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“Probably no symbol in mathematics has evoked as much mystery, romanticism, misconception and human interest as the number Pi” ~William L. Schaaf, Nature and History of Pi. π
The History, Explanation & Love for π
What’s Pi, π, peeeeeeeeeee?
Pi interchangeably shown as the lower-case Greek letter π (correctly or traditionally pronounced “pea” or “peeeee”), is a universal known mathematical constant. π is unique related to it’s association to cycles. A real-world example everyone is familiar with would be rhythmic processes, like a pulsing heart, a planet orbiting the sun, or the ocean’s waves. π is the ratio of a circle’s circumference (distance around the circle, represented by the letter C) to its diameter (distance across the circle at its widest point, represented by the letter D).
This ratio, which equals approximately 3.14, also appears in the formula for the area within the circle. A = πr2, where π is the Greek letter “pi” and r is the circle’s radius (distance from center to rim).
The circumference of a circle is always 3.14 x its diameter. π is only an approximation because the decimals are infinite and ultimately unknowable. Amazingly, π always expresses the mathematical relationship between the diameter and the circumference of a circle, no matter how small or how large the circle.
If the Greek representation isn’t painting a mental picture for you, Type π into a calculator and press ENTER. 3.14159265 pops up, but this isn’t because this is the exact value of π, but because a calculator’s display is often limited so many digits.
History of π
The Universe contains many round and near-round objects; finding the exact value of π helps bring clarity to design, building, manufacturing and understanding to life, as we know it.
Historically speaking, we only had a coarse estimation of π for some time. Gaining a more precise value of π led to more accuracy and development of important concepts and techniques; such as limits and iterative algorithms. Which became fundamental basics to new areas of science and mathematics.
Nearly 4,000 years ago, “the ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3. One Babylonian tablet (ca. 1900–1680 BC) indicates a value of 3.125 for π, which is a closer approximation.” Source
“The Rhind Papyrus (ca.1650 BC) gives us insight into the mathematics of ancient Egypt. The Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for π.” Source
Archimedes drew a polygon inside a circle and drew a second polygon outside of the circle. Then he continuously added more and more sides of both polygons, getting closer and closer to the shape of the circle. Having reached 96-sided polygons, he proved that 223/71 < pi < 22/7.
π was first calculated by Greek mathematician, Archimedes of Syracuse (287–212 BC). Archimedes approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular polygons: the polygon inscribed within the circle and the polygon within which the circle was circumscribed.
Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave upper and lower bounds for the area of the circle.
Archimedes found the value of π to be approximation within those limits. In his experiment, Archimedes proved π is between 3 1/7 and 3 10/71.” Source.
Unfamiliar with Archimedes or his work, Zu Chongzhi (429–501) a Chinese mathematician and astronomer took a similar approach to calculate the value of the ratio circumference of a circle, to its diameter to be 355/113. To compute this accuracy for π, he started with an inscribed regular 24,576-gon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places. Source.
The development of infinite series techniques in the 16th and 17th centuries greatly enhanced people’s ability to approximate pi more efficiently. An infinite series is the sum (or much less commonly, product) of the terms of an infinite sequence, such as ½, ¼, 1/8, 1/16, … 1/(2n ).
The first written description of an infinite series that could be used to compute pi was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji around 1500 A.D., the proof of which was presented around 1530 A.D.
1600 AD & Beyond
Then, in mid 1600’s, using calculus, English mathematician and physicist Isaac Newton used infinite series to compute π to 15 digits. In 1699, Newton and German mathematician Gottfried Wilhelm Leibniz together discovered π reached 71 digits. Those found 100 digits in 1706, and calculated 620 digits in 1956, which was the best approximation achieved manually.
In 1761, Johann Heinrich Lambert, a Swiss mathematician first proved π is an irrational number, has an infinite number of digits, and never follows a repetitive pattern.
German mathematician Ferdinand von Lindemann, in 1882, discovered π cannot be expressed in a rational algebraic equation (such as pi²=10 or 9pi4–240pi2 + 1492 = 0)
In most recent discoveries, Scottish-born mathematician Jonathan Michael Borwein, the world’s leading expert on the computation of Pi. Which has been spelled out to about 2.7 trillion decimal points. In 1984 Jonathan Michael Borwein and brother, Peter Borwein, “produced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step. ” Source. and the discoveries continue…
“Every major civilization had its theories of π and its mathematicians who tried to explain it. Ancient Egypt and Babylon and India. The Greek Archimedes, the Greco-Roman Ptolemy, the ancient Chinese and Indians — all figured out this ratio, which exists both on paper and, as if by some sort of divine plan, throughout nature.” Source
“The beauty of π, in part, is that it puts infinity within reach. Even young children get this. The digits of π never end and never show a pattern. They go on forever, seemingly at random — except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of π.” Dr. Steven Strogatz, Award Winning American Mathematician and Professor says.
He continues, “Pi touches infinity in other ways. For example, there are astonishing formulas in which an endless procession of smaller and smaller numbers adds up to π. One of the earliest such infinite series to be discovered says that π equals four times the sum 1–1⁄3 + 1⁄5–1⁄7 + 1⁄9–1⁄11 + ⋯. The appearance of this formula alone is cause for celebration. It connects all odd numbers to π, thereby also linking number theory to circles and geometry. In this way, πjoins two seemingly separate mathematical universes, like a cosmic wormhole.” Source.
