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In creating a simpler model for particle interactions, scientists created a new elegant pi.
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Representations of pi help scientists use values ​​close to real life without storing a million digits.
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Creating a new pi involved using a series, which is a structured set of terms that either converge to a single expression or diverge.
In new research, physicists use principles from quantum mechanics to build a new model of the abstract concept pi. Or more accurately, they built a new model that will happen include great new representation of pi. But what does this mean and why do we need different representations of pi?
Because quantum mechanics examines the smallest particles, one by one, even simple questions can have complex answers that require massive computing power. Rendering high-tech video games and movies like Avatar it can take days or longer, and that’s still not at the level of reality. In this new paper published in a peer-reviewed journal Physical Review Lettersphysicists Arnab Priya Saha and Aninda Sinha describe their new version of a quantum model that reduces complexity but maintains accuracy.
This is called optimization. Think of how early Internet video memorized bits of similar colors, or how classic animators painted static bodies with individual moving parts on top. Heck, think about how people cut the corners of square footpaths until they take a shortcut down a dirt road. We are surrounded by optimization and optimizing behavior.
As detailed in their paper, Saha and Sinha combined two existing ideas from mathematics and science: the Feynman particle scattering diagram and Euler’s beta function for scattering in string theory. The result is a series – something represented in mathematics by the Greek letter Σ surrounded by parameters.
Series may or may not end up generalizing into overall equations or expressions. And while some series diverge—meaning the terms alternate further apart—others converge toward one approximate, concrete result. This is where pi comes in. The digits of pi go to infinity, and pi itself is an irrational number, meaning it can’t really be represented by a fraction of a whole number (the one we often learn in school, 22/7, isn’t very accurate by 2024 standards).
But it can be represented quite quickly and well with a series. This is because the series can continue to build values ​​down to the smallest digits. If a mathematician compiles the terms of the series, he can use the resulting abstraction to do calculations that are not possible with the approximation of pi, which is truncated to 10 digits on a standard desktop calculator. A sophisticated approximation enables the work of nanoscopic particles that inspired these scientists.
“In the early 1970s,” Sinha said in a statement from the Indian Institute of Science, “scientists briefly explored this line of research but quickly abandoned it because it was too complicated.”
But mathematical analysis like this has come a long way since the 1970s. Today, Sinha and Saha are able to analyze the existing model and redesign it with changed terms. They are able to construct the sequence and see that it converges to the value of pi in far fewer terms than expected, making it easy for scientists to run the series and use it for further work.
All of this requires decades of basic work in the field and a large body of work showing that some mathematical moves work where others don’t. It is a comment on the ongoing and collaborative nature of mathematical theory, even if the results are a working model that could help scientists. Our ability to approach meaningfully has grown in tandem with our ability to solve complex problems head-on.
“Doing this kind of work, even though it may not have immediate application in everyday life, gives the pure pleasure of doing theory for the sake of doing it,” Sinha said in a statement.
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