Archimedes and his Pi: The Great Numerical Hope
Archimedes lived a long life, which, for the third century BCE was an exceptional accomplishment of good health. He might have lived longer had his life not been taken from him by the sword of a Roman soldier. This was even despite General Marcellus’s orders to spare Archimedes.
Archimedes was born in 287 BCE in Syracuse, Sicily but studied in Alexandria, Egypt, along with Conon of Samos and Eratosthenes of Cyrene. All three were Alexandria’s most prominent mathematicians and astronomers at that time. Archimedes was an infamous and remarkable scientist who founded many foundational principles and theories in mathematics, astronomy, physics, and engineering.
Archimedes left a tremendous impression on the world of mathematics and physics that he has 22 mathematical concepts and two physical concepts named after him, including Archimedes’s axiom, Archimedes’s constant, Archimedes’s paradox, and Archimedes’s principle.
He also designed a claw, appropriately called the Archimedes claw that consisted of a giant iron claw attached to a lever and pulley. Its purpose was to capsize approaching boats. Some of his other inventions include the efficacious Archimedes screw that helps to pump rainstorm runoff and propel dry, bulk materials, and the Archimedes bridge, which is a submerged tunnel.
Archimedes also designed inventions for the military. One of his most famous inventions was the parabolic reflector that the Sicilian military used to reflect the sun, which, in theory, would burn the approaching enemy ships.
Archimedes also wrote The Sand Reckoner, wherein he presented his mathematical proof that the amount of sand in the universe is not infinite. In this proof, he stated that the amount of sand in our universe is in very large numbers and as a result can be expressed in exponential form. In his proof, he used numbers in base 100,000,000. As a result, he was able to express numbers using exponents in order to show that values as large as 8,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 could be represented as
8×10^{63}. Archimedes’s Ostomachion dissection was part of his larger treatise called Archimedes Palimpsest. This is a fun puzzle that requires moving 14 pieces, all different shapes around so that it forms a square.
Archimedes wrote his treatise Dimension of the Circle around 250 BCE, which provides three propositions that literally help to describe the dimension of the circle. Each of these three propositions leads up to a method for approximating the value of pi.
Dimension of the Circle – Proposition One
Proposition One states that the area of a circle equals the area of a 90-degree triangle as long as one side of the triangle equals the radius of the circle, and the other side of the triangle equals the circumference.
Triangle ABC has height, BC = 15, and base, AB = 2.388.
Circle D has circumference 15.003, with radius r = 2.388.
As a result,
Area\enspace of \enspace D \approx Area \enspace of \enspace ABC=17.9
As long as
BC=circumference\enspace of \enspace D=15
and
AB=r=2.388
Dimension of the Circle – Proposition Two
Proposition Two states that the area of a circle is proportional to the area of a square. In other words, the area of the circle, divided by the square of the circle’s diameter equals 11/14.
Circle F has 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 as 11/14.
\frac{πr^2}{d^2} =\frac{π\cdotp3^2}{6^2} =\frac{28.27}{36}=.785and
\frac{11}{14}= .785Thus,
\frac{πr^2}{d^2} =\frac{11}{14}Dimension of the Circle — Proposition Three
Proposition Three leads us to pi. In proposition three, Archimedes found the approximation of the square root of three by solving its upper and lower bounds:
\frac{265}{153}\lt \sqrt{3}\lt \frac{1351}{780}.Finally, we come to Pi! If it were not for Pi, we would not be able to design trajectories for rocket ships or wheels for our cars. We would not be able to understand how the eye works in order to make advancements in vision, or understand the structure and the function of our DNA, or look at the ripples in fluid and understand how fluids flow, or perfect our GPS systems. The list goes on!
It is the third proposition that opened up opportunities for mathematicians over the millennia to determine the value of pi, and how many digits it has. When Archimedes was trying to estimate the value of pi, he was only able to estimate pi to two digits accurately ultimately finding that the value of pi was greater than
3 \frac{10}{71} \space and \space 3 \frac{1}{7}.In Proposition Three, Archimedes found that the ratio of the circumference of any circle to its diameter is the value of pi.
So, if we were to take a circle F, with diameter (d = 7), radius (r = 3.5) and circumference
2 \pi r=2 \pi 3.5 = 7 \pi = 21.991
We find that
\frac{21.991}{7}=\frac{7 \pi}{7}=\pi=3.141This value is so close to the value of pi. Since
3 \frac{10}{71} =3.1408 \space and \space 3 \frac{1}{7}=3.143Then pi equals that value between the two proposed values:
3 \frac{10}{71} \lt \pi \lt 3 \frac{1}{7}.Then, 400 years later, around 150 CE, the famous astronomer Ptolemy, used Pi to five digits, which was 3.1416.
The accuracy of Pi and the number of digits slowly grew over the years. In 1593, Francois Viete estimated Pi to an accuracy of nine decimal places. However, a Dutch mathematician outdid him in the same year. This mathematician was Adrian Van Rooman, who employed Archimedes methods, and circumscribed and inscribed a circle with a polygon that had 230 sides.
His calculation of Pi consisted of 15 decimal places. Three years later another Dutchman, Ludolph van Ceulen, also employed Archimedes methods and used a polygon with 6 x 229 sides. This provided him a value of Pi to 20 decimal places.
And so the digits of pi grew and grew. In the late sixteen hundreds, astronomer Abraham Sharp found Pi to 72 decimal digits. In 1706, John Machin found Pi to 100 decimal places. In 1717, French mathematician, De Lagny determined that Pi had 127 decimal places.
In 1797, Carl Friedrich Gauss determined pi to 205 decimal places. Then over the course of 200 years, the digits of pi grew extensively. By 1967, with the help of the computer age, the value of Pi was determined to have 500,000 decimal places. In the early 1990s, in a tiny Manhattan apartment, two brothers, Gregory and David Chudnovsky, calculated Pi to 2,000,000,000 digits using a homemade supercomputer that they had built. A few years after that, they doubled the digits of Pi to 4,000,000,000 digits.
With the age of computers, it became easier and easier to determine how many digits are in the number of Pi. With the help of Y‑cruncher, a program that can compute Pi to trillions of digits, the current world records are:
- 5 trillion digits by Shigeru Kondo in 2010
- 12.1 trillion digits, Shigeru Kondo in 2013
- 13.3 trillion digits, Sandon Van Ness in 2014
- 22.4 trillion digits, Peter Trueb in 2016
- 31.4 trillion digits, Emma Haruka Iwao in January 2019
But, do we really need 31.4 trillion digits of Pi? Even NASA uses only 15 digits of Pi for their calculations in space travel.
Archimedes proved to us that as humans we have the intellect to create a world that is forward moving and future seeking, and capable of so much more. Over two thousand years ago, with the seed of curiosity, the digits of pi grew. Today, its digits are unimaginable. This is hope, in its greatest numerical form. This is hope that in our current age, with the seeds that our ancient mathematicians planted, our tree of knowledge will only grow and take us to great heights and unimaginable destinations.
[i] Thomas Heath, A History of Greek Mathematics, Vol II (Oxford: Clarendon Press, 1921), 50.