Something Special about those Numbers
For those of us interested in the applications of mathematics to the real world, π is indispensable. Whenever we think about rhythms — processes that repeat periodically, with a fixed tempo, like a heart beat, we inevitably encounter π.
The formula is termed Fourier series and looks like this:
This series is an all-encompassing portrayal of any ‘fixed tempo’, x(t), repeating every T units of time. The foundation of the formula is π, the sine, and cosine functions from trigonometry.
Through the Fourier series, π appears in the mathematics that chronicle the delicate breathing of an infant and the peaceful circadian rhythms of nocturnal rest, as well as the energetic wakefulness that govern our bodies.
“When structural engineers need to design buildings to withstand earthquakes, π always shows up in their calculations. π is inescapable because cycles are the temporal cousins of circles; they are to time as circles are to space. π is at the heart of both.” Source.
Dr. James Grime claims that we can calculate the size of the universe with amazing accuracy using pi. The first 39 digits of pi can measure the circumference of the Universe to the accuracy of the size of an atomic nucleus. Source.
π is a infinite number therefore it aides in studying our infinite Universe. π was developed by early astronomers to study the Earth’s rotation and orbital patterns. π can been used in calculating the density of a planet, which in turn tells us about a planets ecosystem (If the atmosphere is solid or gases).
NASA uses π to calculate the trajectory of a spacecraft (termed ‘pi transfer’), for measuring craters, learning about the composition of asteroids. More recently, π was also used in calculating the amount of hydrogen present in the ocean beneath the surface of Jupiter’s moon Europa. NASA, Nov, 2019 Source.
Dr. James Grime claims that we can calculate the size of the universe with amazing accuracy using pi. The first 39 digits of pi can measure the circumference of the Universe to the accuracy of the size of an atomic nucleus. As you can see, π is woven into our descriptions of the innermost workings within the Universe. Photo Source.
Designers Cristian Ilies Vasile and Martin Krzywinski have transformed mathematical numbers, theories and symbols into detailed works of art. The transition probabilities for each 10 digit bin for the first 2,000 digits of pi, phi and e are shown in this image. Photo Source.
π could potentially hold the key to seemingly unanswerable questions about our Universe.
Despite the laws of physics stating the Universe was birthed from chaos, π remains apparently random in nature and can be used to create stability while simultaneously representing a state of infinity. π allows a window into the foundation of reality and showcases the enchanting forces used to form every particle in the Universe.
π’s Slice on Life; Underappreciated & Need a Holiday Dr. Larry Shaw during Pi Day Celebration at The Exploratorium Photo Source.π
Pi Day(π) is an annually celebrated day to praise the numerical constant. The earliest claimed celebration of Pi Day, was organized in 1988 by Larry Shaw, a physicist at the San Francisco Exploratorium. Mr. Shaw and staff publicly marched around one of the buildings circular spaces, while consuming fruit “Pi-es”. (hehe) The Exploratorium continues to hold entertaining Pi Day events. π Day is observed on March 14 related to the North American calendar format’s date sequence. March is the third month, therefore, 03/14 shows the first three significant digits of π. The United States House of Representatives supported the designation of ‘Pi Day’ and made it a National Observance in 2009 by passing a non-binding resolution (111 H. Res. 224). Everything was signed March 12 and two days later, March 14, 2009 was recognized as the first official ‘National Pi Day’.
Google Doodle in honor of Pi Day 2010 on Google.com Photo Source.
The following year Pi Day 2010 brought masses to the fun “Math Party” when Google presented a “Google Doodle” celebrating the holiday, with the word Google laid over images of circles and π symbols; and for the 30th anniversary in 2018, Google held Dominique Ansel pie’s with the circumference divided by its diameter to beautifully articulate
Pi’s 30th Anniversary Google Doodle by Dominique Ansel Photo Source.π Fun Fact π
The entire month of March 2014 (3/14) was observed by some as “π Month”. In the year 2015, March 14 was celebrated as “Super π Day”. This Special National Pi Day had special significance, because the first 10 digits of π was presented. 3/14/15 at 9:26:53, date written in month/day/year format following the time, left many Pi-fectly Partying with lots of Pi-e! #SorryNotSorry #punny
HOW TO OBSERVE
Considering an conducting a π-experiment.
Buffon’s Needle. Experiment to Estimate π
‘Buffon’s Needle Problem’ was question first posed by Georges-Louis Leclerc, Comte de Buffona n eighteenth-century French mathematician. He unintentionally devised a way to calculate π based on probability.
Question. “Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?”
This solution was the earliest problem in geometric probability to be documented; simply using integral geometry.
The solution for the sought probability p, in the case where the needle length l is not greater than the width t of the strips, is;
p = 2/π x l/t
This can be used to design a Monte Carlo method for approximating the number π, although not the original desire for de Buffon to conduct this experiment.
This simple experiment is used to calculate πPlace a piece of lined paper on a flat surface, then drop a needle onto the lined paper, the needle will land on either a blank space or will be touching the lines of the paper. Drop the needle onto the piece of paper 100 times, and then calculate how many times it landed on a line or in a blank space using this formula:
2 x stick length x # of needles dropped _______________________________________________________________ distance between lines x # of needles touching a line
The equation should approximately result to 3.14 with each trial.
Monte Carlo Dinner Math? π-rfect!
You think Olive Garden will give me 200 bread sticks to attempt?
Be a cool kid, with the rest of us STEM geeks and use #NationalPiDay with #GIVENation and post your love for Pi on social media.
I’ll see you all on May 4th….
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