by Martin Erickson
There is probably no mathematician who has never been surprised by a mathematical statement or has not been delighted in a mathematical argument. For those who know, mathematics has beauty, elegance, mystery, and, of course, surprise. Most mathematicians chose this endeavor after having experienced such a moment of enlightenment. The book "Beautiful Mathematics" by Martin Erickson offers ample opportunity for such an experience to everyone with even a slightest interest in mathematics.
The book is a collection of short (1-2 pages long) vignettes that illuminate mathematical beauty from various angles. Mathematicians use imaginative words: lemniscate (a woolen ribbon used in fastening a garland to someone's head), waterfall of primes, golden ratio, triangular numbers ... They visualize intriguing images: binary trees, projective plane, two-colored graph, and come up with captivating formulas: series and products for π, the Riemann Zeta function, the Jacobi identity. Mathematicians prove delightful theorems: Morley's, Monge's, Minkowski's. Mathematics has pleasing proofs and elegant solutions; to pose an interesting problem requires creativity; all of that stands on harmonious foundations. The book ends with eye-opening explorations and these come with solutions, to boot. If pressed for an extra rubric, I would consider a separate part "Engaging Games", as this is something that mathematicians are preoccupied with - literally and metaphorically. (Not that the book entirely overlooks that side of mathematical activities. There are chapters on zero-sum games, nonattacking queens game, transversal achievement game - and more.)
The author does not supplicate the reader to accept his views - tastes will obviously vary from one reader to another. However the book is so rich in its mathematical samplings that a rare reader would not have his or her imagination fired up with a few of them. The language is simple, the prose lucid; Erickson has a clear knack for mathematical exposition.
The book, I believe, serves an antidote to the entrenched perception of mathematics as an utilitarian tool to cram the young minds with. Besides the intended audience of high school and college students, the book will be of interest to their teachers and will be enjoyed by any one with a bent for mathematics. It does make a very pleasant reading.
Unfortunately, the book has - so to speak - a fly in the ointment. The book's Index is an incredible mess to the extent of being entirely useless. I was disappointed every time I tried to look up a topic. The author made available online a correct version of the index. I hope that it will be incorporated into the next edition of the book.
Here is a list of pages where I used Erickson's materials:
- Plane Isometries As Complex Functions
- Increasing or decreasing subsequence
- Monge's Theorem of three circles and common tangents
- Gauss and Euler Integrals
- Sylvester's Problem, a Second Look
- Propositiones ad acuendos juvenes