|Daily Times - Site Edition||Tuesday, December 10, 2002|
POETIC LICENCE: Pi calculated to 1.24 trillion digits
Euclid was born in about 330 BC. He devoted much of his life to writing the “Elements,” the most successful textbook in history
We will now be able to measure the length of rivers more accurately. How come? Well, because the good news is that researchers at a leading Japanese university have set a new world record by calculating the value of pi to 1.24 trillion places. Professor Yasumasa Kanada and nine other researchers at the Information Technology Centre at Tokyo University calculated the value of pi with a Hitachi supercomputer, team member Makoto Kudo said on Friday.
As an Associated Press news agency report pointed out, “The new calculation is more than six times the number of places in the record currently recognised by the Guinness World Records — 206.158 billion places — which Kanada also helped calculate in 1999.”
“We would need to verify it, but it sounds like Professor Kanada has broken his own record,” Guiness World Records spokesman Neil Hayes told reporters. He said a Guinness mathematics expert would need to verify the data.
But does Guinness know just what it’s letting itself in for when it says it wants its own maths boffin to verify the data? One asks this question because, according to Kudo, Kanada’s team spent five years designing the programme used in the September experiment. Given this fact, the verification exercise could take not just a month of Sundays but years of Sundays or even decades of Sundays. Indeed, Guinness could be at it forever — unless, of course, it has a computer as fast as the one used by Professor Kanada and his team.
According to Kudo, the Hitachi computer used by the Tokyo University team is capable of 2 trillion calculations per second, or twice as fast as the one used for the current Guinness record calculation.
Pi, usually given as 3.14, is the ratio between the circumference of a circle and its diameter and has an infinite number of decimal places.
But what, you may well ask, has all this got to do with measuring the length of rivers more accurately? The answer is that Professor Hans-Henrik Stolum, an earth scientist at Cambridge University, has calculated the ratio between the actual length of rivers from source to mouth and their direct length as the crow flies. As Simon Singh notes in his fascinating book “Fermat’s Last Theorem” (1997), although the ratio varies from river to river, the average value is slightly greater than three, that is to say that the actual length is roughly three times greater than the direct distance. In fact, the ratio is approximately 3.14, which is close to the value of pi. So if — thanks to Professor Kanada and his team — pi can now be calculated more accurately, it stands to reason that the length of rivers, too, can now be measured more accurately.
The number pi was originally derived from the geometry of circles and yet it appears over and over again in a variety of scientific circumstances. Singh, in his book, puts it evocatively when he observes: “In the case of the river ratio, the appearance of pi is the result of a battle between order and chaos.”
He tells us that Einstein was the first to suggest that rivers have a tendency towards an ever more loopy path because the slightest curve will lead to faster currents on the outer side, which in turn will result in more erosion and a sharper bend. The sharper the bend, the faster the currents on the outer edge, the more the erosion, the more the river will twist and so on.
“However,” Singh notes, “there is a natural process which will curtail the chaos: increasing loopiness will result in rivers doubling back on themselves and effectively short-circuiting. The river will become straighter and the loop will be left to one side forming an ox-bow lake. The balance between these two opposing factors leads to an average ratio of pi between the actual length and the direct distance between source and mouth.”
The ratio of pi is most commonly found for rivers flowing across very gently sloping plains, such as those found in Brazil or the Siberian tundra. What about rivers like the Indus? Well, the problem there, of course, is that in its upper reaches — that is from its source in Tibet up to the point some 600 miles downstream when it emerges from the mountains and enters the plains of Pakistan — the Indus is too fast-flowing a river (indeed, the fastest-flowing major river in the world) for the pi effect to work.
In 332 BC, having conquered Greece, Asia Minor and Egypt, Alexander the Great decided that he would build a capital city in Egypt that would be the most magnificent in the world. Alexandria, the city he founded, was indeed a spectacular metropolis but not immediately a centre of learning. As Singh notes, “It was only when Alexander died and his half-brother Ptolemy I ascended the throne of Egypt that Alexandria became home to the world’s first-ever university. Mathematicians and other intellectuals flocked to Ptolemy’s city of culture, and although they were certainly drawn by the reputation of the university, the main attraction was the Alexandrian Library.”
As Singh tells us, Ptolemy’s dream of building a treasure house of knowledge lived on after his death, and by the time a few more Ptolemys had ascended the throne the Library contained over 600,000 books. Mathematicians could learn everything in the known world by studying at Alexandria, and there to teach them were the most famous academics. The first head of the mathematics department was none other than Euclid.
Euclid was born in about 330 BC. He devoted much of his life to writing the “Elements,” the most successful textbook in history. In particular Euclid exploited a logical weapon known as reductio ad absurdum, or proof by contradiction. One of Euclid’s most famous proofs by contradiction established the existence of so-called irrational numbers.
The most famous irrational number is pi. As Singh notes, “In schools it is sometimes approximated by 3.14; however, the true value of pi is nearer 3.14159265358979323846, but even this is only an approximation. In fact, pi can never be written down exactly because the decimal places go on forever without any pattern. A beautiful feature of this random pattern is that it can be computed using an equation which is supremely regular: pi = 4(1/1-1/3+1/5-1/7+1/9-1/11+1/13-1/15+...).”
By calculating the first few terms, you can obtain a very rough value for pi, but by calculating more and more terms an increasingly accurate value is achieved. Which is where Professor Kanada and his team come in, with their new record of pi calculated to 1.24 trillion digits — a lot of digits in anybody’s book, even Euclid’s.
Although knowing pi to a mere 39 decimal places is sufficient to calculate the circumference of the universe accurate to the radius of a hydrogen atom, this has not prevented computer scientists from calculating pi to as many decimal places as possible.
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