14th March 2023

Few numbers can match the star status of Pi. Mathematics’ most famous constant - familiar to most of us from high school geometry classes, features in some of the subject’s most celebrated formulae and occurs naturally all around us in the circular patterns of nature. Written about and romanticised more than any other mathematical symbol, Pi evokes a degree of human interest far beyond its applications. From memes to merchandise, competitive recitation to wordplay, Pi has transcended the field of mathematics to become an icon of modern geek culture. In fact, such is Pi’s popularity, it has even found its way on to the international calendar!

Pi is not new. It’s existence has been known for thousands of years, with the ancient Babylonians and Egyptians amongst the first to find close approximations for its value. In the third century BC, renowned Greek mathematician Archimedes, calculated a more accurate value of Pi by using Pythagoras’ Theorem to work out the areas of regular polygons inscribed within a circle and similar polygons in which the circle was circumscribed. Beginning his investigation with hexagons, Archimedes repeated the process for polygons of increasing numbers of sides, noting that as the side number increased the area of the polygon(s) edged closer to the area of the circle. From this work he was able to deduce that the exact value of Pi lay somewhere between the bounds of 3 1/7 and 3 10/71.

In the ensuing years, Chinese and Indian mathematicians continued to improve the accuracy of estimations and with the invention of calculus in the late seventeenth century, hundreds of decimal places of Pi could be determined. Since then, due largely to advances in computational power, we have been able to calculate Pi to an ever increasing number of digits, but given that Pi is an infinite decimal, its exact value will always remain an enigma.

It is perhaps the accessibility of Pi - the relative ease of the mathematics in which we are first introduced to it, that distinguishes this number from other useful constants like e and tau. By its simplest definition, Pi is the ratio of a circle’s circumference to its diameter. That is, for any circle of any size, dividing the circumference (distance around the outside of the circle) by the diameter (line that passes through the centre of the circle from one point on the circumference to another), always yields the same result: 3.14159265359... or what we have come to know as ‘Pi’.

Similarly, Pi is also the ratio of a circle’s area to the square of its radius (a line that extends from the centre of a circle to any point on its circumference) and can be used not only to calculate the properties of cyclic two-dimensional figures, but also the surface area and volume of solids, such as cylinders, spheres, ellipsoids and cones.

Pi is infinite and irrational. Irrationality in mathematics does not refer to something utterly illogical or deprived of reason. Rather it describes a number that is impossible to express as a simple ratio (fraction) of two integers (whole numbers). And Pi is perhaps the most famous irrational number of all! Although modern computing technology allows us to calculate its value to many trillions of digits, Pi’s infinite nature dictates that we will never ascertain its exact value. As well as being irrational, Pi is also a Transcendental number meaning that it is not the solution of any integer polynomial!

Pi is integral to some of mathematics’ most famous formulae including that which is often cited as the most beautiful equation of all. Euler’s Identity is revered for its bringing together the five fundamental mathematical constants in a single, succinct formula. But this is not Leonard Euler’s only association with Pi. The Swiss mathematician is also noted as the man who popularised the use of the lowercase Greek letter π to represent the ratio of a circle’s circumference to its diameter, after it was first used by British mathematician William Jones.

Such is Pi’s modern appeal beyond the realm of mathematics, it has even managed to infiltrate the ranks of literature. Pilish - a form of constrained writing in which the number of letters in each word must match the corresponding digit of Pi is believed to have originated in the early twentieth century with physicist Sir James Hopwood Jeans credited with one of the earliest (and often quoted) examples: ‘How I need a drink, alcoholic in nature, after the heavy lectures involving quantum mechanics!’ In recent years, Pilish has gone beyond simple sentences and short poems to include full length texts in this form.

Since 1988 when Larry Shaw of the San Francisco Exploratorium organised what is thought to be the first large scale event in honour of Pi, numberphiles all over the world have celebrated Pi Day on the fourteenth day of the third month, a date which when written using American date-writing convention gives the most common approximation of Pi: 3.14. Popular activities enjoyed by Pi enthusiasts on this day include record-breaking recitations of its digits - the current record stands at more than 70,000 digits!, composition of poems or prose in Pilish and the consumption of - what else - but pie!

Happy Pi Day!

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