This is the fifth volume in a series of annual anthologies of high-quality articles covering various aspects of mathematics. There are 25 articles in this entry, most of which (but not all; one article here is the text of a speech) appeared in some journal, ranging from the online general-interest journal slate.com, to more scientifically oriented ones like American Scientist, to more mathematically specialized ones like the American Mathematical Monthly and Notices of the AMS.
It would unduly extend the length of this review to list and describe all the articles in this volume, but clicking on the “Table of Contents” link at the top of this review should give a good indication of what’s here. As can be seen, the articles cover a wide spectrum of topics related to mathematics. A number consist of substantive mathematical discussions, by which I mean an exposition of a theorem or solved problem, even if the theorem is not proved in detail. Some of these are quite likely accessible to people without sophisticated mathematical training: see, for example, Jordan Ellenburg’s The Beauty of Bounded Gaps, which in five pages of text, and without even assuming knowledge of the definition of “prime integer”, manages to convey a good sense of what Zhang’s recent proof of the “bounded gap” conjecture, a weaker version of the still-unsolved twin primes conjecture, means. Another example is Conway’s Wizards by Tanya Khovanova, which describes the author’s solution (using nothing more than simple logic and basic properties of numbers) to an interesting puzzle posed by John Conway. (For those who would like to take a crack at it themselves, I’ll reproduce the puzzle at the end of this review.)
On the other hand, some of these substantive articles are likely not so accessible to laypeople; Séquin’s On the Number of Klein Bottle Types, for example, begins with the daunting “A Klein bottle is a closed, single-sided mathematical surface of genus 2…” Likewise, The Fundamental Theorem of Algebra for Artists by Kalantari and Torrence, which describes a geometric approach to the solution of that theorem, does describe from scratch some basic properties of the complex numbers, but at the same time appears to assume at least a passing acquaintance with first-semester calculus.
Not all articles revolve around a theorem or problem, of course. Some are of a philosophical bent; the very first article in this volume, Mathematics and the Good Life, is written by a philosophy professor (Stephen Pollard) and explores the question of what “contributions mathematics makes to our success as individuals and as a species.” Others deal with aspects of mathematical pedagogy: Francis Su argues for a humanistic approach to teaching in The Lesson of Grace in Teaching (this is the article, previously referred to, which arose from a speech he gave when accepting a teaching award); Behar, Grima, and Marco-Almagro discuss Twenty-Five Analogies For Explaining Statistical Concepts; Penelope Dunham, in Food for (Mathematical) Thought explores using food as a teaching aid.
The history of mathematics is also not slighted in this volume. Michael Barant’s Stuck in the Middle: Cauchy’s Intermediate Value Theorem and the History of Analytic Rigor gives some nice insight into the two approaches taken by Cauchy in his calculus text to the intermediate value theorem; Chaos at Fifty by Motter and Campbell surveys the history of chaos theory (the fifty year period referenced in the title stems from the publication of Lorenz’s famous paper in 1963, but, as pointed out by the authors, even Poincaré had, in the late 1880s, experienced a “close encounter” with the subject); the history of space-filling curves is sketched in Crinkly Curves by Hayes.
Then there are the articles that defy easy categorization. There is one (Adventures in Mathematical Knitting) by sarah-marie belcastro that describes some of the mathematical objects (for example, the Klein bottle) that she has knitted (some years ago I read in Experiencing Geometry by Henderson and Taimina about crocheting the hyperbolic plane, but I don’t believe I’ve read about knitting geometric objects previously), and Keith Devlin has an interesting piece (The Music of Math Games) that discusses the qualities that a video game should or should not have to be an effective instrument for teaching mathematics. (I suppose I could have put this article in the earlier pedagogy category, but it seemed to me to be more about the design of the games than about actual classroom teaching.)
One nice feature of this volume (that was not present in the 2012 entry, the only other book in this series that I’ve had occasion to read) is a 16-page insert of color plates for a number of the articles; true, you have to flip back and forth between the article and the insert, but it’s nice to have them there. Another valuable feature of this book is a section called “Notable Writings”, a nice bibliography of other articles, most of which were considered for inclusion in this volume but which, for one reason or another, did not get included.
Of course, any book with a title that begins “The best…” is bound to generate some controversy, if only because judging the “best” of anything is a highly subjective endeavor. And, of course, among the articles in this volume, an individual reader will like some more than others. For these reasons, I won’t even make any attempt to describe which articles I liked the most, but I will say that I didn’t dislike any of them, and I learned something from every one; even articles that I did not completely agree with — I think I tend to put more of an emphasis on achievement than does Su, for example — provided an interesting point of view. So, in summary, this is a pretty good book to find in your Christmas stocking, or to put in somebody else’s.
APPENDIX. As promised, here is Conway’s puzzle, which involves a conversation on a bus between two wizards:
A: I have a positive integral number of children, whose ages are positive integers, the sum of which is the number of this bus, while the product is my own age.
B: How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages?
B: Aha! AT LAST I know how old you are!
The question: what is the number of the bus?
Mark Hunacek (email@example.com) teaches mathematics at Iowa State University.