What Is Pi and Why Does It Turn Up Everywhere?

Pi is approximately 3.14159, and no fraction equals it. In 1768 Johann Lambert proved it is irrational, which means its decimal expansion never ends and never settles into a repeating cycle. In 1882 Ferdinand von Lindemann proved it is transcendental, a stronger property: it is not the solution of any polynomial equation with whole-number coefficients, which is why it is impossible to construct a square with the same area as a given circle using only a ruler and compasses, a problem the Greeks had worked on for two thousand years.

Because it cannot be written exactly, pi is defined by what it is rather than what it equals: the ratio of a circle's circumference to its diameter, or equivalently the area of a circle of radius one. Everything else about it has to be calculated.

The Babylonians used 3.125 and an Egyptian papyrus from about 1650 BCE implies 3.16, both good enough for building. Archimedes, around 250 BCE, made the first rigorous estimate by trapping a circle between two 96-sided polygons, one inside and one outside, and showed that pi lies between 3.1408 and 3.1429. The method was refined for eighteen centuries, with the Chinese mathematician Zu Chongzhi reaching seven correct decimal places in the fifth century and the Dutchman Ludolph van Ceulen spending most of his life computing 35, which were carved on his tombstone.

Calculus changed the game. From the late seventeenth century mathematicians found infinite series that add up to pi, such as Leibniz's formula, in which pi over four equals one minus a third plus a fifth minus a seventh and so on forever, and faster ones followed. By 1873 William Shanks had computed 707 digits by hand, of which the first 527 were right. Computers took over in 1949, and the record now runs to hundreds of trillions of digits. The milestones:

• •Around 250 BCE: Archimedes, between 3.1408 and 3.1429
• •Fifth century: Zu Chongzhi, 3.1415926 to 3.1415927
• •By 1610: van Ceulen, 35 digits, by polygons
• •1706: John Machin, 100 digits, by infinite series
• •1949: the ENIAC computer, 2,037 digits in 70 hours
• •2024: over 100 trillion digits

Pi turns up in places with no obvious roundness. The bell curve that describes heights, measurement errors and exam scores has pi in its formula, because the area under it is the square root of pi. The sum of the reciprocals of the square numbers, one plus a quarter plus a ninth and so on, equals pi squared over six, a result Euler found in 1734 that still surprises people. Pi governs the period of a pendulum, the behaviour of alternating current, the equations of quantum mechanics and Einstein's field equations for gravity.

The common thread is that pi is really about rotation and periodicity, not circles as pictures. Anything that repeats, oscillates or spreads evenly in all directions involves angles, and angles are measured in radians, which are defined by pi. The circle is simply the first place humans met it.

The trillions of digits are a test of computers and algorithms, not a practical need. NASA's Jet Propulsion Laboratory uses 15 decimal places for interplanetary navigation, which is enough to keep an error in the circumference of the solar system below the width of a hand. Thirty-nine digits would compute the circumference of the observable universe to within the width of a hydrogen atom. For any school or engineering problem, 3.14159 is more precision than the measurements it is combined with. The digits are memorised for sport; the record stands at 70,000, recited over ten hours.

The symbol was introduced by the Welsh mathematician William Jones in 1706, as the first letter of the Greek word for perimeter, and popularised by Euler. Pi Day is 14 March, from the American date format 3/14, and is celebrated in schools with pie. A small movement argues that the more natural constant is tau, twice pi, the ratio of circumference to radius, because so many formulas contain two pi; it has a day of its own on 28 June and has not caught on. Pi has had a four-thousand-year head start.

In the eighteenth century the French naturalist Georges-Louis Leclerc, Comte de Buffon, posed a puzzle. Draw parallel lines on a floor, a fixed distance apart, and drop a needle whose length equals that distance. What is the chance that the needle lands across a line? The answer turns out to be two divided by pi, a little under two in three. Pi appears because the result depends on the angle at which the needle falls, and every possible angle of a spinning needle is equally likely, which brings in the geometry of rotation.

That means you can estimate pi without measuring any circle at all. Drop a needle many times, count how often it crosses a line, and rearrange the formula. You can try it at home with a matchstick and lines drawn the same length apart, though you will need thousands of throws to get even two correct decimal places, since random experiments converge slowly. The same idea, using chance to calculate something fixed, underlies Monte Carlo methods, which scientists use to simulate everything from the paths of particles through a detector to the spread of an epidemic.

In 1897 the lower house of the Indiana legislature voted unanimously for a bill about the circle. It had been written by Edwin Goodwin, a country doctor and amateur mathematician who believed he had squared the circle, and it offered his discovery free to Indiana's schools while he would charge royalties elsewhere. The text was so muddled that it implied several different values for pi, including 3.2. Along the way it was referred to the committee on swamp lands, which passed it to the education committee, and the full House approved it by 67 votes to none.

It failed in the Senate largely by chance. Clarence Waldo, a mathematics professor from Purdue University, happened to be at the statehouse on other business, was shown the bill, and explained to senators why it was nonsense; they postponed it indefinitely amid some mockery, and it was never revived. The episode is often retold as Indiana trying to make pi equal 3, which the bill never said. A later story claiming that Alabama had voted to round pi to its biblical value of 3 was an April Fool's joke published in 1998, and it still circulates as fact.

Reciting the digits of pi has become a sport of its own. Guinness World Records has recognised feats of tens of thousands of digits, such as 70,000 recited by Rajveer Meena in India in 2015, and others have claimed even longer recitals. Memorisers usually turn digits into images or stories, using systems that convert numbers into words and then build long imaginary journeys.

A gentler tradition uses sentences in which each word's number of letters gives the next digit. How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics gives 3.14159265358979. Some writers have composed entire poems and even short stories under this rule. None of this is needed in practice, since a few dozen digits already exceed any physical measurement, but it is a playful way to engage with the number.

Pi is the ratio of any circle's circumference to its diameter, about 3.14159, an irrational and transcendental number whose digits never end or repeat. Estimated by Archimedes with polygons and computed to trillions of digits by computers, it appears throughout mathematics and physics because it belongs to rotation and repetition, and for any practical purpose a dozen digits is far more than enough.