|Ivars Peterson's MathTrek|
May 11, 1998
The mathematical landscape has changed considerably in the last 10 years. Computers are responsible for much of that transformation.
In the original, 1988 edition of The Mathematical Tourist, I wrote, ". . . computers are still scarce among mathematicians and seldom used for serious mathematical research. One mathematician, Stanford University's Joseph Keller, even remarked in 1986 that at his university, the mathematics department has fewer computers than any other department, including French literature. Many mathematicians have the feeling that using a computer is akin to cheating and say that computation is merely an excuse for not thinking harder."
In the newly published, updated edition of The Mathematical Tourist, I write, ". . . computers have become commonplace tools used by nearly all mathematicians, not only to communicate with each other but also to explore mathematical questions."
". . . most mathematicians now consider the computer an essential part of their work," I continue. "By providing vivid images that suggest new questions and by allowing a wide variety of calculations of test cases, computers are helping to mend a rift that had developed more than a century ago between pure and applied mathematics."
The recent arrival of print and electronic journals such as Experimental Mathematics and Communications in Visual Mathematics testifies to the power of computation in contemporary mathematical research.
Of the new material added to The Mathematical Tourist, a large proportion involves some sort of computation in the service of mathematics. The revised book updates efforts aimed at identifying gargantuan prime numbers and highlights the astonishing and quite unexpected finding that the use of computers based on quantum logic and quantum bits (qubits) significantly speeds up the factoring of large composite numbers. It also introduces the application of cellular automata models to social questions and, in a recreational math context, to the peregrinations of virtual ants.
Computers are featured prominently in the greatly expanded concluding chapter on mathematical proof, which discusses Turing machines, transparent (or holographic) proofs, and automated reasoning software. Computers are not the whole story, however, as revealed in the account of how Andrew Wiles proved Fermat's Last Theorem. Nonetheless, electronic communication played a central role in the rapid spread of the news of Wiles's landmark achievement.
The interplay between computers and proof also shows up in my account of a new version of a proof of the four-color theorem, which states that four colors are always enough to fill in every conceivable map that can be drawn on a flat piece of paper so that no countries sharing a common boundary are the same color (see Maps of Many Colors). In another example, I describe how a proof that the standard double bubble, familiar to soap-bubble aficionados, is the most economical way of packaging a pair of identical, touching volumes required the use of a computer to check alternative configurations.
One remarkable change in the mathematical landscape concerns access to information. A decade ago, a nonmathematician interested in glimpsing the world of modern mathematical research had relatively few choices. The World Wide Web, in particular, has greatly expanded that menu.
Recently, I stumbled upon an amazing resource called "The Geometry Junkyard" (http://www.ics.uci.edu/~eppstein/junkyard/topic.html), maintained by David Eppstein of the University of California, Irvine. This Web site provides brief introductions and numerous, carefully annotated links related to geometric topics, ranging from circles, spheres, and spirals to tilings, polyhedra, and origami.
That's just one example. You can learn about Archimedes at http://www.mcs.drexel.edu/~crorres/Archimedes/contents.html, find out about the innovative Project MATHEMATICS! approach to high-school math topics at http://www.projectmathematics.com/, visit Ralph Abraham's Visual Math Institute at http://www.vismath.org/ to ponder Euclid and chaos, or delve into the mathematical physics of superstring theory at http://http://www.sukidog.com/jpierre/strings/. There's much, much more!
In general, I can find useful Web references on just about any mathematical topic that I want to write about. I must admit, however, that I often end up going back to the Scientific American "Mathematical Games" articles of Martin Gardner as a starting point. Yet even here, the Web has proved handy. The books containing collections of Gardner's articles don't have indexes, but I have discovered a detailed listing of the contents of his books at http://www.mathematik.uni-bielefeld.de/~sillke/gardner/lit, which partly makes up for that unfortunate oversight.
Another significant innovation that has turned out to be a boon to both mathematician and nonmathematician is the development of symbolic math programs such as Maple and Mathematica. Such software has been extremely useful for doing complicated algebraic manipulations, solving equations, proving theorems, teaching calculus, generating graphic images of mathematical objects, and guiding the creation of mathematical art.
A decade ago, I had to rely on obtaining illustrations for The Mathematical Tourist from the researchers involved. This time, I was able to use Mathematica myself to generate about two dozen new illustrations for the updated edition.
"In recent years, the mathematical community as a whole has displayed a renewed emphasis on applications, encouraged a return to concrete images, and allowed an increasing and more explicit role for mathematical experiments," I remark in the updated edition of The Mathematical Tourist. "Computation and computer graphics have provided a wide range of colorful, exotic images to illuminate mathematical ideas and suggest new mathematical questions. These changes are making the forbidding territories of mathematics more accessible than before to outsiders."
For the mathematical tourist, a trek across abstract terrain can be both revealing and rewarding. Peeking into the minds of mathematicians, reveling in the intricacies of number and shape, and tangling with intriguing concepts provide much food for deep thought.
May the journey continue!
Copyright 1998 by Ivars Peterson
Peterson, I. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. New York: W.H. Freeman.
______. 1988. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W.H. Freeman.
You can check out Experimental Mathematics at http://www.expmath.org/ and Communications in Visual Mathematics at http://www.geom.uiuc.edu/~dpvc/CVM/.
Information about Mathematica 3.0 is available at http://www.wolfram.com/.
Comments are welcome. Please send messages to Ivars Peterson at email@example.com.