It was finally the weekend! After my long mathematics presentation, I came home to watch my favorite tv show, Person of Interest, to de-stress. Surprisingly, the episode was about the most famous mathematical constant, pi (π), equal to the ratio of a circle’s circumference to its diameter, commonly approximated as 3.14159. Mr. Finch (the main character) acted as a substitute teacher and wrote on the chalkboard 3.1415926535. He then asked the students, “What does this mean?”
I answered the question in my mind, thinking, “If I have a bicycle tire with a diameter of 1, then one full revolution of the bicycle tire would travel a distance equal to pi.” However, in the show, no one answered. Then Mr. Finch responded to the question himself, saying:
Pi, the ratio of the circumference of a circle to its diameter — 3.1415926535 — is just the beginning. It keeps going forever without ever repeating, which means that contained within this string of decimals is every other number; your birth date, the combination to your locker, your social security number, etc. It’s all in there somewhere. And if you convert these decimals into letters, you would have every word that ever existed in every possible combination; the first syllable you spoke as a baby, the name of your latest crush, your entire life story from beginning to end, and everything we ever say or do. All of the world’s infinite possibilities rest within this one simple circle. Now, what will you do with that information; what it’s good for? Well, that would be up to you!Although that scene was actually inaccurate, I loved it. This scene is beautiful because most educators in the world struggle to be as clever and interesting of a teacher as Mr. Finch. His knowledge about the subject expands the discussion beyond the textbooks and keeps the students focused throughout the lecture.
Unfortunately, Mr. Finch was wrong because mathematicians have not yet proved that pi has the characteristic of “normality.” In other words, mathematicians are unsure if π contains all the finitely long permutations of digits from 0 to 9. They are not sure if every digit is used after a certain amount of time or unlimited times in pi’s decimal representation.
Nobody knows what we will find in the digits of π if we keep going. For instance, when we check the first billion digits of π, 7 occurs almost 100 million times. This makes π an excellent random number generator because the probability of getting the digit seven should be 1/10 or 10%. However, after some points, π may not contain the digit seven and instead have a non-repeating number with just two or three digits such as 010203112233000111222333… .For instance, after the first 761 digits of π, there is a famous mathematical coincidence where six nines occur in a row called Feynman point.
But we are sure that the digits of π keep going on forever and in random order. This randomness makes π interesting because pi’s value is finite; however, its decimal value is infinitely long. You might think this is a contradiction, but it is not. The number π is a constant number because it is the ratio of a circle’s circumference and its diameter, finite values. Still, we need an approximate value for π.
In 1768, Johann Lambert proved that the value of π is an irrational number and cannot be written as a simple rational fraction. 22/7 is a commonly used approximation but does not contain all of the digits of π. This is because irrational numbers cannot be written as a ratio of two numbers, such as a/b. After all, irrational numbers continue to infinity and do not follow a pattern.
In 1882, Ferdinand Lindemann proved that π is a transcendental number because it is not algebraic; it is not a non-constant polynomial equation with rational coefficients.
I think we can safely say that π is transcendental because the mathematician Yasumasa Kanada found that the first trillion digits of π appear to be statistically random. His calculation took more than 600 hours. If you check the table below, you see that the event of each digit occurring is independent, and its probability is one-tenth of the time.
Digit Occurrences
0 99,999,485,134
1 99,999,945,664
2 100,000,480,057
3 99,999,787,805
4 100,000,357,857
5 99,999,671,008
6 99,999,807,503
7 99,999,818,723
8 100,000,791,469
9 99,999,854,780
Total 1,000,000,000,000After many years, Emma Haruka Iwao found 34.1 trillion digits of π in 2019. It took 121 days for Haruka and her computer because calculating π requires a lot of power, even for a computer. You can picture it in your mind like this; if you were to print a billion decimal values of π in normal-sized, ordinary font, it would stretch from New York to Kansas.
However, 34.1 trillion digits are still not enough to prove whether π is normal or not. Supercomputers are still crunching the numbers. If you check the graph below, you will see the number of known digits of pi, by year, since 250 B.C.
Going back to Mr. Finch, we see that he is not 100% wrong. We can find our birthdays in π easily. If you go to mypiday.com and type your birthday, it will give you the decimal place in pi. For example, my birthday occurs at the 675,097th decimal place.
