The Story of Pi (π)

Introduction

The ratio between a circle's circumference and diameter is a mathematical constant known as pi. (i.e. C∕D=π) It is estimated to be around π ≈ 3927⁄1250 = 3.1416 (accuracy 2·10^-6).

Two of the very earliest civilizations, the Ancient Egyptians and Babylonians both investigated pi around 4000 years ago. The Babylonians (ca. 1900-1680 BC) estimated the value of pi to be about 3 1/8 or 3.125. The Egyptians made a similar approximation as well. The oldest evidence of pi is found in the Rhind Papyrus which dates from about 1650 BC and estimates the value of pi slightly less accurately, as 3.1605.

However, the puzzle was far from being solved, as mathematicians would spend centuries uncovering the true extent of the mystery of pi.

Archimedes' Approach to π

The ancient mathematicians were struck by the problem of calculating the length of a curved surface. Sure, there existed a much simpler method of using a string to measure the length and then finding the value by comparing it with a linear scale. However, there was another way of approaching this problem, pioneered by Greek mathematician Archimedes in 250 B.C.E. It was to think of the circle as a regular polygon, just with an extremely large number of sides. So large, in fact, that each side is considered to be infinitesimally small, meaning that the sides are so small that the shape looks round.

If you think about it, a square is a little like a circle, but when a side is added to form a pentagon, it comes closer to resembling a circle, and when another side is added it gets even closer, and so on. Archimedes implemented this thinking with the idea that he could draw a regular polygon inscribed within the circle to approximate π and that the more sides he drew on the polygon, the better approximation to π he would get. This initial method of calculating pi is known as exhaustion.

Archimedes' method finds an approximation of pi by determining the length of the perimeter of a polygon inscribed within a circle, (which is less than the circle's circumference) and the perimeter of a polygon circumscribed outside of a circle (which is greater than the circle's circumference). The value of pi lies between those two lengths.

Archimedes started off with two hexagons. Using some basic geometry, he observed that the length of each of the sides of the inscribed hexagon would be 1/2. The six sides of the hexagon all have length 1/2, so this perimeter is 6 × 1/2 = 3. And the lengths of the sides of the circumscribed hexagon would be 1/√3. This perimeter is 6 × 1/√3, which is about 3.46.

This gives the inequalities 3 < π < 3.46, already a close approximation to 3.14.

But how could this be achieved using polygons with exponentially more sides?

The Mathematical Approximation

Archimedes started off by drawing a circle of radius 1 unit centred at A and inscribed an n-sided polygon within it. (Think of n as a variable that can be any number you decide.) The estimate for π is half the circumference of the polygon (circumference of a circle is 2πr, r = 1, giving 2π)

Figure 1 shows only one segment of the polygon (square) ACE. The side of the polygon CE has the length d_n (d stands for the length or distance of one side, and n stands for the number of sides in the polygon). Assuming we know the d for an n-sided polygon, we can find an expression for the length CD = d_2n (the edge length of a polygon with 2n sides.) Thus improving our estimate for π. This process is repeated for d_3n and eventually d_567n. Remember, the larger the number of sides, the closer to pi we get.

It is important to note that the advanced mathematics we have today did not exist during Archimedes' time, which is why the Pythagoras Theorem was implemented despite modern methods existing today that could have facilitated the approximation of π.

Using the Pythagorean theorem on the right-angled triangle, ABC, it is shown that:

Given that:

Archimedes knew that he had not found the value of π but only an approximation within those limits. In this way, Archimedes showed that π is between 3 1/7 and 3 10/71. He also proved that the ratio of the area of a circle to the square of its radius is the same constant.

A Brief Timeline

Archimedes was able to come up with a method that could more accurately approximate the value of pi, however only to a certain extent. For centuries to follow, pi continued to puzzle scientists and mathematicians. However, by performing extensive calculations, they were able to extend the number of known decimal places of pi. Chinese mathematician and philosopher Zu Chongzhi made very notable contributions to these calculations, approximating pi to be around 355/113.

The specifics of his work remain unknown as his book is long gone. Though it is believed that he used a method similar to that of Archimedes, inscribing a regular polygon with 24,576 sides inside a circle and performing lengthy calculations to arrive at his answer.

By the end of 17th century, new mathematical methods of analysis in Europe provided improved ways of calculating pi, some of which involving infinite series. Isaac Newton, as an example, used his binomial theorem to calculate 16 decimals with relative haste.

Mathematicians began using the Greek letter π in the 17oos and was introduced by William Jones in 1706. The use of the symbol was popularized by Leonhard Euler, who adopted it in 1737.

In the early 20th century, Indian mathematician, Srinivasan Ramanujan developed efficient ways of calculating pi that were later incorporated into computer algorithms.

Now, in the 21st century computers have calculated pi to 100 trillion decimal places, as well as its two-quadrillionth digit when expressed in binary.

The Importance of π

So why all the fuss over a mere symbol?

Well, π occurs in various mathematical problems involving the lengths of arcs and other curves, the areas of ellipses, sectors and other curved surfaces, and the volumes of many solids. Additionally, it is used in various formulas of physics and engineering to describe various phenomena and scientific concepts. Whether its calculating the vastness of space or understanding the double helix formation of DNA, π is involved in some shape or form. Because of π, we have managed to make significant progress within the sectors of mathematics, physics, technology and engineering etc. We use pi to calculate the stress and strain on different materials, design bridges and other structures, and calculate the speed and momentum of particles.

In reality, π has always existed. We just had to look for it.