Smile of the Day Sunday, March 14, 2021

Sunday March 14, 2021’s Smile of the Day: Pi

On this Day:

This day in 1592 was the “Ultimate Pi day”: At 6.53 am the largest correspondence between calendar dates and significant digits of pi occurred since the introduction of the Julian calendar (3.14159265358).

The number π (/paɪ/) is a mathematical constant. It is defined as the ratio of a circle’s circumference to its diameter, and it also has various equivalent definitions. It appears in many formulas in all areas of mathematics and physics. The earliest known use of the Greek letter π to represent the ratio of a circle’s circumference to its diameter was by Welsh mathematician William Jones in 1706. It is approximately equal to 3.14159. It has been represented by the Greek letter “π” since the mid-18th century, and is spelled out as “pi”. It is also referred to as Archimedes’ constant.

Being an irrational number, π cannot be expressed as a common fraction, although fractions such as 22/7 are commonly used to approximate it. Equivalently, its decimal representation never ends and never settles into a permanently repeating pattern. Its decimal (or other base) digits appear to be randomly distributed, and are conjectured to satisfy a specific kind of statistical randomness.

It is known that π is a transcendental number: it is not the root of any polynomial with rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.

Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematics approximated π to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques. The first exact formula for π, based on infinite series, was discovered a millennium later, when in the 14th century the Madhava–Leibniz series was discovered in Indian mathematics.

The invention of calculus soon led to the calculation of hundreds of digits of π, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits. The primary motivation for these computations is as a test case to develop efficient algorithms to calculate numeric series, as well as the quest to break records. The extensive calculations involved have also been used to test supercomputers and high-precision multiplication algorithms.

Because its most elementary definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. In more modern mathematical analysis, the number is instead defined using the spectral properties of the real number system, as an eigenvalue or a period, without any reference to geometry. It appears therefore in areas of mathematics and sciences having little to do with geometry of circles, such as number theory and statistics, as well as in almost all areas of physics. The ubiquity of π makes it one of the most widely known mathematical constants—both inside and outside the scientific community. Several books devoted to π have been published, and record-setting calculations of the digits of π often result in news headlines. Adepts have succeeded in memorizing the value of π to over 70,000 digits.

Perhaps because of the simplicity of its definition and its ubiquitous presence in formulae, π has been represented in popular culture more than other mathematical constructs.

In the 2008 Open University and BBC documentary co-production, The Story of Maths, aired in October 2008 on BBC Four, British mathematician Marcus du Sautoy shows a visualization of the – historically first exact – formula for calculating π when visiting India and exploring its contributions to trigonometry.

In the Palais de la Découverte (a science museum in Paris) there is a circular room known as the pi room. On its wall are inscribed 707 digits of π. The digits are large wooden characters attached to the dome-like ceiling. The digits were based on an 1874 calculation by English mathematician William Shanks, which included an error beginning at the 528th digit. The error was detected in 1946 and corrected in 1949.

In Carl Sagan’s novel Contact, it is suggested that the creator of the universe buried a message deep within the digits of π. The digits of π have also been incorporated into the lyrics of the song “Pi” from the album Aerial by Kate Bush.

In the United States, Pi Day falls on 14 March (written 3/14 in the US style), and is popular among students. π and its digital representation are often used by self-described “math geeks” for inside jokes among mathematically and technologically minded groups. Several college cheers at the Massachusetts Institute of Technology include “3.14159”. Pi Day in 2015 was particularly significant because the date and time 3/14/15 9:26:53 reflected many more digits of pi. In parts of the world where dates are commonly noted in day/month/year format, 22 July represents “Pi Approximation Day,” as 22/7 = 3.142857.

During the 2011 auction for Nortel’s portfolio of valuable technology patents, Google made a series of unusually specific bids based on mathematical and scientific constants, including π.

In 1897, an amateur mathematician attempted to persuade the Indiana legislature to pass the Indiana Pi Bill, which described a method to square the circle and contained text that implied various incorrect values for π, including 3.2. The bill is notorious as an attempt to establish a value of scientific constant by legislative fiat. The bill was passed by the Indiana House of Representatives, but rejected by the Senate, meaning it did not become a law (per Wikipedia).

First, a Story:

Come to the nerd side. They have pi.

Second, a Song:

Today’s topic on Pi virtually called out for a song by Tom Lehrer.

Thomas Andrew Lehrer (/ˈlɛərər/; born April 9, 1928) is a retired American musician, singer-songwriter, satirist, and mathematician. He has lectured on mathematics and musical theater. He is best known for the pithy and humorous songs that he recorded in the 1950s and 1960s. His songs often parodied popular musical forms, though he usually created original melodies when doing so. A notable exception is “The Elements”, in which he set the names of the chemical elements to the tune of the “Major-General’s Song” from Gilbert and Sullivan’s Pirates of Penzance.

Lehrer’s early musical work typically dealt with non-topical subject matter and was noted for its black humor in songs such as “Poisoning Pigeons in the Park”. In the 1960s, he produced a number of songs that dealt with social and political issues of the day, particularly when he wrote for the U.S. version of the television show That Was the Week That Was. The popularity of these songs has far outlasted their topical subjects and references. Lehrer quoted a friend’s explanation: “Always predict the worst and you’ll be hailed as a prophet.” In the early 1970s, Lehrer largely retired from public performances to devote his time to teaching mathematics and musical theater history at the University of California, Santa Cruz.

New Math is a 1965 song by American musician Tom Lehrer. Found on his album That Was the Year That Was, the song is a satire of the then-contemporary educational concept of New Math.

The song is composed in the key of C major in a 2/4 time signature. It correctly describes the step-by-step process for subtracting 173 from 342 in decimal and then subtracting the numbers 1738 and 3428 having the same digits in octal. The song features a spoken-word intro by Lehrer, followed by “piano played at a quick tempo and brisk lines”.

Lehrer, at the time a doctoral student of mathematics at Harvard University, used the song to satirize the then-new educational concept of New Math, introduced in American schools in the late 1950s and early 1960s as an attempt to reform education of mathematics. According to the book The New Math: A Political History, the song “purported to be a lesson for parents confused by recent changes in their children’s arithmetic textbook”. The same book states that by the time of the song’s release in 1965, the concept was at its peak in American education.

Lehrer’s song has been described as “well-informed and literate … enjoyed by new math proponents and critics alike”. Historian Christopher J. Phillips writes that, by including this song among other songs of great political and social import on That Was the Year That Was, Lehrer “seamlessly—and accurately—placed the new math among the major events of the mid-twentieth-century United States” (per Wikipedia).

Here is Tom Lehrer performing “New Math”.  I hope you enjoy this!

(https://www.youtube.com/watch?v=UIKGV2cTgqA)

Thought for the Day:

“I recited Pi to 22,514 decimal points in five hours and nine minutes. I was able to do this because of weeks of study, aided by the unusual synaesthesic way my mind perceives numbers as complex multidimensional coloured and textured shapes.” – Daniel Tammet

© 2021 David J. Bilinsky and Colleen E. Bilinsky