If π is a normal number, then we can say that our whole destiny is encoded in π. The pictures we will take will be in π because there are binary numbers behind images. All digital products are in π, and even this article has been in π for thousands of years. Furthermore, the DNA of every creature is in π. Mr. Finch was probably right, but we are not yet sure.There is an interesting and artistic way to show the randomness of π. Some scientists might be happy with their tedious scatter plots. Still, many artists use colors for data visualization to communicate with the public. Martin Krzywinski is one such artist who found beauty and artistry in the randomness of π. He took the digits of π and gave each digit a different color. For instance, he gave three the color orange, one the color red, four the color yellow, etc. Then he made a beautiful poster, and if you look at it carefully, you do not see any particular pattern to the colors.
Besides, there are so many fascinating facts about π. It is also the most studied number in math history. Many people want to memorize the digits in π, but not other irrational numbers, which drives people into madness and chaos. Mathematicians have been struggling to calculate π for centuries precisely.
So, should we stop working on π, or should we continue looking for a better approximation? Is assuming π to be equal to 3.14 good enough? Or is it sufficient to use 40 digits of π to find the circumference of the Milky Way galaxy to an error less than the size of a proton? Are the first 152 digits enough to find the observable universe's circumference at 93 billion light-years?Hundreds of mathematicians have been trying to figure out more digits of π for years. It is like trying to get to the moon and then to the next planet, and so on. But why? Why do mathematicians bother calculating any more digits? Why aren’t 34.1 trillion digits of π not enough? Is it because π lurks in every circle?
The logical reason seems cryptic because π is a beautiful source to generate random numbers. However, the real reason is that countries can show off their technology to other countries because calculating the trillion digits of pi needs powerful computers. For instance, in the Star Trek episode “Wolf in the Fold,” Spock foils the evil computer by commanding it to “compute to the last digit, the value of π.” So asking a computer to compute π is called “a stress test” and may make it crash.
On the other hand, we humans are awkward creations. Staying at home and drinking tea is a beautiful activity. Still, when we get bored, we try to climb the highest of mountains, befriend a tiger, or l try to memorize the digits of π like Chao Lu, who correctly memorized the first 67,890 digits of π. We will keep doing these things because we like to understand our world.
The most common mnemonic technique is to memorize the first digits of π. "May I have a large container of coffee?"
On September 12, 1962, John F. Kennedy gave a speech about the space program. He said:
There is no strife, no prejudice, no national conflict in outer space as yet. Its hazards are hostile to us all. Its conquest deserves the best of all mankind, and its opportunity for peaceful cooperation may never come again. But why, some say, the moon? Why choose this as our goal? And they may well ask why climb the highest mountain? We choose to go to the moon. We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one that we are willing to accept, one we are unwilling to postpone, and one which we intend to win, and the others, too.We are inevitably connected to the past, and π is a thread that’s gone through all of human history. That’s why we can say that as long as there are people, there will always be somebody who wonders what’s next. And I assure you that there is a mathematician or scientist using π for something important for our universe somewhere in the world because π is still the mysterious constant of nature.
Finding Pi
The previous statement is utterly true because there has always been someone who works on π. Mathematics is as old as civilization, and the human race has studied π for almost 4000 years. When the last mammoths were going extinct, people were exploring π. As far as we know, Archimedes from ancient Greece was one of the first humans to calculate π. He was most likely helping wheel makers. But how did he estimate the value of π?
Firstly, he saw all polygons as circles. According to Archimedes, if you keep increasing the number of a polygon’s sides, you would get closer to the perfect circle. In other words, a pentagon is more circle than a square, but a hexagon is more circle than a pentagon, and so on. Thus, the legendary mathematician Archimedes defined a circle as a regular polygon with an extremely large number of sides, more than two thousand years ago.
His definition was helpful because measuring a curved surface was hard to do accurately. He found a way to find the circumference of a circle. First, he drew a square with its corners touching the perimeter of a circle and found the inscribed square's perimeter. Then, he drew another square with its sides touching the circle's perimeter and found the perimeter of the circumscribed square. He concluded that the circle's circumference had to lie somewhere between the perimeter of those squares.However, using this method, the difference between the two values was pretty big. So, he drew pentagons to see the upper and lower bounds of the circle's circumference. He then got a smaller range of bounds. After that, he kept increasing the number of faces of the polygon that he was drawing inside and outside of the circle. Each time he did this, his estimation got more accurate. Archimedes worked hard and got up to a 96 sided regular polygon called an enneacontahexagon until he got exhausted. The lower and upper bounds that he found were 3.1408 and 3.1429. Thus, he calculated π to two decimal places.
However, Archimedes’ method needed improvement because even his entire life span would not be enough time to find the other digits of π by hand. Mathematicians were required to discover more efficient formulas and new techniques.
