The Best Writing on Mathematics 2015
The concept has not changed since the previous edition. This time, we get 29 contributions which, as you may recall from previous volumes, are not per se mathematical, but reflect on mathematics. The writings are on mathematics, not of mathematics. They are addressing mathematicians as well as non-mathematicians. The book is copyright 2016, and from the title, one could expect reprints of papers that were published in 2015. However, all the papers are re-formatted in a uniform lay-out and that takes some time, so the papers that are collected in this volume are all from 2014. In that sense the title is a bit misleading.
The topics are covering again mathematical games and recreation, mathematical education, some history and some geometry, in short mathematics (also, and most prominently so) for the non-mathematician. Let me pick some items that caught my attention, implying that this is a personal selection, and other readers may have a totally different choice.
M. Barany and Donald McKenzie emphasize the importance of the blackboard in mathematical education and research. There may be some benefits in using PowerPoint or other digital presentation tools, but abandoning chalk and blackboard has definitely a great impact on the organization of our thoughts in mathematics communication. P. Mutalik points to the psychological impact of an Aha! experience when solving a puzzle or a mathematical problem and hence moulds our minds and shapes our development as humans. The reform of mathematical education is a permanent struggle, if not war, worldwide. Strong emotions, political, economic, and emotional forces tear in all directions. J. Fey and S. Garfunkel give five tenets that should be taken into account to reform math education in high schools (in the US). Highly instructive is the paper by G. Zhang and M. Padilla who compare the mathematical education in China and the US which are almost opposites, yet with a result that is favorable for the Chinese.
In the realm of puzzles and games, I mention yet another celebration of Martin Gardner (C. Mulcahy and D. Richards), the morphing game needed when juggling (R. Tou), billiards of all possible shapes (M. Freiberger), the generation of magic squares (A. Benjamin, E. Brown), the mathematics behind candy crush (T. Walsh), and the historical roots of the game Nim (L. Rougetet).
Some of the papers can be classified as related to geometry. Several of the classical curves such as spirals, (epi-)cycloids, etc. are discussed by E. Maori and E. Jost. A quite remarkable study is B. Polster's paper about non-circular shapes of constant width and how they have been applied. J. Conway and A. Ryha give an amusing discussion of different proofs for an old geometric problem in triangular geometry. The perspective used by Dürer in his drawings duscussed by A. Cranell, M. Frantz, and F. Futamura, and unseen visualizations of symmetry groups such as the quaterion group by V. Hart, and H. Segerman.
But as I am enumerating the papers that I found personally interesting, I realize that I will end up with most of the content, if not all of it. There are the ones that relate mathematics and art, philosophy, biology, the papers with a statistical aspect, etc. Certainly, I have to mention the introduction by the editor Mircea Pitici explicitly. As before, he does not only summarize the contributions, but since there are only papers in the book, he also gives a list of books recently published that have a similar content and are intended for a broad readership. New is that he started a Twitter account (@mpitici) and supplementary material is available on the website of this book at Princeton University Press that can be found elsewhere on this review page. This extra material consists e.g. of an extended version of the introduction of the book –—so you can also find there his summary of the contents of all the papers in the book which is more complete than mine—– and there are many links to extra online information. This is provided in pdf form, so it is a bit disappointing that the links have to be copy-pasted since they are not directly clickable. Everyone is invited to make suggestions and contribute to future volumes.
Every year I look forward to the new volume of Best writing on mathematics and just like the previous volumes, this one, even thicker than the previous, fulfills all the expectations.