Wednesday, 31 August 2011

Penetrating the Inner Circle

The Joy of π, David Blatner

A delightful little book about a delightful big number: the ratio of the circumference of a circle to its diameter, aka π. The Bible says it's three and though we knew far better by the nineteenth century, we still had fewer than a thousand digits. We had 707, in fact, and it wasn't until 1945 that we discovered that some of them, calculated with enormous labor and dedication by the English mathematician William Shanks, were wrong.
1945 was the year someone set to work calculating π with the aid of a desk calculator, and was the start of the electronic race to find π with greater and greater accuracy. Fifty years later, in 1995, the Japanese mathematician Yasumasa Kanada had calculated 6 billion digits thats 6,000,000,000 only for the Russian-American brothers David and Gregory Chudnovsky to hit back the following year with 8 billion. Kananda took the lead again in 1997 with 51·5 billion digits (and holds the record as of May 2005 with 1·2 trillion digits).
The story of π is a story of competition too, you see, and Blatner devotes a chapter to the Chudnovskys and their attempts to build ever more powerful computers to win and then win back the π-digit record. For almost all practical purposes, the competition is useless, and this quotation from the nineteenth-century Canadian astronomer Simon Newcomb tells you why:
Ten decimals [of π] are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimals would give the circumference of the whole visible universe to a quantity imperceptible to the most powerful microscope. But the quest for more and more digits does test computers and their software and programmers to their limits and mathematically speaking the digits are interesting because they can be tested for what is called normality. That is, are the digits of π effectively random, like those one would expect from rolling a perfect ten-sided die (or n-sided die in base n)? So far it seems that they are, and that is one of the paradoxes of π. A circle is the complete opposite of a random shape, and the ratio of its circumference to its diameter has a completely fixed value. Yet the digits of that ratio seem to be completely unpredictable.
But the quest for more and more digits is valuable for two other reasons symbolic ones. The English mountaineer George Malory said that he wanted to climb Everest because it was there. If π-nauts try to find the digits of π because they are there, they are only there because we have in fact found ways of predicting them. Mathematicians have discovered many finite formulae for an infinite sequence of digits. [cut bit of mathematical ignorance] The second symbolic value of the quest for ever more digits of π is that the quest is being carried out by men. The story of π is a male story, or rather, the story of the human relationship with π is a male story. Mathematics is beyond sex and personality, but for various biological reasons mathematics, as practised and applied by human beings, is overwhelmingly dominated by men. The ethnicity of Kanada and (I presume) the Chudnovsky brothers is symbolically important too: East Asians like the Japanese have a higher-than-average IQ and Ashkenazi Jews have a much higher-than-average IQ Ashkenazim are hugely over-represented among mathematicians, just as they are hugely over-represented among grandmasters of chess.
Blatner, who I presume is himself Jewish, doesnt comment on race and biology, but its one of the most interesting aspects of mathematical contingency: the way the necessary truths of mathematics are discovered by and influence human beings. Much less interesting, for me, are other aspects of mathematical contingency: the appearance of π in popular culture, for example. Blatner looks at these too in passing, and includes a list of π mnemonics in various languages. My favorite is this one in Spanish, in which the number of letters in each word stands for a digit of π:
Sol y Luna y Mundo proclaman al Eterno Autor del Cosmo.
(Sun and Moon and Earth acclaim the Eternal Creator of the Cosmos.)
With no accents and digraphs and every letter standing for exactly one sound, its about as close as language gets to the clarity and concision of mathematics. This book is an excellent popular insight into that clarity and concision, and more beside.

No comments:

Post a Comment