However, before mathematicians could do this, they needed to wait for Al-Khwarizmi to discover Algebra. In the beginning, people used signs for digits. For instance, let’s say you and your neighbor had 75 horses, 35 of which were yours. You need to find the number of horses your neighbor has. Without algebra, the solution would take a long time to find. But after the discovery of algebra, we just used equations to solve the problems. In this specific example, we can write 75 = x + 35, where x is your neighbor’s number of horses. Writing such an equation and using a variable instead of a number was revolutionary to the classical world. Algebra allowed for much easier calculations across all of mathematics.
The adoption of algebra by great mathematicians inspired a whole new way of looking at the world. The next significant jump in calculating π was the invention of calculus. After that, mathematicians started working on infinite series.
An infinite series is an expression with numbers added together, one after another, until infinity. Sometimes, however, these endless series converge to a particular value.There are many methods available now to calculate π. Gottfried Leibniz found π in infinity. James Gregory found the equation below for π. He worked on one of the astonishing infinite series for the inverse tangent function below. He added infinitely many small numbers together and found π.
He put x = 1 into the inverse-tangent series. He showed us the further we go, the closer to the estimation of π we get. However, to get ten digits of π, we need to add about 5 billion fractions.
Borwein Integrals are pretty interesting, and we can see a beautiful pattern of π in them. Unfortunately, the pattern eventually breaks down.
Another great mathematician, Leonhard Euler — who officially adopted the Greek letter “π” as a symbol to represent the value- found a more efficient equation for π when he was 28. The character became iconic. Euler’s π equation calculates an infinite sum. The Basel Problem was named after him.
Euler also used π to write another beautiful equation, Euler’s Identity. I have a whole article dedicated to it, which you can read below.
Although imaginary numbers are not real, π can make imaginary numbers real. If you put i^i into a calculator, you will get 0.2078795763507619085469… . The proof is totally related to π. Below, you can see why i^i is real.
Thanks to the genius mathematician Ramanujan’s obsession with π, we have many new formulas to find it. When he arrived at Cambridge, he brought a notebook with 400 pages of formulas to find π.
After the invention of the mechanical computers, mathematicians used Leibniz’s, Euler’s, and Ramanujan’s infinite series to calculate a trillion decimal digits of π. Without a supercomputer, finding the digits of π that far would have been difficult. For example, the mathematician William Shanks managed to calculate the first 707 digits of π by hand, but unfortunately, he made a mistake after the 527th.
Pi is everywhere.
Children start learning about π when they are in 7th grade and use it until they graduate from college. Even after that, most people use π again when their children go to school. Pi appears everywhere in the universe and in our lives. It is literally woven into our universe; the orbits of planets, electromagnetic waves, rivers, the colors of auroras, the structure of DNA, the Great Pyramids of Giza…
Since anything involving a circle or a sphere is about π, if a scientist wants to describe the structure of the universe or finds a relationship between planets, they need to use π. Circles appear throughout the natural world, whether a soap bubble or the moon in the night sky. This explains why mathematics is essential in all areas of science, and Pi helps us see the mathematical ideas underlying diverse physical processes.
The Connection Between Pi & Sinuosity of Rivers
Pi has a direct relationship with rivers on Earth. But how? To figure this out, we need to measure the length of a river in two different ways.
Assume that we know the starting and ending point of the river. First, we need the actual length to see how bendy the river is. In other words, the actual length is the distance that you need to swim from the beginning point to the ending point. This whole length will be "L." Second, we need to find the straight length. In other words, we need to fly from the beginning to the end. And this direct route will be a lowercase "l." Now we can write the formula for the river's sinuosity by dividing L by l. The sinuosity is a ratio that measures how bendy the river is.
What is important here is there’s no limit to how high sinuosity can be, and the river can be really bendy. However, Hans-Henrik Stølum proved that the average sinuosity of rivers around the world is π. If you find all the rivers’ sinuosity and take the average sinuosity, you should get π.
There is another interesting fact about sinuosity. Rivers can be very bendy at some points and therefore, we would expect a high sinuosity, but suddenly, those rivers become straighter and make the sinuosity π. For example, it is hard to find a river with sinuosity equal to 7 because of fluid dynamics. Mathematicians found the highest sinuosity around 3.5 and the lowest sinuosity to be around 2.7.Rivers can start acting very chaotic after some time, and then they suddenly go back to normal. At the extremely bendy point, rivers cut off and make a shortcut to become straight again. This phenomenon is known as an oxbow lake, which controls the sinuosity of rivers. This keeps the sinuosity of a river around π.
Pi in Space
There is a mathematical order inherent in our universe. For instance, to understand our solar system, we need π. We know that our planet moves in front of its host star, and the light comes from host stars. We need to know how big the host star is to talk about that light. In other words, we need the surface area of the host star. A sphere’s surface area formula is 4πr², with r being the star’s radius. The size of a planet also helps scientists guess whether it is habitable or not.
Another good example of the relationship between π and the universe is electrostatic force, which is the force between two electric charges. An electron exerts a force in all directions and forms a sphere field. Electrons also interact with each other on an electric field. To figure out that interaction, we need to find the surface area of spheres, where π comes up again.
There is also a connection between π and gravity. If you have had a chance to see Einstein’s field equations, you might notice that π is there too.
The formula above calculates how objects with a large mass, such as stars and galaxies, can curve space and time with their gravity. Einstein says that just like a ball sitting on a bedsheet, any form of momentum and energy can also curve space-time around it. In other words, the formula is saying:
Gravity = 8 x π x Energy & MomentumSo it π and not any other irrational number that is part of gravity, energy, and the momentum of the universe and all of its objects. If you take the square root of the gravity of Earth, you almost get π. I don’t think this is a coincidence.
√9.8 = 3.1305
Finding Pi in Nature
Infinite series are not the only way to find π. You can do some cool and fun activities to estimate π on your own, and one of them is called the Monte Carlo method.
Assume that you are working on a 1x1 grid. You generate pairs between 0 and 1 to plot the points on a coordinate plane. If you keep plotting the points, you will see that the distance of some points to the origin will be less than 1, and some of them will be greater than 1. After some point, you will see that you are getting a quarter circle. If you find the area of that quarter circle, it will be almost π/4. There is an example below of 1,000 points. You can try it from here.
If you don't want to deal with computer programming, you can use just a pencil and paper to do this. You just need to draw a circle with a radius of 1 and then draw a square around the circle. The square area has to be four because the circle's diameter is 2. Now, if you take your pencil and close your eyes and put random dots on the paper many times, eventually, the percentage of times your dot landed inside the circle will approach π/4, and you can feel like Archimedes.Buffon’s Needle
When there was no internet, kids used to play by tossing a coin on the floor and seeing whether the coin crossed a line or not. A French philosopher and mathematician, Georges-Louis Leclerc, decided to determine the probability that the coin would cross the line. Remarkable idea!
He started by dropping a needle on a lined sheet of paper and determining the probability of the needle crossing one of the lines on the paper. Then he tried his experiment with many needles many times, and he got a remarkable result. The probability was directly related to the never-ending number pi because two times, the number of needles he dropped divided by the number of needles crossing a line was almost equal to pi all the time. So he made a formula:
P: the probability | n: the number of needles | c: the number of needles crossing a line. Then;P = 2n/c
After Leclerc, an Italian mathematician, Mario Lazzarini, threw needles almost 4000 times to perform this experiment. He got pi with perfect accuracy and got to the first six decimal places of π.
If you don’t believe me, you can throw thousands of toothpicks and test the experiment yourself, or you can check the Monte Carlo simulation below to see. The gif shows the π estimation with a different number of toothpicks.
My Pi Day
Due to its inherent nature, Pi is an irrational number, which means that it has neither an end nor a certain pattern that it follows. That means that all of the numbers in the world exist somewhere inside of Pi. My Pi Day is a tool that locates and shows you where your date of birth lies inside the realm of the Pi number.
Movie: Pi
“Pi” is a movie where director Darren Aronofsky shows us how a $60,000 budget can be used to shoot a masterpiece. In fact, the black and white color scheme couldn’t have been better suited for the film.
The main character is a man named Max Cohen, a mathematician who takes mathematics from beyond simple pleasure and utilizes it as a tool in order to understand every facet of life.
There are dull and monotonous parts of the movie that may get tedious for the audience, so it is not a film suited for relaxation purposes. However, it is a must-watch for those who are even slightly intrigued by the relationship between mathematics and philosophy.
The Pi Bicycle
In 2017, two good friends, Martijn Koomen and Tadas Maksimovas, designed a fully functional pi bicycle built in the shape of the mathematical symbol, π.
The Pi Day
After a long history of studying π, people decided to organize an official celebration of π on March 14. Since 1988, people have celebrated the magical constant on March 14. Interestingly, Albert Einstein was born on Pi Day, March 14, 1879, and Einstein also published his theory of general relativity on Pi Day.
To summarize, mathematics is a language inscribed in all human beings’ brains, and Pi is just a word in that language. John F. Kennedy knew that the moon was not infinitely far away, and he got there. I believe that one day the great mathematicians will reveal all the mysterious digits of π.
I wish Finch had been my teacher when I was a student.
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