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Pi, a mathematical constant denoted by the Greek letter π, is the ratio of a circle’s circumference C to its diameter d: π = C/d. The circumference of a circle is, in turn, equal to 2πr, where r is the circle’s radius. No matter how big or small a circle is, this relationship will always be the same, which is why it’s called a constant.
Pi shows up everywhere, from particle physics to sacred geometry. But why?
History of Pi
As a concept, pi has been in use since at least the time of the Babylonians and ancient Egypt: both approximated modern values of pi to within one percent (3.125 and 3.16, respectively). Archimedes is the attested author of the first rigorous definition of pi, which he did by way of geometry, using polygons with ever-greater numbers of sides. By 150 CE, Ptolemy had the value of pi at 3.1416, which he may have done using the Archimedean method.
Archimedes used inscribed and circumscribed polygons such as these in his approximation of pi.
Credit: Public domain
Ancient Greek mathematicians, including Pythagoras and his followers, knew the constant relationship between radius and circumference. Pi shows up in the sacred geometry of the mathematically inclined mystics of ancient Greece. But like the square root of 2, pi is an irrational number, a neverending string of numbers that never repeats; it can’t be precisely expressed as a ratio of two integers. That idea was anathema to Pythagoras’ sense of cosmos, the knowable order of things. Instead, followers of Pythagoras doubled down on polygons. Polygonal proofs, like the ones Archimedes and Pythagoras favored, were de rigeur in mathematics for another millennium and a half—that is, until calculus swept in and stole the show.
The earliest recorded use of the Greek symbol π comes not from Greek antiquity but from 1706, when Welsh mathematician William Jones used it to stand for periphery. However, calculating pi to ever-increasing precision was already a pastime beloved of mathematicians. Isaac Newton spent many hours engrossed in calculating a 15-digit value for pi as a constant in 1665 or 1666. “I am ashamed to tell you to how many figures I carried these computations,” Newton wrote, “having no other business at the time.” Newton and Leibniz both used their approaches to calculus to approximate pi, converging on a value accurate to dozens of decimal places. Swiss polymath Leonhard Euler later adopted the single-letter form, beginning with his 1727 Essay Explaining the Properties of Air.
Pi Is in Everything
Everywhere there’s a circle or a spiral, you can find pi. Pi is hiding in the meandering of rivers, in light and sound waves, and in eclipses and solar flares and the double helix of DNA.
Pi is a constant relationship between aspects of a circle—but it shows up in trigonometry too. Euler realized there was a deep connection between circles and triangles, by way of cycles that repeat over time. In his foundational 1748 work Introductio in analysin infinitorum (Introduction to Analysis of the Infinite), he articulated that connection through a formula that would eventually bear his name. Euler’s use of π in his definition of e, the natural logarithm, is a famous example of mathematical beauty: a statement known as Euler’s identity.
Carl Friedrich Gauss was a student of Euler, and Euler’s sense of cycles informed the work Gauss did with electromagnetism, periodicity, and noise. Scientists such as Faraday and Maxwell soon invoked pi to derive the principles of electromagnetics. By the turn of the twentieth century, Edison, Westinghouse, and Tesla had waged their Current Wars to bring electricity into the home.
Today, pi is visible in every aspect of the digital age, from the band gap that enables semiconductors to the platters of hard drives and the A/C and D/C electricity that powers our devices. Pi even crops up in a mathematical workhorse called a Fourier transform, which helps digital circuits and software make sense of analog waveforms. Fourier transforms are central to the algorithms we use to encode and broadcast voice, music, and video.
Pi Facts
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The first 100 digits of pi are 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679.
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An Indiana lawmaker once tried to declare a value of π = 3.2 by legislative fiat. However, a professor of mathematics and fellow legislator proved him wrong in front of Congress and scuttled the bill.
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Archimedes used a 96-sided polygon to calculate his most precise value of pi.
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In 2008, an elaborate crop circle appeared in a barley field in the English countryside, leaving conspiracy theorists abuzz. Eventually, an astrophysicist realized that the circle’s complex structure encoded the first ten digits of pi.
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Scientists have calculated pi to a precision of a hundred trillion digits. However, mathematicians have estimated that an approximation of pi to 39 digits will suffice for any cosmological calculation humans might attempt. At that level of precision, which mathematicians achieved in 1630, you can calculate the circumference of the observable universe with an error smaller than the diameter of a single atom of hydrogen.
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In 2009, Congress officially declared March 14th National Pi Day.
A Precision-Cut Slice of Pi
As of 2023, the world record holder for the most precise calculation of pi is Jordan Ranous of StorageReview. Ranous used y-cruncher, a benchmarking program that uses the Chudnovsky algorithm for main computation, to calculate pi to a precision of one hundred trillion decimal places. The calculation and validation took a few hours shy of two months to complete, using a pair of AMD EPYC 9654 processors that drew on just under 600TB of QLC flash memory and more than 1.5 terabytes of RAM between them.
In 2022, Google Cloud Developer Advocate Emma Haruka Iwao announced that she and her team had calculated the value of pi to a precision of 100 trillion decimal places—a flex on her previous record of 31.4 trillion digits, set in 2019. “We were impressed by the achievement of Emma and the Google Cloud,” Ranous wrote, “but we also wondered if we could do it faster, with a lower total cost.”
Still, they had to solve one critical problem: How to create a storage volume of sufficient size to hold a text file a hundred trillion digits long. In the end, the team used a RAID 0 array, finishing their calculation in about a third of the compute time it took the team from Google Cloud. (“While RAID 0 might raise some eyebrows,” Ranous wrote, “in our defense, the file server storage was carved out of a mirrored Windows Storage Spaces pool, so redundancy was available on the remote host.”)
Ranous and his team also published their y-cruncher validation file and the last 100 of those 100 trillion digits—just in case anyone wants to check their work.