Hidden Connections and Double Meanings: A Mathematical Exploration

The book under review is a Dover reprint of a book originally published in 1988 by Cambridge University Press. So, if it sounds awfully familiar, you may have a first edition sitting on your shelf or have given one to someone else in the late 1980s.

I have read (and reviewed) many books similar to this one, and I always start reading with the primary question: “To whom would I recommend this book?” As with many such books, this one presents a series of mostly independent expository chapters each of which conclude with a selection of problems/puzzles for the reader to try out. Solutions for all problems are provided and hints for many problems are also given.

The overall theme of the book is that there is great value in mathematics from seeing the same problem from two different points of view. The author states in the introduction:

The mathematician who spots that two apparently different problems are ‘the same’, solves both by solving one. Realise that a million problems are ‘essentially’ the same, and you can solve a million problems by solving one. Now there is power indeed!

The topics in the book derive mostly from geometry and tessellations, with a little bit of algebra and discrete math thrown in here and there. As a “professional mathematician,” I cannot say that I learned anything significantly new from the book, but I did pick up one or two ideas that I am going to try to incorporate into my geometry course for future teachers. On the other hand, I am pretty certain that I am not part of the target audience for this book, which returns me to the question I asked myself when I started reading the book.

Given the highly visual nature of the topics, the level of algebra used, the complete absence of calculus, and a theme that attempts to instill in the reader the ways that mathematicians think, I believe that the best audience for this book would be mathematically talented and curious students in the age range 11–16. While other people of a higher age could certainly enjoy the book, I think that the students in that age range will have the most to gain and are the most likely to find that the topics and problems are pitched at a level that matches where they are.

The topics and the writing are engaging, with a lighthearted edge that allows the book to be inviting and not intimidating. Overall the quality of writing is excellent. The book does show its age in a few small but forgivable instances. Some of the photos and artwork appear dated, a reference to the number of known perfect numbers is no longer accurate, and the phrase “advent of electronic computers” (what other type of computer is there?) sounds strange to my 21st century ears. I am also curious whether a “door answerphone” is a now-forgotten device from the 1980s or just the British expression for a front-door intercom.

Setting aside those nit-picky criticisms, the book is a fine general audience mathematics book, but one that I think is particularly well-suited to younger, curious readers who have a basic knowledge of geometry and a beginning understanding of algebra.

Geoffrey Dietz is a Professor of Mathematics at Gannon University in Erie, PA. He is married and has six children.