Pi

March 19, 2010

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.

You can’t comb a coconut (and other important facts)

March 11, 2010

Apologies to my hordes of devoted followers for my recent absence…I’ll make up for it by including some (shamelessly copied and pasted) pictures today.

I’m going to talk a bit about topology, which is a kind of generalisation of geometry; if we imagine every object to be made of some kind of incredibly pliable substance which enables it to be morphed into any other vaguely similar object, then we can study general properties of space rather than being constricted by such trifling matters as shape and form.  The basic rule is: if you can stretch, squeeze or twist (no cutting or glueing allowed!) a shape into another one, then they have the same topological properties.  The classic – and rather over-used – saying is that a topologist can’t tell his coffee cup from his doughnut (they both have one hole).

Topologists spend a lot of their time considering holes, and trying to work out things like how many different ways there are to wrap spheres of different dimensions around each other, and which knots are actually unknots (not knots!).  It is a very visually appealing branch of mathematics (see the link on the right to “sketches of topology” for some nice images) and there are some wonderfully odd constructions with special properties that topologists like to study.