Pi, a mathematical constant denoted by the Greek letter π, is the ratio of the circumference of a circle. *do* to its diameter *d*: π = C/d. The circumference of a circle is, in turn, equal to 2πr, where *to* is the radius of the circle. No matter how big or small a circle is, this ratio will always be the same, which is why it is called constant.

Pi appears everywhere, from particle physics to sacred geometry. But why? In honor of Pi Day on March 14, let's examine the history of this fascinating number.

## Pi's Story

As a concept, pi has been in use at least since the time of the Babylonians and ancient Egypt: both of these values 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 geometrically, using polygons with increasingly larger numbers of sides. By AD 150, Ptolemy had established the value of pi at 3.1416, which he may have done using Archimedes' method.

Archimedes used inscribed and circumscribed polygons like these in his approximation of pi.

Credit: Public domain

Ancient Greek mathematicians, including Pythagoras and his followers, knew of the constant relationship between radius and circumference. Pi appears in the sacred geometry of mathematically inclined ancient Greek mystics. But, like the square root of 2, pi is an irrational number, an endless string of numbers that never repeats; it cannot be precisely expressed as a ratio of two integers. That idea was anathema to Pythagoras' sense of truth. *cosmos*the knowable order of things. Instead, the followers of Pythagoras redoubled their efforts on polygons. Polygonal proofs, such as those favored by Archimedes and Pythagoras, were *of rigor* in mathematics for another millennium and a half, that is, until calculus came along and stole the show.

The earliest recorded use of the Greek symbol π comes not from Greek antiquity but from 1706, when the Welsh mathematician William Jones used it to represent the periphery. However, calculating pi with ever-increasing precision was already a much-loved pastime for mathematicians. Isaac Newton spent many hours absorbed 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 calculations,” Newton wrote, “I had no other business at the time.” Newton and Leibniz used their methods of calculation to approximate pi, converging on a value accurate to dozens of decimal places. The Swiss polymath Leonhard Euler later adopted the single-letter form, beginning with his 1727 work. *Essay explaining the properties of air*.

## Pi is in everything

Anywhere there is a circle or a spiral, you can find Pi. Pi hides in the meanders of rivers, in the waves of light and sound, and in eclipses and solar flares and the double helix of DNA.

Pi is a constant relationship between aspects of a circle, but it also appears in trigonometry. Euler realized that there was a deep connection between circles and triangles, through cycles that repeat over time. In his foundational work of 1748, *Introduction to infinitorum analysis* (Introduction to the Analysis of Infinity), articulated that connection through a formula that would eventually bear his name. Euler's use of π in his definition of *my*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 the sense of Euler's cycles informed Gauss's work on electromagnetism, periodicity, and noise. Scientists such as Faraday and Maxwell soon invoked the number pi to derive the principles of electromagnetism. By the early 20th century, Edison, Westinghouse, and Tesla had waged their current wars to bring electricity into homes.

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 alternating current and direct current electricity that powers our devices. Pi even appears in a mathematical tool called the Fourier transform, which helps digital circuits and software interpret analog waveforms. Fourier transforms are fundamental to the algorithms we use to encode and transmit voice, music, and video.

## Facts about Pi

The first 100 digits of pi are 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679.

An Indiana legislator once attempted to declare a value of π = 3.2 by legislative fiat. However, a mathematics professor and fellow legislator proved him wrong before Congress and killed the bill.

Archimedes used a 96-sided polygon to calculate his most accurate value of pi.

In 2008, an elaborate crop circle appeared in a barley field in the English countryside, leaving conspiracy theorists in an uproar. Eventually, an astrophysicist realized that the circle's complex structure encoded the first ten digits of the number pi.

Scientists have calculated pi to within a hundred trillion digits. However, mathematicians have estimated that a 39-digit approximation of pi will suffice for any cosmological calculations humans might attempt. With that level of precision, which mathematicians achieved in 1630, the circumference of the observable universe can be calculated to within 0.001. smaller than the diameter of a single hydrogen atom.

In 2009, Congress officially declared March 14 as National Pi Day.

## A precisely cut slice of Pi

As of 2023, the world record holder for the most accurate calculation of pi is Jordan Ranous of StorageReview. Ranous used y-cruncher, a benchmarking program that uses the Chudnovsky algorithm for the core calculation, to calculate pi to an accuracy of one hundred trillion decimal places. The calculation and validation took a few hours less than two months to complete, using a pair of AMD EPYC 9654 processors utilizing just under 600TB of QLC flash memory and over 1.5 terabytes of RAM in total.

In 2022, Google Cloud Developer Advocate Emma Haruka Iwao announced that she and her team had calculated the value of pi to an accuracy of 100 trillion decimal places, a major advance on their previous record of 31.4 trillion digits set in 2019. “We were impressed by Emma and Google Cloud’s achievement,” Ranous wrote, “but we also wondered if we could do it faster, at a lower overall cost.”

Still, they had to solve a critical problem: how to create a storage volume large enough to hold a hundred-trillion-digit text file. In the end, the team used a RAID 0 array and finished their calculation in about a third of the processing time it took the Google Cloud team. (“While RAID 0 may surprise some,” Ranous said, wrote“In our defense, the file server storage was drawn from a mirrored Windows Storage Spaces pool, so redundancy was available on the remote host.”)

Ranous and his team also published your y-cruncher validation file and the last 100 of those 100 trillion digits, just in case anyone wants to check your work.

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