## Pi

You may or may not know that Sunday this week was Pi Day in the US.  This occurs on March 14th every year, is so named because in the American system* the date reads 3.14$(\pi$to two decimal places), and in the past has been little more than an excuse to eat lots of pie in a knowing manner.  But this all changed last year, when the U.S. House of Representatives passed a non-binding resolution – whatever that means – designating it as “National Pi Day”, capital letters and all.  The curmudgeons among you might argue that they should have more important things to do with their time…well, my aim in this post is to persuade you otherwise!

$\pi$is probably the most famous of the mathematical constants (fixed numbers)…indeed a sure sign of its importance and ubiquity is that it is one of the few mathematical constructs that most people have actually heard of.  Constants like$\pi$and$e$have always fascinated mathematicians and non-mathematicians alike, and for good reason.  Mathematics is arguably the most pure and direct way we have of describing physical reality, and the pivotal role these numbers play in the subject hints strongly at some deep significance to us.  It has been suggested a number of times in popular culture – including in Darren Aronofsky’s film$\pi$and Carl Sagan’s novel Contact – that there is a message hidden in the digits of$\pi$which holds the key to understanding the nature of the universe.  And it is easy to see why this is such an enduring idea:$\pi$is definite, fixed…yet mysterious and unknowable (in fact we will never know exactly what is).  It seems like such an arbitrary number, but at the same time could not really be anything else (circles would not be circles otherwise!).  Like the distribution of the prime numbers, there is no pattern contained in the digits of$\pi ,$but this doesn’t stop people looking for one anyway (you might say it positively encourages some).   As a race we have computed it, analyzed it, memorized it, and generally celebrated its existence ever since we first really thought about circles.

To date,$\pi$has been calculated to 2.7 trillion digits – far more than the average computer even has hard disk space to store.  And the record for reciting the number from memory is held by a Chinese graduate named Lu Chao, who took 24 hours and 4 minutes to recite$\pi$to the 67,890th place.  This record has recently been challenged by a Japanese engineer named Akira Haraguchi, who has claimed to have recited 100,000 places – this is yet to be verified.  One method of memorizing large numbers is to use a mnemonic in which each digit is replaced with a word containing that number of letters.  Here is a well-known example for the first 15 digits of$\pi ,$attributed to the physicist Sir James Jeans:

“How  I need a drink (alcoholic of course), after the heavy lectures involving quantum  mechanics!”

There is even an entire adaptation of Edgar Allen Poe’s The Raven written in this rather constraining language…quite a feat, especially as it actually reads very well.

So I suppose I should explain what$\pi$actually is before I go any further!  It is simply the ratio of the circumference of a circle to its diameter.  In other words, the circumference of a circle is$\pi$times its diameter.   Or, if you prefer,

$\pi=3.14159 26535 89793 23846 26433 83279 50288...$

…etc.  The decimal expansion never ends, because$\pi$is irrational.  This means that it cannot be expressed as a fraction, and that it has an infinite non-repeating decimal expansion.

Find the hidden pattern!

It has been known since at least 2000BC that the ratio of every circle’s circumference to its diameter is the same, and is slightly greater than 3.  It was Archimedes who first accurately calculated$\pi ,$although rational approximations were used long before this…in fact the Great Pyramid at Giza is built with proportions based on one such approximation.  The perfect shape of circles, along with their ubiquity in nature, has led them to be associated with the divine in various cultures, and they do certainly have some close connection with both unity and the infinite.  They have infinitely many lines of symmetry, for example, all of which are equivalent under rotation.  They have one side, but we could think of them instead as polygons with infinitely many sides and vertices; indeed the method Archimnedes used to compute$\pi$was to consider the limit of a series of polygons.  They are often used to symbolise something eternal, whether it be in wedding bands, prayer wheels, or stone circles.  In mathematics, we add the “point at infinity” to create a projective plane; this is the point at which all lines – including parallel ones – intersect, and as an infinitely long line is really no different from an infinitely large circle, adding this point thus in effect turns all lines into circles.

This is all rather non-rigorous and unmathematical so far!  So I will briefly discuss another place in which$\pi$makes a surprising appearance.  The following equation is known as Euler’s Identity:

$e^{i\pi}+1=0.$

It is a particular favourite of mathematicians, and is consistently voted the most beautiful or greatest theorem in all of mathematics.  The reason for this is the way in which it combines what are probably the five most important constants ($1,0,e,\pi,$and$i),$with the three basic arithmetic operations (multiplication,addition and exponentiation) and the single most important relation (equals) in a disarmingly simple and yet very deep way.  At first glance it seems completely paradoxical: how can it possibly be that we can combine two irrational numbers (not just irrational but transcendental!) with an imaginary number in such a way as to produce an integer?

It would be too difficult to present the proof succinctly here, but it is based on the behaviour of trigomometric functions in the complex plane – functions which can be constructed geometrically in terms of a circle with radius one.  Raising the constant$e$to succesively larger multiples of$i$has the effect of tracing out a unit circle in the complex plane, and$2\pi$happens to be the precise multiple which defines a full circle.

But I feel the attention of the non-mathematicians beginning to wander, and I think I have proved my point that National Pi Day is worthy of a non-binding resolution in the House of Representatives, so I’ll sign off for now.

* Brits fear not!  You can instead celebrate Euler Day on 27th January

1. Dan says: