Pi Day:What is pi? How is it calculated?

March 14, or 3/14 as per the American convention, is celebrated as ‘Pi Day’ worldwide as an ode to the most well-known approximation (3.14) of the mathematical constant Pi. But what is pi?

Pi Day was started by physicist Larry Shaw of the Exploratorium museum in San Francisco in 1988. On the day, mathematicians try to raise awareness on their subject among lay persons, through lectures, museum exhibitions and pie-eating competitions.

With the power of modern computers, mathematicians have calculated pi up to 31 trillion decimal places. 

In 2019, UNESCO’s 40th General Conference designated Pi Day as the International Day of Mathematics

What is Pi?

Pi, often represented by the Greek letter π, is the most famous of all mathematical constants. It represents the ratio of a circle’s circumference (boundary) to its diameter (a straight line between two points on the circle’s boundary, passing through its centre). Regardless of the circle’s size, this ratio always remains constant.

Pi is an irrational number — it is a decimal with no end and no repeating pattern — which is most often approximated to the 3.14, or the fraction 22/7.

How is Pi calculated?

The importance of Pi has been recognized for at least 4,000 years. Petr Beckman in A History of Pi (1970), wrote that “by 2,000 BC, men had grasped the significance of the constant that is today denoted by π, and that they had found a rough approximation of its value.’

Ancient Babylonians and Egyptians came up with their own measurements, probably by drawing a circle of some diameter, and then measuring its circumference using a rope.

Babylonians settled at 25/8 (3.125) as the value of Pi, while ancient Egyptians settled at (16/9)^2 (approximately 3.16)

Greek polymath Archimedes (circa 287-212 BCE) was the one who came up with the method to calculate Pi that remained in use till the 17th century.

Archimedes realized that the perimeter of a regular polygon of ‘n’ sides inscribed in a circle is smaller than the circumference of the circle, whereas the perimeter of a similar polygon circumscribed around the circle is greater than its circumference.

He used this to calculate the limits within which the value of Pi must lie and arrived at the approximate value of pi. Notably, he started with a hexagon (6-sided polygon), then doubled the number of sides till he reached a 98-sided polygon.

Now, as one keeps adding more and more sides to this polygon, it gets closer and closer to the shape of a circle. Having reached a 96-sided polygons, Archimedes proved that 223/71 < Pi < 22/7 (in decimal notation, this is 3.14084 < π < 3.142858).

Following Archimedes, mathematicians constantly increased the number of sides of the polygon to calculate pi, which meant that the number of decimals after the point kept increasing.

By 1630, Austrian astronomer Christoph Grienberger calculated 38 digits of pi using polygons with 10^40 sides, but this method was extremely labour-intensive.

Indeed Dutch mathematician Ludolph van Ceulen (1540-1610) took 30 years to calculate pi to 35 decimal points.

It would be Isaac Newton (1643-1727) who significantly simplified the process of calculating Pi.

In 1666, Newton calculated pi up to 16 decimal places using calculus, which he discovered along with mathematician Gottfried Wilhelm Leibniz (1646-1713).

With Newtonian calculus, the value of pi which had taken previous mathematicians years to calculate now could be done in a matter of days.

By 1719, French mathematician Thomas Fantet de Lagny (1660-1734) had already calculated pi up to 112 correct decimal places.

Today, with the help of modern computers, pi has been calculated up to 31 trillion (1012) decimal places.

What is the use of pi?

Circles are everywhere in the world, they are one of the most common shapes of things found in everyday life.

Three-dimensional shapes like cylinders, spheres, and cones, all carry the proportion of pi.

Knowing Pi’s value, thus, has some crucial practical benefits in architecture, design, and engineering.

From constructing tanks to bridges to taps, to engineering satellites, the value of pi is indispensable in all sorts of areas.

Pi has gone on to answer the very questions of our existence, from the Big Bang to the discovery of black holes, answering the deepest workings of the universe — from calculating the vastness of space to understanding the spiral of DNA.

While Archimedes’ calculation was fairly adequate for all practical purposes that pi was used for in his time, in more modern times, 31 trillion decimals is a bit overkill, although it might help us further advance technology at a quicker rate than what is happening already.

Experts suggest pi needs to be calculated to about 39 decimal places in order to perform all calculations in the observable universe with virtually no error in modern times.

Why then are mathematicians so fixated on the number?

Pi is so alluring because it puts infinity within reach. The digits of Pi never end and never show a pattern. They go on forever, seemingly at random, because they embody the order inherent in a perfect circle and seek to get closer to the truth and answer questions about the world we live in.