Happy 𝜋 Day!

Mar 14

Today has been officially known as 𝜋 (Pi) day since 2009 when the US House of Representatives passed a non-binding resolution recognizing March 14 as National Pi Day. Aside from being a great day to eat pie, it’s a great day to celebrate how mathematics, and one specific concept, has shaped our world.

Pi (𝜋) is a mathematical constant that represents the ratio of the circumference of a circle to its diameter. It is an irrational number, which means it cannot be expressed as a finite decimal or fraction, and its decimal representation goes on infinitely without repeating. Pi is commonly approximated as 3.14. [1]

By John Reid - Edited version of Image: Pi-unrolled.gif., CC BY-SA 3.0

This illustration depicts pi by showing the rolling of a circle on a baseline. No matter the size of the circle, the circumference will always be π. The challenge, since ancient times, has been to come up with the exact value of pi.

The pi ratio is essential in mathematics, particularly trigonometry, geometry, calculus, and physics. Its applications are incredibly broad and are key to engineering, surveying, architecture. Since antiquity, mathematicians have been working to define and calculate pi.

Ancient Pi

Babylonia

In the 17th century BCE, the Babylonians had developed advanced mathematics. They created tables that expressed squares, fractions, square roots, cube roots, reciprocal pairs, and linear and quadratic equations. They also estimated the value of pi to be 3 1/8 = 3.125.

Egypt

The Rhind Papyrus, which dates from about 1650 BCE, demonstrates a sophisticated use of mathematics for engineering and construction. This document shows that the Egyptians calculated the value of pi as 4 × (8/9)² = 3.16.

Interestingly, the Great Pyramid of Giza (it was built back in 2500 BCE) has a perimeter of 1760 cubits and a height of 280 cubits – a ratio that's approximately twice pi. [2]

Greece

The Greeks significantly advanced the study of geometry. In the 3rd century BCE, Archimedes, the great engineer and inventor, devised the first known theoretical calculation of pie. According to Archimedes the value of pi was 223/71 < 𝜋< 22/7. At this point, Archimedes' calculation was around 3.1418, which was by far the closest approximation up to this point in history.

Archimedes Thoughtful (also known as Portrait of a Scholar) by Domenico Fetti, 1620. Archimedes is considered the most important physicist in antiquity.

About 400 years later, another Greek, Ptolemy, further refined the estimate of the pi using the chords of a circle with a 360-sided polygon to obtain 𝜋 = 3 17/120 = 3.14166.[3]

China

After Archimedes, it took several centuries before any significant breakthroughs came in the search to accurately calculate pi. Near the end of the 5th century, Tsu Ch'ung-chih and his son Tsu Keng-chih came up with astonishing results when they calculated 3.1415926 < pi < 3.1415927. The father and son duo used inscribed polygons with as many as 24,576 sides.[4]

The work continues

Over the centuries many mathematicians throughout the world worked to accurately calculate pi. In 1596, a German named Ludolph Van Ceulen presented 20 digits, using the Archimedean method with polygons with more than 500 million sides. Van Ceulen spent a great part of his life hunting for pi, and by the time he died in 1610, he had accurately found 35 digits. His accomplishments were considered so extraordinary that the digits were cut into his tombstone in St. Peter's Churchyard in Leyden.[3]

Monument for Ludolf van Ceulen (+ 1610) in the Leiden Pieterskerk ; it is a copy of his tombstone, which was already lost in 1800. The text gives the value of π in 35 decimals. Photograph by A.L. Boon

The symbol for pi (π) was first used by Welsh mathematician William Jones in 1706.

The work of calculating pi continues. With the help of computers, pi has been calculated to more than 31 trillion digits!

What is pi used for?

Besides being foundational for mathematics, physics, astronomy, architecture, engineering and surveying, pi is also key to many biological processes and patterns. In 1952, the mathematician and father of computer science, Alan Turing, proposed a mathematical model describing the simple biophysical principles of pattern formation during morphogenesis (the embryonic stage when cells specialize to become various organs and tissues).

This model explains a vast array of patterns we see in nature, from leopard spots to stripes on a zebra. These patterns all have one constant: pi. Pi is also intimately woven into periodic processes. It appears in the governing biophysical laws of cell division timing, heart beats, breathing cycle, and circadian rhythms controlling sleep-wake cycles.[5]

For example, surveying also uses 𝜋, including calculating the length of the arc in a curve; total stations rely on 𝜋 in when evaluating the precision of measurements to include the curvature of the earth, distortions in wavelengths and so on.

Some examples of daily uses of Pi:

Electrical engineers used pi to solve problems for electrical applications
Statisticians use pi to track population dynamics
Medicine benefits from pi when studying the structure of the eye
Biochemists see pi when trying to understand the structure/function of DNA
Physicists looking into the behavior of fluid ripples see pi and use it in their calculations
Clock designers use pi when designing pendulums for clocks
Aircraft designers use it to calculate areas of the skin of the aircraft
Signal processing and spectrum analysis (finding out what frequencies are in the wave you are using) uses pi since the fundamental period of a sine wave is 2*pi.
Navigation, such as global positioning (GPS)
Calculating the number of death in a population
Solving mathematics problems in geometry like finding the area of circle etc.
— copied directly from https://amazingarchimedes.weebly.com/real-life-application-of-pi.html

Enjoy a slice of pie to celebrate pi!

Today, as we enjoy a generous slice of pie, we can use pi to calculate the top surface area of our delicious dessert.

The shape of the slices in a pie is like a sector of a circle. The graphic shows a pie sectioned into 8 equal slices where each slice is a sector, and the radius of the pie is 4 inches.

We can find the area of the sector formed by each slice by using the sector area formula.

The area of a circle is pi x r^2, where r is the radius.

4 ^2 = 16

3.14 x 16 = 50.24

If we divide a circle into eight equal sectors, then the area of each sector is (pi / 8) x r^2 because we are evenly dividing the area across each sector.

(3.14 x 16) ÷ 8 = 6.28 sq. inches of surface area of a delicious slice of pie.