Underground Pi Digging Out from Under Pseudoscience

By Mark Cowan

In the Washington Park Station of the Westside Light Rail Tunnel, in the 16-million year recitation of history that runs alongside a 260-foot core sample of Portland’s West Hills, you will find, etched in granite, the first 107 digits of the transcendental constant pi (figure 1).

Figure 1. Pi as it appears in the Washington Park Station of the Westside MAX Light Rail.

3.1415926535  821480865144  288109757245  870066330572  703698336733  620005681271  420199561160  244592555982  534904897932

     Pi, as every schoolchild knows, is the ratio of the circumference of a circle to its diameter - one of the fundamental ratios of the space that fills our universe. Its digits never repeat, and they follow no pattern. Other than its role in geometry, and the fact that it pops up in many diverse branches of mathematics, pi has no special significance. It does not mean anything...so far as we know.

     Yet pi has exerted a steady pull on the human imagination. The Babylonians and Egyptians knew its value to within a half a percent some 4000 years ago. By the 3rd century BC, Archimedes had rectified the circle, nearly invented the calculus, and established pi’s value to about one part in 100,000 by the use of regular polygons. And by the 5th century AD a Chinese father and son, using a variation of this method, pinned down eight digits-a precision unequaled in Europe until the 16th century. Their laborious extraction of square roots was aided by the early Chinese introduction of a blank for zero.

     By diligent use of Archimedes’ method, in a 1596 paper entitled “On the Circle,” Dutch mathematician Ludolph van Ceulen single-handedly delivered the first 20 digits - then challenged anybody to top it. None did, but he soon extended his claim on history first to 32, and then to 35 digits (figure 2), of which the last three were engraved on his tombstone. To this day pi remains “the Ludolphine number” in Germany.

Figure 2. Pi as per van Ceulen (1596).

3.1415926535 897932384626 433832795028

     But the big guns were ready to fire. The methods known since Archimedes’ time could, theoretically at least, calculate pi to any desired degree of accuracy, the only limits being the calculator’s fortitude. When European mathematics began to flourish, the methods themselves were improved.

     In 1665 and 1666, during the Plague, Issac Newton developed the calculus-and offhandedly produced an efficient infinite series for calculating pi . Evaluating only 22 terms of the series yielded 16 digits. He saw no practical value to this effort, however, and later apologized for how far he had carried his computation “Having,” as he wrote, “no other business at the time.”

     But if a giant like Newton could fall under pi’s transcendental spell just for lack of anything better to do, was anyone truly immune?

     Heedless of such reservations, the hunt continued. With various modifications to improve efficiency, by 1719 the French mathematician de Lagny had sweated his way through 127 decimal digits (figure 3), a record that would stand for 75 years.

Figure 3. Pi as per de Lagny (1719).

3.1415926535 897932384626 433832795028 841971693993 751058209749 445923078164 062862089986 280348253421 170679821480 865132823066 470938446

     Further progress required more efficient tools. Around 1755 Leonhard Euler, perhaps the greatest mathematician of all time, discovered the fastest converging series yet known. Using it, he worked out the first 20 decimal places of pi in a single hour. But, doubtless mindful of the limited value of this pursuit, like Newton he went no further. Others, of course, were more than willing to extend the tally-and naturally they used his methods.

     But one wonders: if any of these early pi hunters were somehow to wander down the Washington Park MAX Station today, what might they think of those 107 digits etched in cold granite? Would Newton sneer? Would Euler wince?

     For the physical accuracy implied by 106 decimal places of pi has no counterpart in reality. If you inscribed a circle the diameter of the known universe (which has varied recently, but we’ll use 24-billion light-years), and then calculated its circumference by use of those 106 digits, the error due to truncation would be 1/1061 of the width of an atomic nucleus!

     Still the hunt went on. Calculating prodigy Johann Dase produced 200 decimal digits of pi in just two months in 1844, with others ringing up slightly larger tallies - until finally William Shanks published 707 digits of pi in 1873-74. This record stood until 1945-when he was shown to have gotten the last 180 digits wrong.

