Archimedes and his Pi

Archimedes was born in 287 BCE in Syracuse, Sicily. He lived 75 years, which, for that time, was quite an accomplishment of good health. Sadly, his life ended with a Roman soldier’s sword despite General Marcellus’s orders to spare him.

Historians note that Archimedes studied in Alexandria, Egypt, along with the mathematicians and astronomers Conon of Samos and Eratosthenes of Cyrene. Archimedes was a notoriously ingenious scientist who founded many foundational principles and theories in mathematics, astronomy, physics, and engineering. The Archimedes principle refers to his method of determining the volume of an object that has an irregular shape.

Archimedean Spiral animation with Gears in and out tracing

The efficacious Archimedes screw, still used today, helps to pump rainstorm runoff and propel dry, bulk materials.

During the siege of Syracuse, he designed military hardware. One such tool included the claw of Archimedes, which was a giant iron claw attached to a pulley and lever that attempted to capsize approaching boats. He also designed his parabolic reflector, which the Sicilian military used to reflect the Sun and burn approaching enemy ships.

However, Archimedes’s most significant contribution to science included his groundbreaking mathematics, a subject to which he devoted his life. He authored the book The Sand Reckoner to prove that the amount of sand in the universe is not infinite. As a result, he stated that we could define the amount of sand in our universe in vast numbers expressed in exponential form. His process utilized values in base 100,000,000.

His work, The Ostomachion, is a dissection puzzle, which was part of the larger treatise called Archimedes Palimpsest. The Ostomachion is a puzzle that utilizes combinatorics to explain how many ways we can assemble fourteen different shaped pieces to form a square.

## Method of Archimedes

Around 250 BCE, Archimedes wrote his treatise Dimension of the Circle, a brief body of work describing a circle’s measurements, including an approximation of the value of π. This treatise, consisting of three propositions, was originally part of a larger work.[1] Archimedes proves the three propositions through Menaechmus’s method of exhaustion.

### Proposition One

Proposition One states that the area of any circle is equal to the area of a 90-degree triangle if one side of the triangle is equal to the circle’s radius and the other side of the triangle is equal to the circumference.

In the triangle ABC, we see that the height = BC = 15 and the base = AB = 2.388.

Circle D has circumference = 15.003 and radius = r = 2.388.

As a result, BC = Circumference of D = 15 and Area of D ~ Area of ABC = 17.9

as long as AB = r = 2.388

## Proposition Two

Proposition Two states, “The area of a circle is to the square on its diameter as 11 to 14.” Proposition Two utilizes the results found in Proposition Three, which violates the process of deductive reasoning. As a result, this order of propositions might not have come from the original Archimedes text.[2] [3] In other words, Proposition Two relies on knowing the value of π to understand how the ratio of the area of the circle to the square of its diameter equals 11/14.

Circle F, Archimedes’s Dimensions of the Circle, Proposition Two

Circle F shows Radius = r = 3, Diameter = d = 6 and area = AC = 28.27.

If we solve for the area of the circle divided by the square of the circle’s diameter, we obtain the same result if we were to divide 11 by 14.

\frac{\pi r^2}{d^2}=\frac{\pi3^2}{6^2}=\frac{28.27}{36}=.785

And

\frac{11}{14}=\ .785

Thus,

\frac{\pi r^2}{d^2}=\frac{11}{14}

As a result, we see that Proposition Two requires establishing Proposition Three first, as he would need to know the value of π to find this result.

## Proposition Three

In Proposition Three, Archimedes presented two findings. In one result, he discovered an approximation of

\sqrt3

by providing upper and lower bounds to show that

\frac{265}{153}<\sqrt3<\frac{1351}{780}\ .

Finally, we come to π! The value of π helps us to design trajectories for rocket ships, make advancements in ophthalmology, evaluate the flow of liquids, and perfect our GPS systems. The list is lengthy.

The third proposition produced two thousand years of opportunities for mathematicians to determine an accurate value of π. When Archimedes tried to estimate the value of π, he could only estimate π to two digits accurately.

## What is π?

Pi is a beautiful number. It is a mathematical constant equal to the ratio of the circumference of a circle to its diameter. In other words, it is the circumference of a circle divided by its diameter.

Archimedes determined that when we compare the circumference of any circle to its diameter, the ratio is greater than

3\frac{10}{71}

but less than

3\frac{1}{7}

.

So, for circle F, we see that the diameter = d =6, the radius = r = 3, and the circumference = 2πr = 2π3 = 6π = 18.849.

When we divide the circumference by the diameter, we get

\frac{6\pi}{6}=\pi=3.1416

Since

3\frac{10}{71}=3.1408\ and\ 3\frac{1}{7}=3.1433

then

3\frac{10}{71}\ <\pi<3\frac{1}{7}

and

3.1408<3.141<3.143

In Proposition Three, Archimedes presents the earliest definition of π as the value that lies between these lower bound and upper bound values.

