authored the new paper with fellow IISc mathematician Arnab Priya Saha.
“All we were doing was studying high-energy physics in quantum theory and trying to develop a model with fewer and more precise parameters to understand how the particles interact. We were excited when we got a new way to look at pi. “
Being a mathematical constant, the value of pi has not changed, however irrational a number it is; over time we simply got a more accurate rendering of its exact value, reaching 105 trillion numbers at last count.
This new work by Sah and Sinha presents a new series representation of pi that they say provides an easier way to extract pi from the calculations used to decipher the quantum scattering of high-energy particles thrown in particle accelerators.
In mathematics, a series interprets the components of a parameter such as pi, so mathematicians can quickly arrive at the value of pi from its components. It’s like following a recipe, adding each ingredient in the right amount and order to create a delicious meal.
Except that if you don’t have a recipe, then you don’t know what ingredients the food is made 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 quickly abandoned it because it was too complicated,” Sinha explains.
Sinha’s group was concerned with something entirely different: ways to mathematically represent the interactions of subatomic particles using as few and as simple factors as possible.
Saha, a postdoctoral researcher in the group, tackled this so-called “optimization problem” by trying to describe these interactions — which emit all kinds of strange and hard-to-observe particles — based on different combinations of particle mass. , including vibrations and a wide spectrum of their erratic movements.
What helped unlock the problem was a tool called a Feynman diagram, which represents mathematical expressions describing the energy exchanged between two interacting and scattering particles.
Not only did this yield an efficient model of particle interactions that captured “all the key fiber elements up to a certain energy”, but it also produced a new formula for pi that closely resembles the first ever serial representation of pi in recorded history. handed down 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 their appropriate modifications to test the experimental data for hadron scattering,” Saha and Sinha write in their published paper.
“Our new representation will also be useful in conjunction with celestial holography,” the pair add, referring to an interesting 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 be satisfied that scientists can more accurately describe what exactly makes up that famous irrational number.
The research was published in Physical Review Letters.