This fine book is a compilation of selected articles from The Mathematical Intelligencer, Springer's mathematical magazine "about mathematics, about mathematicians, and about the history and culture of mathematics". According to Springer's Web site, this magazine aims to "inform and entertain a broad audience of mathematicians, including many mathematicians who are not specialists in the subject of the article".
This selection of 40 articles is a good representation of the variety that The Mathematical Intelligencer has presented to its readers over the last twenty years. The articles are categorized by their subject as parts of the book: 1. Interviews and Reminiscences, 2. Algebra and Number Theory, 3. Analysis, 4. Applied Mathematics, 5. Arrangements and Patterns, 6. Geometry and Topology, and 7. History of Mathematics. Each of these categories contains 5-6 articles.
If you read mathematical books like I do, you will enjoy "the first reading", i.e., browsing, of this book, because it covers so many interesting topics, has a lot of illustrations (including photographs), and displays formulas in a clear and readable format. "The second reading", i.e., reading the sections and articles that currently interest you, is even more enjoyable, because each gem in this collection was created by an expert in the respective field to be appreciated by the general mathematical audience. Every article has a historical overview of the problem it covers, shows the development of ideas from their early stages all the way to the current state of affairs, and includes references to important works from the past and papers and books for those interested to dig deeper. Again, rich graphics help a lot. However, "the third reading", i.e., from cover to cover, is much more difficult. This is nobody's fault, it is due to the nature of the book: in a diverse collection of interesting but difficult articles you are likely to find some articles too hard to digest. Also, you may find some article enjoyable until it becomes too technical for your taste. But that is OK too, that's also mathematics.
The following is a brief overview of the seven parts of this book:
In Part 1, Interviews and Reminiscences, you will find interviews with Fields Medallists Michael Atiyah and Jean-Pierre Serre and Nobel Laureate Chen Ning Yang in which they tell us about their early encounters with mathematics, their main achievements, and friendship and collaboration with other mathematicians. Similarly, in the reminiscences written by Steve Smale (another Fields Medallist) and Yuri Matijasevich (who solved Hilbert's Tenth Problem) you will learn about Smale's activities against the American involvement in Vietnam and about fruitful collaboration between Matijasevich and Julia Robinson. If you love anecdotes about famous mathematicians, you will enjoy the article by Steven G. Krantz.
Part 2, Algebra and Number Theory, begins with the article on Faltings' proof of the Mordell Conjecture:
If f(x,y) is any irreducible polynomial in two variables with rational coefficients and genus greater than or equal to 2, there exist at most a finite number of rational pairs (p,q) such that f(p,q)=0.
To prove the Mordell Conjecture, you will learn from this article, Faltings first had to prove two equally important conjectures, the Shafarevich Conjecture and the Tate Conjecture.
Other articles in Part 2 show us relations between the Riemann Hypothesis, Fermat's Last Theorem, and the Goldbach Conjecture; ways to solve systems of polynomial equations; methods of attack on the Artin's Conjecture for primitive roots (which takes us back to the Riemann Hypothesis); a history of the representation theory of finite groups from Frobenius to Brauer; and different ways to define determinants for quaternions.
The articles in Part 3, Analysis, are equally diverse. Here, among other topics, you can learn about the surfaces of Delaunay, see Xia's Example which proved the Painlevé's Conjecture, and follow the proof of the famous Banach-Tarski Theorem, here "formulated" as follows
It is theoretically possible, believe it or not, to cut an orange into a finite number of pieces that can then be reassembled to produce two oranges, each having the same size and volume as the first one.
Part 4, Applied Mathematics, begins by a discussion of applications of topology to biology (oscillations in the metabolism of yeast) and chemistry (circular waves in the Belousov-Zhabotinsky reaction) and continues with articles on strings and solitons, among other topics.
Part 5, Arrangements and Patterns, is particularly rich in illustrations. In this part you can learn about various disguises in which the Problème des Ménages appeared; read about how the discovery of quasicrystals intrigued both mathematicians and the solid state physicists; enjoy the mathematics in the architecture of one of the oldest surviving buildings in Florence; appreciate the mathematics of the Celtic knotwork; and read an article by the doyen of the "polyhedronists", H.S.M. Coxeter.
Part 6, Geometry and Topology, is also rich in graphics. Here, among other topics, you will learn about the role the computer graphics played in finding certain classes of minimal surfaces.
Part 7, History of Mathematics, begins with the article on Kurt Gödel and continues with a speculation on who would have won the Fields Medal if it had been awarded in 1885. Then you can read about the last 100 days of the Bieberbach conjecture, about the origins of the Max-Planck-Institute for Mathematics in Bonn, the war of the frogs and mice between Hilbert and Brouwer, and a rare update on the status of the 23 Problems of David Hilbert and the recent developments around them.
At the very end, the book contains a very detailed Index of Names.
If you ask me about the purpose of this book and the intended audience, I would have to guess, because the book itself doesn't say. It could simply be a tribute to the twenty years of a great mathematical magazine, The Mathematical Intelligencer. Or it could be a good starting point towards a thesis (or at least a general field) for an undecided graduate student. But since all these articles are already available in The Mathematical Intelligencer, i.e., in every good mathematics library, it seems to me that whatever the purpose of this collection, it would be better appreciated if it was published online! Isn't it great that we have a free access to the articles from the Notices of the AMS or the online columns here at the MAA Online?
Branislav Kisacanin (firstname.lastname@example.org) is Advanced Development Engineer with Delphi Automotive Systems in Kokomo, IN. He wrote Mathematical Problems and Proofs, which was reviewed recently on MAA Online.