HAPPY 3.14!:

A slice of Pi (SAM JORDISON, 3/14/07, The Scotsman)

The practical applications of p are, in fact, legion. The number has all kinds of relevances outside the world of square-ruled exercise books and smudged equations, above and beyond the fairly obvious usefulness of being able to accurately work out the circumference of circles in the tunnelling and construction industries. Pi is used in just about every manufacturing process you can think of, from loo rolls to fighter jets. Everywhere there's a circle that needs to be measured, in fact, and that's an awful lot of places, if you think about the number of screws there are in the world (not to mention lenses, tubes and wheels). It's also vital in telecommunications. Radio, TV, telephone and radar signals can all be described as sine waves and p is fundamental in calculating their size and frequency, as it is in calculating the size of the waves in the sea. The magic number is also used in an unutterably complex way to stimulate unknown factors and loading conditions in engineering, wind gusts on a plane, and even random variables in computer-game manufacture.

In short, p is one of the foundation stones of our way of living and we'd be in a lot of trouble without it. It's not overstating things too much to say that the history of our civilisation can be traced in the history of p. Arguably the first technological society, the Babylonians had calculated p using the value of 25/8. It was this level of accuracy that enabled them to produce some of the first serious construction marvels - and to build all those towers that so annoyed the writers of the Old Testament.

Meanwhile, in spite of its claims to be the infallible word of an omnipotent God, the references to p in the Old Testament are distinctly underwhelming. Verses in Kings and II Corinthians about the construction of Solomon's Temple give p a value of "3". Proof at least that the concept had broad currency by the first millennium BC, but nothing like as impressive as the earlier Ancient Egyptian figure of 3.160, written down by a scribe called Ahmes in 1650BC, and which no doubt helped the people of the pharaohs build all those magnificent temples.

The single biggest leap in the evolution of p came, as with so many things, thanks to the ancient Greeks. In the 3rd century BC Archimedes of Syracuse work-ed out the first known theoretical calculation of the number. His idea was that by drawing a polygon outside a circle and then a smaller one inside and calculating the perimeters of both, he'd be able to approximate the circumference of the circle somewhere in between the two figures and thus work his way back to p. This gives a pretty rudimentary value if you draw a four-sided polygon outside and inside the circle, but very accurate when you draw, as Archimedes did, two 96-sided polygons. He worked out the value of p as lying somewhere between 223/71 and 22/7. The average of these two values is roughly 3.1419.

Now, don't worry if you don't completely understand Archimedes's calculations. If I'm being honest, I don't either - which just goes to show how impressive his achievement was. And, just as it took almost two millennia for modern civilisation to catch up with that attained by the ancient Greeks and Romans, it also took almost 2,000 years for Europeans to come up with a better calculation of p. In other words, you aren't alone if you have trouble following the man in the toga. The Indians and Chinese had both produced more accurate approximations by the 15th century AD, but the first modern (ahem) pioneer of Western civilisation was Ludolph van Ceulen, who managed to work the number out to within 35 decimal places. So proud was he of this achievement that he supposedly had them inscribed on his tombstone.

Calculations became steadily more accurate as the Renaissance gave way to the Enlightenment and, in 1706, a Welshman called William Jones also became the first known person to use the actual symbol "p" when discussing the magic number. He did so in a text with the snappy title Synopsis Palmariorium Mathesios. Unfortunately, he didn't record for posterity the reason he opted for this symbol. The best explanation is that it was a little tribute to Archimedes, being the first letter of the Greek word, perimetron, from peri (around) and metrein (to measure). Nobody really knows, but all the same, the symbol stuck, and calculating it accurately became something of a Holy Grail for eggheads around the world as technology advanced during the Age of Steam.

Rather tragically, one William Shanks devoted the 20 years of his life leading up to 1873 to calculating p to 707 decimal places, only to have a DF Ferguson come along in 1944 and prove that his predecessor had made a mistake. Shanks had got the figure at the 528th decimal place wrong, which meant that all his subsequent figures were also incorrect.

Petr Beckmann's terrific little book, A History of Pi (often available in the Barnes & Noble remainder bin), includes that story and many other enchanting ones. And the recent film, Pi, is delightfully counterzeitgeist.

Posted by Orrin Judd at March 14, 2007 8:10 AM