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Beautiful Mathematics

Martin Erickson
Mathematical Association of America
Publication Date: 
Number of Pages: 
[Reviewed by
Alex Bogomolny
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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. Beautiful Mathematics 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 section on “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 require the reader to accept his views — tastes will obviously vary from one reader to another. However the book offers so rich a sampling that it will be a rare reader who will not have his or her imagination fired up with a few of them. The language is simple, the prose lucid; Erickson clearly has a knack for mathematical exposition.

The book, I believe, serves an antidote to the entrenched perception of mathematics as an utilitarian tool to cram into young minds. 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, there is 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 has made available online a corrected version of the index. I hope that it will be incorporated into the next edition of the book.

Alex Bogomolny is a former associate professor of mathematics at University of Iowa. He lives in New Jersey, maintains a popular site Interactive Mathematics Miscellany and Puzzles, with a server somewhere in Michigan, and blogs at CTK Insights.

The table of contents is not available.

Comments's picture

In Beautiful Mathematics pages 106–108, section 5.17, "A Group of Operations," Martin Erickson does not acknowledge any source. That section, on a group of 322,560 permutations generated by permuting the rows, columns, and quadrants of a 4x4 array, is based on the Cullinane diamond theorem. See that theorem (published in an AMS abstract in 1979) at and, and elsewhere on the Web.

Details of the proof given by Erickson may be found in "Binary Coordinate Systems," a 1984 article on the Web at

— Steven H. Cullinane