To celebrate Pi Day 2016, we have prepared a collection of key technical papers that have appeared in the past half century on topics related to Pi and its compution. The collection, entitled Pi the Next Generation: A Selection, is soon to be published by Springer, with ISBN 978-3-319-32377-0. Details are available at the Springer site.
Here is a synopsis of the book, as taken from the Springer site:
This book contains compendium of 25 papers published since the 1970s dealing with pi and associated topics of mathematics and computer science. The collection begins with a Foreword by Bruce Berndt. Each contribution is preceded by a brief summary of its content as well as a brief key word list indicating how the content relates to others in the collection. The collection includes articles on actual computations of pi, articles on mathematical questions related to pi (e.g., “Is pi normal?”), articles presenting new and often amazing techniques for computing digits of pi (e.g., the “BBP” algorithm for pi, which permits one to compute an arbitrary binary digit of pi without needing to compute any of the digits that came before), papers presenting important fundamental mathematical results relating to pi, and articles presenting new, high-tech techniques for analyzing pi (i.e., new graphical techniques that permit one to visually see if pi and other numbers are “normal”).
This volume is a companion to Pi: A Source Book whose third edition released in 2004. The present collection begins with 2 papers from 1976, published by Eugene Salamin and Richard Brent, which describe “quadratically convergent” algorithms for pi and other basic mathematical functions, derived from some mathematical work of Gauss. Bailey and Borwein hold that these two papers constitute the beginning of the modern era of computational mathematics. This time period (1970s) also corresponds with the introduction of high-performance computer systems (supercomputers), which since that time have increased relentlessly in power, by approximately a factor of 100,000,000, advancing roughly at the same rate as Moore’s Law of semiconductor technology. This book may be of interest to a wide range of mathematical readers; some articles cover more advanced research questions suitable for active researchers in the field, but several are highly accessible to undergraduate mathematics students.
Here is a brief description of the individual articles:
- A 1976 paper by Eugene Salamin, who exhibited an an algorithm for computing pi based on formulas with the remarkable property that each application of the formula approximately doubles the number of correct digits in the result.
- A 1976 paper by Richard Brent, who presented his independently discovered version of the algorithm described above, plus similar “quadratically convergent” algorithms for computing other transcendental functions and constants.
- A 1984 paper by David Cox, who gives some interesting background on Gauss’ discovery of the arithmetic-geometric mean and its connection to the theory of elliptic modular functions.
- A 1984 paper by one of us (Jonathan Borwein) and his brother (Peter Borwein), presenting some even faster algorithms for computing Pi and various elementary functions such as sine, cosine, exponential and logarithm. This paper was read by the other of us (Bailey), which led to some large-scale computations of Pi and a 32-year-and-counting collaboration.
- A 1985 paper by D. J. Newman, who presented simplified proofs of the algorithms of Salamin and Brent.
- A 1985 paper by Stan Wagon of Macalester College in Minnesota, who highlighted the long-standing and unresolved question of whether (and why) the digits of Pi are statistically random in a certain specific sense.
- A 1988 paper by one of us (Bailey) describing the computation of Pi to 29,360,000 digits, which at the time was a world’s record. This computer program was used as part of an integrity test of a new supercomputer and, in fact, did disclose some hardware bugs in the system.
- A paper by Gert Almkvist and Bruce Berndt on the historical roots of Gauss’ arithmetic-geometric mean, which is the basis of several state-of-the-art methods for computing Pi. Among these roots was the writings of a 19th century British mathematician who published articles for the Ladies Diary, a popular womens magazine that featured a regular mathematics column.
- A 1988 article by Japanese computer scientist Yasumasa Kanada, who computed Pi to 201,326,000 digits (a record at the time).
- A 1988 Scientific American article describing how the work of Indian mathematician Srinivasa Ramanujan has been applied to the computation of Pi.
- A 1989 paper by the present authors and Peter Borwein from the American Mathematical Monthly that presents the historical and mathematical background of Pi, including a formula discovered by Ramanujan that was used, after suitable modification, to compute up to two billion digits of Pi.
- A 1989 paper that highlighted and explained the remarkable fact that when Gregory formula for Pi, a simple formula dating back to 1671, is used to compute decimal digits of Pi using, say, 500,000 terms, the errors exhibit a curious pattern, with correct digits interspersed by errors.
- A 1995 paper by Stanley Rabinowitz and Stan Wagon that exhibited a simple “spigot algorithm” for the digits of pi that generates the digits one by one, much like drops of water dripping from a water spigot.
- A 1997 paper by Bailey, Peter Borwein and Simon Plouffe that presented what is now known as the “BBP formula” for Pi. This formula has the remarkable property that it permits one to calculate binary or base-16 digits of Pi beginning at an arbitrary starting position (such as the millionth or the billionth position), without needing to compute any of the digits that came before. Included in this paper were base-16 digits of Pi beginning at position ten billion, a record at the time.
- A 2001 paper by Dirk Huylebrouck that presents, in a relatively simple, self-contained fashion, proofs of irrationality (i.e., proofs that the constant cannot be written as the ratio of two whole numbers) for Pi, the natural logarithm of 2, and the Riemann zeta function evaluated at 2 and 3.
- 2006 paper by Jeremy Gibbons, who found “unbounded” variations of of the Rabinowitz-Wagon “spigot” algorithm.
- Two selections from a 2008 book by the present authors that deal with Pi: one chapter deals specifically with formulas (with proofs) for computing Pi; the other deals with the persistent and still-unresolved question of whether Pi and various other mathematical constants are “normal” (i.e., have statistically random digits in a certain sense).
- A 2009 paper by Stephen Lucas, who showed how approximations for Pi such as 22/7 and 355/113 can be seen to follow straightforwardly from integral formulas.
- A 2009 paper by Baruah, Berndt and Chan discussing a formula of Ramanujan for Pi and a related formula that was employed by David and Gregory Chudnovsky to compute over two billion decimal digits of Pi.
- A 2013 paper by the present authors, Andrew Mattingly and Glenn Wightwick describing the computation of base-64 digits of Pi2, base-729 digits of Pi2 and base-4096 digits of Catalan’s constant, in each case beginning with position ten trillion.
- A 2013 paper by Francisco Aragon Artacho, the present authors and Peter Borwein analyzing the normality (a form of statistical randomness) of the digits of Pi and numerous other mathematical constants using new techniques based on computer graphics.
- A 2013 article by Ravi Agarwal, Haans Agarwal and Syamal K. Sen presenting a detailed chronology of the computation and analysis of Pi from its earliest origins in India and Babylonian mathematics up to and in eluding the most recent results and computations.
- A 2014 article by us from the American Mathematical Monthly giving a brief overview of the history of Pi and decimal arithmetic, up to and including the high-tech methods, such as graphical analysis, that are now being employed by researchers investigating the normality of Pi.
- A 2014 by Jonathan Borwein, entitled “The Life of Pi,” which presents numerous interesting and important facts about Pi in their historical context.
- A 2014 paper by Jonathan Borwein and Scott Chapman, entitled “I prefer Pi” (a palindrome), listing and briefly describing the many papers on Pi that have appeared in the American Mathematical Monthly, the world’s most widely read mathematics journal.