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This is a two-parter. Part I explains the backstory of what π is and how it came to be.
There's a little bit of geometry and some calculus to boot. But the pie is anything but dry (pun intended). |
“There is a famous formula, perhaps the most compact and famous of all formulas — developed by Euler from a discovery of de Moivre: e^(i*π) + 1 = 0! It appeals equally to the mystic, the scientist, the philosopher, the mathematician.” - Edward Kasner and James R. Newman, "Mathematics and the Imagination" (1940)
There are two kinds of people in this world: those who appreciate it and its wonders, and those who think it revolves around and is subject to them. The following is the story of how some people, drunk with power, attempted to defy mathematical truths, and how they are humbled by The Man Who Saved π.
We begin not with the story of The Man, but the concept of which this battle was fought over: π.
To most people, π is simply one of those pesky things we have to endure in secondary school, a number to plug in every time the circumference, area or volume of some circle or sphere had to be calculated for us to squeak by with a C in maths. But what actually is it? It had to mean something. It had to come from somewhere, no?
Pharaohs and Philosophers
Back in the days when philosophy was more widely appreciated (henceforth referred to as “the good old days”), the old master Archimedes decided to further what the Egyptians and Babylonians discovered, the fact that there is a number one has to plug in to draw a decent circle – which is important to said monument-building civilisations.
The Egyptians and Babylonians stopped at a couple of decimal places, due mostly to satisfy architectural practicality. But Archimedes was anything but a practical man, he was curious on whether the ratio between a circle’s circumference and its diameter are as accurate as how the Egyptians had taught him. “Why” you ask?
We begin not with the story of The Man, but the concept of which this battle was fought over: π.
To most people, π is simply one of those pesky things we have to endure in secondary school, a number to plug in every time the circumference, area or volume of some circle or sphere had to be calculated for us to squeak by with a C in maths. But what actually is it? It had to mean something. It had to come from somewhere, no?
Pharaohs and Philosophers
Back in the days when philosophy was more widely appreciated (henceforth referred to as “the good old days”), the old master Archimedes decided to further what the Egyptians and Babylonians discovered, the fact that there is a number one has to plug in to draw a decent circle – which is important to said monument-building civilisations.
The Egyptians and Babylonians stopped at a couple of decimal places, due mostly to satisfy architectural practicality. But Archimedes was anything but a practical man, he was curious on whether the ratio between a circle’s circumference and its diameter are as accurate as how the Egyptians had taught him. “Why” you ask?
For Science!...or rather...For Maths!
He had devised a way where he compared the height of two triangles derived from two polygons (representing the upper and lower limits of a circle’s circumference) to the perimeter of said polygons. His solution was simple and intuitive, but ultimately inaccurate; this technique, called the polygon approximation method, was later independently rediscovered in China, India and Persia.
For almost 1,900 years the polygon approximation method had been the gold standard for finding π, with varying degrees of accuracy, the most by far was by Austrian astronomer Christophe Grienberger, using a 10^40 sided-polygon to derive a π accurate within 36 digits.
Functions and Frenchmen
It wasn’t until the 17th century that a new method was devised from the use of infinite series; again, independently discovered throughout the globe. The first well documented usage of this method is by Madhava Sangamagrama, a 14th century Indian mathematician, using crude techniques that were to be Newton and Leibniz’s calculus. This was difficult to be accepted by mainstream mathematicians as no definite evolution is seen from Archimedes’ method. A missing link of sorts.
This was solved by a French mathematician by the name of Francois Viete using an infinite product series. Remember, π is a ratio, and as such can be derived from other ratios, which is what Viete did. Just as Archimedes compared the ratio between a polygon’s perimeter and its triangular height and repeating it ad nauseam, Viete compared the ratio between area of a polygon and its progressively complex counterparts.
What Archimedes did however is to repeat the process from scratch (drawing new polygons), which is inefficient and time-wasting. Viete skips this process by predicting and multiplying the ratios as he progresses – cutting out Archimedes’ geometrical middle-man, so to speak.
For almost 1,900 years the polygon approximation method had been the gold standard for finding π, with varying degrees of accuracy, the most by far was by Austrian astronomer Christophe Grienberger, using a 10^40 sided-polygon to derive a π accurate within 36 digits.
Functions and Frenchmen
It wasn’t until the 17th century that a new method was devised from the use of infinite series; again, independently discovered throughout the globe. The first well documented usage of this method is by Madhava Sangamagrama, a 14th century Indian mathematician, using crude techniques that were to be Newton and Leibniz’s calculus. This was difficult to be accepted by mainstream mathematicians as no definite evolution is seen from Archimedes’ method. A missing link of sorts.
This was solved by a French mathematician by the name of Francois Viete using an infinite product series. Remember, π is a ratio, and as such can be derived from other ratios, which is what Viete did. Just as Archimedes compared the ratio between a polygon’s perimeter and its triangular height and repeating it ad nauseam, Viete compared the ratio between area of a polygon and its progressively complex counterparts.
