Ivars Peterson's MathLand

March 11, 1996

## A Passion for Pi

I consider myself a loyal member of the Ancient and Honorable Society of Pi Watchers. During the last year, I've written about the discovery of an algorithm for calculating individual, isolated digits of pi, the computation of the value of pi to a record 4.3 billion decimal digits (now up to 6.4 billion digits), and the use of the distribution of bright stars across the sky to approximate the value of pi.

Representing the ratio of a circle's circumference to its diameter, pi turns up in an astonishing number of settings. So, when I received a press release a few weeks ago about a new mnemonic device for remembering the first 167 digits of pi, I was naturally intrigued.

Some of you may be familiar with the sentence: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics! The number of letters in each word represents successive digits of pi: 3.14159265358979.

Over the years, pi enthusiasts have created mnemonic devices for encoding pi in just about any language you can imagine -- from ancient Greek to modern Icelandic. These sentences, poems, miniature dramas, comic episodes, and so forth reflect not only the digits of pi but also the considerable ingenuity of their authors. Even going beyond the 31st decimal digit requires invoking some new rule -- such as using 10-letter words -- to encode the zeros of pi.

The news release I had received came from Alexander Volokh, a writer and amateur mathematician in Los Angeles. The memory aid described in his announcement involved the use of many sentences, with the end of each sentence representing a zero.

I was curious whether anyone else had come up with something similar. A quick tour of the World Wide Web turned up a host of mnemonic devices, some even more clever than the effort offered by Volokh and his friends. One astounding example encodes 740 digits of pi in a lengthy poem modeled on Edgar Allan Poe's "The Raven."

I also remembered having come across another remarkable memory feat involving pi. I had heard about it when I was researching the discovery by David Bailey, Peter Borwein, and Simon Plouffe of a truly amazing formula for computing any given hexadecimal (or binary) digit of pi without being forced to calculate the preceding digits. "No one had previously even conjectured that such a digit-extraction algorithm for pi was possible," notes Steven Finch of MathSoft, Inc.

Plouffe works at the Centre for Experimental and Constructive Mathematics, a research institute at Simon Fraser University in British Columbia. He is the coauthor with Neil J. A. Sloane of the Encyclopedia of Integer Sequences. He also once held the world title for memorizing decimal digits of pi -- a total of 4,096 digits. This achievement was duly recognized in the 1977 French edition of the Guinness Book of World Records.

Actually, Plouffe had memorized 4,400 digits but settled on 4,096 as a nice round number (2^12) to report to others interested in his feat. Back then, "I was young and I had not much else to do, so I did it," Plouffe recalls. He simply liked numbers and was fascinated by pi.

Having a good memory for numbers and the ability to recognize numbers by sight has proved useful to Plouffe in his mathematical work, which often involves looking for relationships between different mathematical series or different number sequences. He is now working on a project to develop an automated system for doing the kind of numerical pattern recognition that he himself does so naturally.

To Plouffe, memorizing the digits of pi was close to a mystical experience. He worked with blocks of 100 digits. He started by writing out a block five or six times. He then tried to recite these digits in his head. To preserve the numbers in his long-term memory, he periodically isolated himself in a room -- no lights, no noise, no coffee, no cigarette. "Like a monk," Plouffe says. As he recited the digits to himself, they would gradually seep into his mind. After a day or two, he would be ready to go on to the next block.

When Plouffe got to 4,400 he decided to stop. "You can continue ... forever," he explains. "You stop mainly because it is boring to do that all the time."

Two years later, the person who had held the previous record of 3,025 digits came back with 5,050 memorized digits. "I knew I could beat him, but ... I had had enough," Plouffe says. The record now stands at 42,000 digits!

There is something delightfully irrational about this enduring interest in -- or perhaps obsession with -- pi. It's the kind of passion that can sometimes lead to interesting, even useful mathematical discoveries. By analogy, you have only to consider how a fascination with prime numbers and factoring on the part of a few eccentrics long ago has now played out in the development of cryptographic schemes for maintaining the security of computer systems and networks.

In an article in Mathematics Magazine about the never-ending fascination with the number pi, Dario Castellanos wrote:

"This is a field of endeavor that has attracted some of the greatest minds of mankind.... the studious pursuer of the many curious and fascinating properties which surround this number will forever meet new results and new algorithms related to 'the mysterious and wonderful pi.'"

### References:

Browne, Malcolm W. "Mathematicians Turn to Prose in an Effort to Remember Pi." The New York Times, July 5, 1988.

Castellanos, Dario. "The Ubiquitous Pi." Mathematics Magazine, 61 (April 1988): 67-96, (June 1988): 148-164.

Davis, Philip J. and William G. Chinn. 3.1416 and All That (2nd Edition). Boston: Birkhauser, 1985.

Peterson, I. "A new formula for picking off pieces of pi." Science News, 148 (Oct. 28, 1995): 279.

Peterson, I. "Spying pi in the sky." Science News, 147 (May 20, 1995): 319.

Peterson, I. "Next number, please." Science News, 147 (May 20, 1995): 319.

Sloane, N. J. A., and Simon Plouffe. The Encyclopedia of Integer Sequences. San Diego, Calif.: Academic Press, 1995.

You can visit Simon Plouffe at http://www.cecm.sfu.ca/~plouffe/.

A good place to start exploring pi-mania on the web is the "Uselessness of Pi and Its Irrational Friends" site at http://www.go2net.com/internet/useless/useless/pi.html. Mike Keith's poetic mnemonic for encoding 740 digits of pi is at http://users.aol.com/s6sj7gt/mikerav.htm.

MathSoft has a nice introduction to "The Miraculous Bailey-Borwein-Plouffe Pi Algorithm" at http://www.mathsoft.com/asolve/plouffe/plouffe.html.