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Posted June 30, 2024 By Presh Talwalkar. Read about me, or email me.
While studying quantum theory, two scientists using string theory have accidentally discovered a new formula for pi, the most famous constant in mathematics. Arnab Saha and Aninda Sinha developed a formula to optimize certain calculations and they were surprised the formula could also calculate the digits of pi. Even more, under a certain limit, the formula equals Madhava’s formula for pi, which was the first recorded infinite series for pi. Here is a link to the study published in Physical Review Letters.. While the work is theoretical at this point, it may find practical applications in the future.
Check out the video for more details on the story.
Scientists just discovered a new formula for pi accidentally!
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Scientists just discovered a new formula for pi accidentally
By way of history, pi is a concept that equals the ratio of a circle’s circumference to its diameter. One of the first known estimates comes from Archimedes, who used perimeters of inscribed and circ*mscribed polygons to bound the circumference of a circle. He came up with 3.14 or 22/7 using a 96 sided polygon. About 500 years later Liu Hui used a slightly different method and tricks using areas of polygons instead of perimeters. He and subsequent mathematicians came up with pi as 3.14159 or 355/113 which was the most accurate approximation of pi for about 100 years. Calculating pi by geometric methods had essentially reached its limit.
Around the late 1300s to early 1400s the mathematician Madhava came up with an entire different approach. He discovered the most wonderful infinite series
π/4 = 1 – 1/3 + 1/5 – 1/7 + …
(The series is unfortunately “named after” Leibniz who was born hundreds of years later than Madhava. The very mathematicians who scrutinize every word of a proof do not have the same zeal for crediting originators, though some are now saying Madhava-Leibniz, so perhaps the convergence on this process is just very slow.)
What an incredible formula that relates pi to the alternating sum of reciprocals of odd numbers! This formula is amazing but is has a flaw: it is very slow to converge. It takes about 5 billion terms to get pi accurate to 10 digits. Madhava actually modified the formula and was able to estimate pi to 10 to 14 digits. In modern times, other infinite series are used and computers do the calculations, so the modern record has over 100 trillion digits of pi computed.
The new formula
In the current paper, the Indian scientists developed the following formula:
The meaning is beyond me, but if we take this as a pure mathematical formula, we can try to understand a bit about it. On the right hand side is the familiar factorial n! like 4! = 4x3x2x1 = 24. The factorial can be generalized to negative values and fractional values with the gamma Γ function, which appears on the left hand side. For positive integral values the gamma function is a shift of the factorial function with Γ(n) = (n – 1)!. Also on the right hand side is a closed parenthesis with a subscript n – 1, and that indicates the Pochhammer symbol which is a ratio of gamma functions (a)b = Γ(a + b)/Γ(a).
The gamma function is an integral
It can be evaluated to particular values like Γ(1/2) = √π and Γ(2) = 1.
So going back to their main formula
If we substitute s1 = s2 = -1/2, and simplify, we get the new formula for pi with the parameter λ
Interestingly this formula relates to an ancient pi formula. If you take the limit as λ goes to infinity and divide both sides by 4 you get exactly the Madhava series for pi!
π/4 = 1 – 1/3 + 1/5 – 1/7 + …
While Madhava’s formula is very slow to converge, the new formula can calculate pi to 10 digits using only 30 terms! You can even try it out as the factorial and gamma functions are built-in to spreadsheets. Taking λ = 10 we get:
The formula does rely on the gamma function, and if one could quickly calculate that, one could perhaps calculate Γ(1/2) = √pi and get perhaps a more direct way to get the digits of pi.
There is no claim the formula has practical uses at the present, or that it will break any records. But it is amazing that a string theory formula can relate to a historically important series for pi. It is these accidental discoveries that remind us of the beauty and wonder of mathematics and science.
References
Press release
https://iisc.ac.in/events/iisc-physicists-find-a-new-way-to-represent-pi/
Paper
https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.132.221601
Sabine Hossenfelder video
https://www.youtube.com/watch?v=rGd7Db52w1Q
References
https://www.iflscience.com/physicists-accidentally-discover-a-whole-new-way-to-write-pi-74768
https://phys.org/news/2024-06-physicists.html
https://www.indiatoday.in/science/story/indian-iisc-physicists-untangle-new-pi-series-that-could-change-maths-forever-2555611-2024-06-20
https://www.msn.com/en-in/entertainment/tv/indian-physicists-untangle-new-pi-series-that-could-change-maths-forever/ar-BB1oyGde
https://www.ynetnews.com/health_science/article/bk7d5ibir
https://mathoverflow.net/questions/473931/possible-new-series-for-pi
https://math.stackexchange.com/questions/4937730/is-the-new-series-for-a-big-or-even-medium-deal
https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
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PRESH TALWALKAR
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