     But now the electronic computer was on the horizon, and by 1949 ENIAC had churned out 2037 places in 70 hours. The digital floodgates opened. Pi’s current world record now stands at 51,539,600,000 decimal digits set in June of 1997 by Kanada and Takahashi (1) at the Tokyo Computer Centre after 29 hours on a machine with 1024 processors. That’s 61 million times faster than ENIAC per digit. Interestingly enough, the two digits beginning at position 49,999,999,999 in both pi and 1/pi are 42.

     Ivars Peterson’s online Mathland delivers more information on pi mania - which, of course, continues unabated - with Internet links to get you started (2). There you can marvel at people who memorize great hunks of pi . You can also learn of a new formula that delivers specific hexadecimal digits of pi -without knowing any of the preceding ones! This completely unanticipated result is being put to use to calculate, via an Internet network (3), both the 5 and 40 trillionth binary digit of pi . No equivalent formula yet exists for decimal digits. Seems like there’s an argument against creation in there somewhere...

     If, by now, you just can’t live without your own big piece of pi, running Piw131 (4) overnight on a decent PC with 32 megabytes or more of memory will get you a cool million digits by morning. If that’s not enough, you can search (5) the first 50 million digits for any string of numbers up to 127 digits. But consider this: in about an hour I wrote, from scratch, a simple program (6) that computes the circumference of a unit circle using nothing more advanced than square roots. Run under QBASIC it delivers Newton’s 16 digits after only 26 iterations.

     Sure, it was fun to do - but is there any real point to any of this, after all?

     Well, that’s where it gets interesting. The distribution of digits in the first 50 billion digits of pi is statistically normal (6). But a recent study (7) has found that the distribution of repeating strings of digits is not. And nobody knows why that should be so! So pi , it would seem, still contains some curious implications for number theory. And the digital expression of it is the source of a new kind of mathematical analysis.

     Which brings us, uhm, full circle-and back to “Pi Underground.”

     According to Rebecca Banyas of Tri-Met’s Westside Light Rail, the artist, Bill Will, “got his information on pi from a reference book called The History of Pi (8). The numbers that appear on the wall are the same as those in the book.”

     Well, sort of. You may have already noticed, however, a slight discrepancy between the values carved into the tunnel and those worked out nearly 300 years earlier by de Lagny. This discrepancy was first spotted by a MAX engineer who had memorized pi to 12 places as a child. But the reason for the error remained obscure. Was it Art? A bad job of typesetting? Deliberate? Just to see if anybody was paying attention?

     After I searched strings of the Washington Park Station digits against the half-million pi digits on my computer, the source of the error became clearer. And checking out a copy of A History of Pi made it obvious. Artist Bill Will wasn’t taking liberties with a constant of the universe. He was just unfamiliar with the format of mathematical tables (figure 4).
 

     And they’ll think you’re both loonies, and won’t let either of you on the train.
 
 

1.  Details (not the result, though!). Some statistical analysis of the first 50 billion digits is provided.

2. Search here on "pi" in the Mathland columns.

3. More info.

4. Harry J. Smith has all kinds of stuff to help you have “Fun With Mathematics!”

5. Pi served daily. Observe the ubiquitous nature of 42 yet again.

6. A QBASIC program to compute pi :

     REM CIRCLE

     H# = 2

     FOR I = 1 TO 26

     L# = 1 - SQR(1 - H# * H# / 4)

     H# = SQR(L# * L# + H# * H# / 4)

     PRINT “H=”; H#, “I=”; I, “Value=”; H# * 2 ^I

     NEXT I

7. Unfortunately I’ve been unable to find where I saw this; you’ll have to trust me. :)

8. Beckmann, Petr. 1971. A History of Pi. The Golem Press. I owe a great debt to this excellent book and have drawn much from it for this article.

9. “How I need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics” gives you 15 digits. Other mnemonics exist, including a poem somewhat reminiscent of Poe’s The Raven that delivers 740 digits. See (2).