What Archimedes did however is to repeat the process from scratch (drawing new polygons), which is inefficient and time-wasting. Viete skips this process by predicting and multiplying the ratios as he progresses – cutting out Archimedes’ geometrical middle-man, so to speak.
Only AFTER surpassing your master do you have the right to sport such a glorious beard!
Ever since then, the method had been relatively the same, what had changed are the means to do it. Between the 18th century and the mid-20th century, scores of mathematicians have competed to derive the most accurate π. One of the more famous attempts is by English mathematician William Shanks in 1874, who took 15 years to calculate down to the 707th decimal place, unfortunately he was only correct within the first 527.
It was also during this period that the irrationality and transcendentality of π are proven.
As your maths teacher may have explained (or not), irrational numbers are those that can’t be expressed as a definite fraction. For example, 3.1 can be expressed as 31/10, 1.833… can be put as 11/6; however π cannot be treated as such. The Egyptians were wrong when they state that π = 25/8, and a modern myth was created when Archimedes approximates π as around 22/7. Both are wrong, so don’t blame me if you lose some marks on that geometry quiz, you have been warned!
As for the transcendentality of π, it simply states that it cannot be expressed in algebraic terms. For any number (except for transcendental numbers, that is), you can write out an algebraic equation that results in that number. For instance, if we were to solve “x^2 - 9 = 0” for x, we’d get “x = 3”. Explaining and proving transcendental numbers is quite long-winded, and we’ll cover that in another article, but trust me on this.
As for further explaining π, there is nothing left but record-breaking history since. With the development of computers, π have been calculated to over 13 trillion decimal places – take that, Egyptians!
π, in and of itself, is of course useful. Other than classical geometry (i.e. circles and spheres), it pops out in physics, from Heisenberg’s uncertainty principle in quantum mechanics, to calculating the maximum load of a vertical column in civil engineering. Practically anything involving circles and spheres, the nose cone of an airplane, the GPS in your phone, the curvature of spacetime when it interacts with large cosmic masses. And you think it’s just for your school maths test?
Although, despite (or maybe because of) its importance, π was one day attacked by a group of ignorant politicians; and when the ignorant is given power, bad things happen.
The sky may not fall, but in this case, satellites, buildings and bridges might - as you’ll find out in Part II.
It was also during this period that the irrationality and transcendentality of π are proven.
As your maths teacher may have explained (or not), irrational numbers are those that can’t be expressed as a definite fraction. For example, 3.1 can be expressed as 31/10, 1.833… can be put as 11/6; however π cannot be treated as such. The Egyptians were wrong when they state that π = 25/8, and a modern myth was created when Archimedes approximates π as around 22/7. Both are wrong, so don’t blame me if you lose some marks on that geometry quiz, you have been warned!
As for the transcendentality of π, it simply states that it cannot be expressed in algebraic terms. For any number (except for transcendental numbers, that is), you can write out an algebraic equation that results in that number. For instance, if we were to solve “x^2 - 9 = 0” for x, we’d get “x = 3”. Explaining and proving transcendental numbers is quite long-winded, and we’ll cover that in another article, but trust me on this.
As for further explaining π, there is nothing left but record-breaking history since. With the development of computers, π have been calculated to over 13 trillion decimal places – take that, Egyptians!
π, in and of itself, is of course useful. Other than classical geometry (i.e. circles and spheres), it pops out in physics, from Heisenberg’s uncertainty principle in quantum mechanics, to calculating the maximum load of a vertical column in civil engineering. Practically anything involving circles and spheres, the nose cone of an airplane, the GPS in your phone, the curvature of spacetime when it interacts with large cosmic masses. And you think it’s just for your school maths test?
Although, despite (or maybe because of) its importance, π was one day attacked by a group of ignorant politicians; and when the ignorant is given power, bad things happen.
The sky may not fall, but in this case, satellites, buildings and bridges might - as you’ll find out in Part II.
Ponder this
Assuming we were to use the same method as Archimedes to determine π. However, rather than working with a circle, we use a sphere instead. How would we do this? Don't use the equations you've learned in class, imagine you're a Greek philosopher who's only just discovering this.
What about ellipses? They aren't exactly circles, but can we use Archimedes' method on them?
Discuss
π is one of those universal constants that have always existed. In a way, it is one of the most basic laws of the universe.
According to the Many Worlds interpretation of quantum mechanics, there is an infinite number of universes that may be fundamentally different from one another. Would constants such as π be the same in these other universes?
Further readings
"Why Pi matters", by Steven Strogatz, Professor of Mathematics at Cornell University
One million digits of Pi, at the Pi Day website (Pi Day is celebrated on March 14th annually, comment if you get the joke)
5 ways NASA uses Pi, at CalTech's Jet Propulsion Laboratory
