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The Best Writing on Mathematics 2015

Mircea Pitici, editor
Princeton University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

Every year about this time, a new volume in Mircea Pitici’s Best Writing on Mathematics series, which started in 2010, appears in print; I was fortunate enough to review the 2012 and 2014 entries for this column, but in 2013 Sandra Keith beat me to the punch. This year, however, my luck improved. I like expository articles on mathematics, but seldom have the time during the academic year to seek them out, so it is always a pleasure to have somebody like Pitici assemble a collection of good ones for me. This year’s assortment, like those of the last few years, did not disappoint.

As in other years, the articles in this book (there are 29 of them this year, all published in 2014) cover a wide array of topics, not only in substantive mathematics but also in philosophy (one entry here discusses the nature of mathematical beauty; another, by Mark Balaguer, offers “a guide for mathematicians who don’t know much about the philosophy of mathematics — a guide that explains how to read philosophers of mathematics”), history (two articles discuss the history of the pigeonhole principle and the game of Nim, respectively), mathematics education (one article compares and contrasts mathematics education in the United States and China, another addresses proposals for high school education reform), and biography (a nice overview of the career of Martin Gardner).

The emphasis in this volume appears to be on recreational mathematics, games and puzzles. In addition to the previously mentioned historical discussion of Nim, there are, for example, pieces on the mathematics of juggling, the use of certain geometric shapes in games, mathematical issues related to the game Candy Crush, and how billiards leads to interesting mathematical questions.

Several entries here might prove to be of particular interest to teachers of undergraduate combinatorics or discrete mathematics (magic squares); probability and statistics (one article by Jeffrey Rosenthal talks about the role statistical analysis played in exposing a lottery retailer scandal in Canada; another article discusses how data gathering and interpretation may lead to inaccurate claims of statistical significance); abstract algebra (see, for example, the discussion of the quaternion group as a symmetry group); or Euclidean geometry (the Steiner-Lehmus theorem, a theorem I teach in my undergraduate course and which is notoriously hard to prove if you don’t “know the trick”, is discussed at some length, including the question of whether direct proofs of it are possible).

Several articles discuss geometry and art: for example, there is an article on mathematical aspects of the work of Albrecht Durer, and another, by Maor and Jost, a combination of text and original art, discusses various spirals and their role in mathematics. Readers of their book Beautiful Geometry, reviewed in this column about two years ago, will experience a strong sense of déjà vu here.

Some of the articles were of an interdisciplinary nature. One discusses the role of mathematics in a new discipline called synthetic biology, and another talks about the Tracy-Widom distribution, a “complex cousin of the familiar bell curve” that pops up in several different contexts involving complex systems.

And then there are the inevitable articles that simply defy easy characterization. The lead article in this collection, for example, discusses the role that the chalkboard has played in mathematical culture.

The above sampling of articles is not exhaustive, but should give a rough idea of the issues discussed in this volume. My selection is somewhat arbitrary, and the omission of an article should not be viewed as a comment on my opinion of its merits.

The articles here vary not only in subject matter but also in length and level of mathematical sophistication. Regarding the former, most seem to be about ten pages long, give or take a couple; some are longer, though, and one is about 25 pages long. As for the latter, there are some articles that require no knowledge of mathematics at all to understand, but a number seem to require at least a nodding acquaintance with standard undergraduate-level mathematics at, say, the junior level. For example, the article The Quaternion Group as a Symmetry Group is clearly intended for people who already know what a group is (although the quaternion group is defined from scratch). Ironically, the one article in the book that I felt that I lacked the prerequisites for was the one on Candy Crush, because the author does not begin the article with a discussion of the game or how it is played.

Some of the older editions in this series began with a Foreword by an eminent mathematician or physicist (Roger Penrose in 2013, David Mumford in 2012, Freeman Dyson in 2011, William Thurston in 2010), but this feature, unfortunately, seems to have disappeared as of 2014. Pitici continues to contribute, as he has in past volumes, a useful Introduction that, among other things, contains a rather complete list of other mathematics-themed books, usually pitched at a level accessible to a general reader, that were published in 2014.

In her review of the 2013 edition, Sandra Keith was critical of the fact that Pitici did not attempt to discuss his criteria for determining which mathematical writing was “best”. He doesn’t do that here either, but this omission didn’t bother me at all. Perhaps the difference in our reactions can be traced to a difference in our expectations. People who approach this book wanting to learn what makes mathematical writing the “best” may well be disappointed, but I assumed at the outset that words like “best” are subjective (and also likely hyperbole), and was really just looking for a collection of interesting articles with which to kill an hour or two at a time. I got what I wanted, and am looking forward to the 2016 volume.

Mark Hunacek ( teaches mathematics at Iowa State University.

Introduction - Mircea Pitici xi
A Dusty Discipline - Michael J. Barany and Donald MacKenzie 1
How Puzzles Made Us Human - Pradeep Mutalik 7
Let the Games Continue - Colm Mulcahy and Dana Richards 14
Challenging Magic Squares for Magicians - Arthur T. Benjamin and Ethan J. Brown 26
Candy Crush's Puzzling Mathematics - Toby Walsh 38
Chaos on the Billiard Table - Marianne Freiberger 47
Juggling with Numbers - Erik R. Tou 63
The Quest for Randomness - Scott Aaronson 69
Synthetic Biology, Real Mathematics - Dana Mackenzie 93
At the Far Ends of a New Universal Law - Natalie Wolchover 99
Twisted Math and Beautiful Geometry - Eli Maor and Eugen Jost 107
Kenichi Miura's Water Wheel, or The Dance of the Shapes of Constant Width - Burkard Polster 119
Dürer: Disguise, Distance, Disagreements, and Diagonals! - Annalisa Crannell, Marc Frantz, and Fumiko Futamura 132
The Quaternion Group as a Symmetry Group - Vi Hart and Henry Segerman 141
The Steiner-Lehmus Angle-Bisector Theorem - John Conway and Alex Ryba 154
Key Ideas and Memorability in Proof - Gila Hanna and John Mason 167
The Future of High School Mathematics - Jim Fey, Sol Garfunkel, Diane Briars, Andy Isaacs, Henry Pollak, Eric Robinson, Richard Scheaffer,
Alan Schoenfeld, Cathy Seeley, Dan Teague, and Zalman Usiskin 181
Demystifying the Math Myth: Analyzing the Contributing Factors for the Achievement Gap between Chinese and U.S. Students - Guili Zhang and Miguel A. Padilla 187
The Pigeonhole Principle, Two Centuries before Dirichlet - Benoît Rittaud and Albrecht Heeffer 201
A Prehistory of Nim - Lisa Rougetet 207
Gödel, Gentzen, Goodstein: The Magic Sound of a G-String - Jan von Plato 215
Global and Local - James Franklin 228
Mathematical Beauty, Understanding, and Discovery - Carlo Cellucci 241
A Guide for the Perplexed: What Mathematicians Need to Know to Understand Philosophers of Mathematics - Mark Balaguer 265
Writing about Math for the Perplexed and the Traumatized - Steven Strogatz 280
Is Big Data Enough? A Reflection on the Changing Role of Mathematics in Applications - Domenico Napoletani, Marco Panza, and Daniele C. Struppa 293
The Statistical Crisis in Science - Andrew Gelman and Eric Loken 305
Color illustration section follows page 316
Statistics and the Ontario Lottery Retailer Scandal - Jeffrey S. Rosenthal 319
Never Say Never - David J. Hand 332
Contributors 337
Notable Writings 349
Acknowledgments 359
Credits 361