Archimedes geometrically determined this by bisecting the angular measure of one-third of a right angle to observe the ratios of the tangent line that lies outside and inside the circle.

Archimedes began by bisecting a 30-degree angle. He continued this process to determine the perimeter of a 6‑sided polygon, a 12-sided polygon, a 24-sided polygon, a 48-sided polygon, and a 96-sided polygon. He did this process in two steps. Archimedes first determined the perimeter of the circumscribed polygon and then determined the perimeter of the inscribed polygon.

By doing so, he was able to compare the approximation of the perimeter of the inscribed polygon to the approximation of the circumscribed polygon. This process allowed him to prove an approximation of the value of π, which he determined as 3.14.

Then, 400 years later, around 150 CE, the astronomer Ptolemy used π to five digits, which was 3.1416.

The accuracy of π and the number of digits slowly grew over the years. In 1593, Francois Viete estimated π to an accuracy of nine decimal places. However, Dutch mathematician Adrian Van Rooman outdid Viete by employing Archimedes’s methods by circumscribing and inscribing a circle with a polygon of 230 sides. His calculation of π consisted of fifteen decimal places. Three years later, another Dutchman, Ludolph van Ceulen, also employed Archimedes’s methods and used a polygon with 6 x 229 sides. Using a polygon with this many sides, van Ceulen determined a value of π to twenty decimal places.

And so, the digits of π grew. In the late sixteen hundreds, astronomer Abraham Sharp found π to seventy-two decimal digits. In 1706, John Machin found π to one hundred decimal places. In 1717, French mathematician De Lagny determined that π had 127 decimal places.

In 1797, Carl Friedrich Gauss determined π to 205 decimal places. Then over the course of 200 years, the digits grew extensively. By 1967, with the help of the computer age, the value of π was determined to have 500,000 decimal places. In the early 1990s, in a tiny Manhattan apartment, brothers Gregory and David Chudnovsky calculated π to 2,000,000,000 digits using a homemade supercomputer they had built. A few years after that, they doubled the digits of π to 4,000,000,000 digits.

With the age of computers, it became more accessible and easier to determine how many digits are in the number of π. With the help of Y‑cruncher, a program that can compute π to trillions of digits, the current world records are:

- 5 trillion digits by Shigeru Kondo in 2010
- 12.1 trillion digits by Shigeru Kondo in 2013
- 13.3 trillion digits by Sandon Van Ness in 2014
- 22.4 trillion digits by Peter Trueb in 2016
- 31.4 trillion digits by Google’s Seattle software developer Emma Haruka Iwao in January 2019

Then, three years later, in June of 2022, Iwao and her team at Google broke her record again by determining π to one hundred trillion digits. In an interview with GeekWire, Iwao stated that they were “able to break a record again, and not just by a few digits, but by a good margin.” She says that she and her team believed that “One hundred trillion sounds reasonable, and a significant advancement over the past record.”[4]

The last 100 digits are as follows:

4658718895 1242883556 4671544483 9873493812 1206904813

2656719174 5255431487 2142102057 7077336434 3095295560

What is fantastic about this one hundred trillion value determination of π is that zero is its final value. If you want to see all the digits, you can also visit Pi.delivery, a Google website that explains how the digits were created.

But do we really need one hundred trillion digits of π? Even Newton said, “The value of π to 40 digits would be more than enough to compute the circumference of the Milky Way galaxy to an error less than the size of a proton.” Newton was right. Even NASA uses only fifteen digits of π for their calculations.

Archimedes showed that we humans have the intellect to create a forward-moving, future-seeking world capable of much more. So, here we are, in an age where the stuff of our imagination is becoming tangible, where the dreams of our future are closer than they seem, and where we are fully aware that as a society, we can use the foundations of math and science to accomplish more than what is currently evident and tangible to us. Over two thousand years ago, with the seed of curiosity, the digits of π grew. Today, its infinite digits are unimaginable. This unimaginable value of π is hope in its most extraordinary numerical form. The expression of π reminds us that in our current age, with the seeds that our ancient mathematicians planted, our tree of knowledge will only grow and take us to greater heights and indescribable destinations.

[1] Thomas Heath, A History of Greek Mathematics, Vol II (Oxford: Clarendon Press, 1921), 50.

[2] Thomas Little Heath Archimedes, The Works of Archimedes (Cambridge: University Press, 2007), 93.

[3] Wilbur R. Knorr, “Archimedes dimension of the circle: A view of the genesis of the extant text,” Archive for History of Exact Sciences 35, no. 4 (1986): 281, http://www.jstor.org/stable/41133787.

[4] Boyle, Alan. “Google Developers Set Another Record for Calculating Digits of Pi: 100 Trillion!” GeekWire, June 8, 2022. https://www.geekwire.com/2022/google-developers-set-another-record-for-calculating-digits-of-pi-100-trillion/.