Celebrating Pi Day: The Evolution Story

Gunjan Sah Avatar

 

Before you read this article on the history of π, I recommend you go through this article that introduces the superhero constant π. In this article, we will briefly discuss how the calculation of π has evolved through the centuries. In the video published earlier, you must have learned how the ancient Egyptian civilization 4000 years ago compared circles and squares to estimate the area of round figures. Well, π was not discovered and coined at that time. But had there been a constant like π used to find the area of circles, the Egyptians would have approximated it as (frac{16}{9})^{2}simeq 3.16

Most of us might remember Archimedes from the Archimedes’ principle we learned at school for the first time. The lesser-known fact is that Archimedes is also credited for devising a robust geometric algorithm to compute π for the first time around 250 BC. Archimedes computed the upper and lower bounds of π by calculating the perimeters of two regular polygons: one inscribed within the circle and the other within which the circle was circumscribed. Archimedes came to a conclusion that frac{223}{71}<pi<frac{22}{7} – this upper bound for the value of π became popularly mistaken for its actual value. We will see how this algorithm worked in a soon-to-be-published video. 

A Chinese Mathematician Liu Hui used a similar approach but used the area of polygons instead of their perimeter. Using a 3072-sided polygon, Liu calculated π up to 5 digits. Another Chinese Mathematician Zu Chongzi later used a 12288-sided polygon to calculate π correct to 6 significant figures. Several other figures including the likes of Aryabhata, Fibonacci and Dante also used polygons to approximate π. 

Then came the era of using convergent infinite series (and products), starting from the 16th century. This was a much more powerful method as compared to the ones devised by Archimedes and other Mathematicians before. The most commonly known infinite series involving π has to be the Basel problem, i.e. 

 

begin{aligned}frac{1}{1^{2}}+frac{1}{2^{2}}+frac{1}{3^{2}}+frac{1}{4^2}+…=frac{pi^{2}}{6}end{aligned}

 

But the sad part is, this series is not efficient when it comes to computing π as it takes an insanely large number of terms to find even 10 digits of π. Instead, there are other infinite series or products that have a greater rate of convergence, i.e. fewer terms are sufficient to compute π to a reasonable degree of accuracy. Hopefully, this will be a topic of discussion in a future article where the rapidly convergent series developed by Ramanujan and Chudnovsky will be discussed. 

Now, let’s come to the modern era of computing π using computers. The use of computer calculation was a game-changer. A team led by John Von Neumann (who is popular for his contribution to computer architecture) calculated 2037 digits of π on an ENIAC computer in 1949. It took a whopping 70 hours for the computer to calculate that many digits. Today, using computers and efficient iterative algorithms, π has been calculated up to a ridiculously high number of digits. On 19 August 2021, the University of Applied Sciences in Switzerland computed π to 62,831,853,071,796 digits. Crazy right? 

 

 

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