November 3, 2003
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Berkeley Lab Mathematician Coauthors Two New Books on Experimental Mathematics
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BERKELEY, CA   David Bailey, a senior scientist and applied mathematician at Lawrence Berkeley National Laboratory, who has made his mark studying problems ranging from the digits of pi to the performance of supercomputers, has coauthored a pair of new books on experimental mathematics.

Scheduled to hit the shelves later this year, the books are Mathematics by Experiment: Plausible Reasoning in the 21st Century by Jonathan M. Borwein and David H. Bailey, and Experiments in Mathematics: Computational Paths to Discovery by Borwein, Bailey and Roland Girgensohn.

"Figure-Eight Knot Complement II," by sculptor and mathematician Helaman Ferguson, embodies a novel mathematical truth about knots in hyperbolic space unsuspected by the sculptor himself -- but discovered by physicist David Broadhurst using Ferguson's own PSLQ algorithm.
(From Mathematics by Experiment)

Published by A. K. Peters, Ltd., the experimental mathematics books generated an enthusiastic review in the May 2003 issue of Scientific American, well in advance of publication, and are selections of Bookspan, a scientific book club.

Bailey is well known for his work in computational mathematics and has written numerous papers on using modern computer technology in mathematical research. His best-known result in this area is a new formula for the mathematical constant pi, which was found by Bailey and two other researchers in 1996 using a computer program they devised. He is currently the chief technologist for the Computational Research Division and the National Energy Research Scientific Computing Center (NERSC) Division of the U.S. Department of Energy's Berkeley Lab.

In Mathematics by Experiment, Bailey and his long-time collaborator Borwein, who is professor of science at Simon Fraser University in Vancouver, B.C., write that applied mathematicians and many scientists and engineers were quick to embrace computer technology but that pure mathematicians -- whose field gave rise to computers in the first place, through the work of beautiful minds like Alan Turing's -- were slower to see the possibilities. Two decades ago, when Bailey and Borwein started collaborating, "there appeared to be a widespread view in the field that 'real mathematicians don't compute,'" they write.

These two books are a testament to a paradigm shift in the making. Hardware has "skyrocketed in power and plummeted in cost," and powerful mathematical software has come on the market. Just as important, the authors remark, "a new generation of mathematicians is eagerly becoming skilled at using these tools" -- people comfortable with the notion that "the computer provides the mathematician with a 'laboratory' in which he or she can perform experiments: analyzing examples, testing out new ideas, or searching for patterns."

Working in this virtual laboratory with Borwein and other colleagues, Bailey was among the first to discover a number of remarkable new algorithms. One was an extraordinary, simple formula for finding any hexadecimal or binary digit of pi without knowing any of the preceding digits. Further research led to proof that a wide class of fundamental constants are mathematically "normal" -- probably including pi, although that remains to be proved.

A section of Mathematics by Experiment on "proof versus truth" is an example of the gems even a mathematical tyro can find among these equations. Bailey and Borwein don't claim computers can supply rigorous proofs; rather, the computer is a way to discover truths -- and avenues for approaching formal proofs. But often, the authors add, "computations constitute very strong evidence ... at least as compelling as some of the more complex formal proofs in the literature."

Drawing on their own work and that of others, Bailey and Borwein and, in the second volume, their German colleague Roland Girgensohn not only explain experimental mathematics in a lively, surprisingly accessible fashion, but give many engaging examples of the "new paradigm" in action.

For those who just can't wait, what Bailey calls a "Reader's Digest version" -- 60 pages of "some of the more engaging material" from both volumes -- is freely available online at http://www.expmath.info/expbook-C.pdf, with a cornucopia of related resources at http://www.expmath.info.

Prior to joining Berkeley Lab in 1998, Bailey was a computer scientist at the NASA Ames Research Center. While at NASA, he coauthored the "NAS Parallel Benchmarks," which has been widely used to measure high-end scientific computer performance. He is currently heading a new eight-institution research effort, the U.S. Department of Energy's Performance Evaluation Research Center (PERC), aimed at understanding and improving performance of scientific computations on high-end computer systems.