This is a lovely little book, both literally and figuratively. The main body of the text consists of ten chapters, each reasonably independent of the others and each covering some interesting aspect of geometry. These chapters are complemented by seven appendices that expand upon the material of the main text. Most of the chapters contain beautiful pictures, often in color, as well as a collection of stereo images. Designed to be looked at from a distance of about one foot, these consist of two images side-by-side, with the left figure designed to be viewed by the left eye and the right figure by the right eye. The author suggests placing a card in between the pictures so as to train the appropriate eye to look at the appropriate picture. (I have to confess that all this didn’t really work for me, but perhaps I wasn’t doing it right.)

As for the particular topics that are covered, I doubt I can improve upon the succinct paragraph-long description offered by the author in the preface:

There are ten basic lectures. We start with the Golden Number which leads naturally to considering regular Shapes and Solids in two and three dimensions and then a foray into the Fourth Dimension. A little amusement with Projective Geometry follows (a necessity in my youth for those going to university) and then a dabble in Topology. A messy experience with Soap Bubbles stimulated by Boys’ Little Book is next. We then look at circles and spheres (the Harmony of the Spheres) and especially Steiner’s porism and Soddy’s hexlet, which provide opportunities for pretty diagrams. Next is a look at some aspects of Chaos and Fractals. We then with some trepidation look at Relativity — special relativity can be appreciated relatively easily (groan — sorry about the pun) but general relativity is a bit tricky. The Finale then picks up a few loose ends.

The prerequisites for understanding the book are, for the most part, minimal, with no college-level mathematics assumed. (At times, though, as in the chapter on soap bubbles, some background in physics would come in handy.) The author states in the preface that this is the kind of book that he wishes he had been given when he was 16. Perhaps this would be true of a 16-year old who is destined to become a professional mathematician, but I suspect that most high-school students of this age would find this book rough going, just because there is a lot of material covered and a certain maturity and attention to detail is necessary for full understanding of it. Nevertheless, it should generally be accessible to a college-age student with prior high school experience in geometry and trigonometry.

The book is, we are told, based on a series of lectures the author gave to an audience of adult students. It was not clear to me whether each chapter corresponded to one lecture; if so, the author certainly manages to pack a lot more material into his lectures than I can comfortably do in mine, as each chapter contains a lot of information.

Not surprisingly, I enjoyed some chapters more than others, but that is undoubtedly a function of my own interests and tastes and does not reflect any unevenness in quality of writing, which is uniformly high throughout the book. The chapter on soap bubbles was, for example, a bit too physics-y for my taste, but I especially enjoyed the chapter on projective geometry (perhaps because, not long before reading it, I had talked about that subject in a class I am currently teaching). This chapter begins with Pappus’ and Desargues’ Theorems, with the latter proved by resorting to a new dimension, and proceeds to talk about such stalwarts of the subject as duality, cross-ratio, homogeneous coordinates, finite planes, and conics.

Chapter 7, entitled “The Harmony of the Spheres”, was also one of my favorites; despite its 3D-sounding title, much of the chapter took place in the plane, stating and proving Steiner’s Porism (by way of inversions, which are developed from scratch) and then extending this ideas to three dimensions via a discussion of Soddy’s hexlet, which was new to me: if three spheres touch each other, then there is a gap between them, and, no matter what the configuration of the original spheres, it is possible to have a closed chain of spheres passing through the gap, each sphere in the chain touching each of the original spheres as well as its neighbors in the chain; moreover, there will always be six spheres in the chain. (Soddy, by the way, was a chemist, and he was the person who coined the word “isotope”, another fact I didn’t know before.)

Other chapters I particularly enjoyed were the first (on the golden number, including discussions of the Fibonacci sequence and continued fractions; fairly standard material, perhaps, but presented clearly and elegantly) and the ninth (a nice little survey of both special and general relativity). The Finale (chapter 10) was also an interesting potpourri of topics, including, among other things, a statement (but no proof) of Morley’s theorem on the equilateral triangle formed by the points of intersection of adjacent angle trisectors of the three angles of a triangle and a discussion of the use of complex numbers in geometry.

This is not a book in which numbered theorems and proofs follow each other relentlessly. There are results that are proved, of course, but the author takes pains to incorporate the statements of the results and their proofs into a smoothly flowing narrative, one characterized by a pleasant informality that is at times positively chatty. More than in most books, it seems that the author is actually talking to the reader. Every chapter of the book except the last one contains at least one or two exercises, sometimes more; some of them seemed nontrivial, and none were accompanied by solutions.

I noticed nothing that I would call a mathematical error, but did occasionally come across a sentence that I might quibble with. In the chapter on projective geometry, for example, the author writes that Desargues’ theorem “cannot be proved in two dimensions without the help of the third dimension.” I suspect that what is intended here is the statement that Desargues’ theorem is not true in all projective planes (which is certainly true), but I think this sentence may be read to convey the idea that it cannot be proved in the extended Euclidean plane without using a third dimension (which is not true).

One other feature of the book that deserves mention is the author’s use of books and art to illustrate geometric ideas. *Flatland* is mentioned in the chapter on the fourth dimension; a book by Charles Dodgson (aka Lewis Carroll) is mentioned in the chapter on topology, which also references a short story by William Upson called *Paul Bunyan versus the Conveyor Belt* and a story called *The Last Magician* by Bruce Eilliott (both of these stories appear in anthologies edited by Clifton Fadiman, which are also mentioned). One of these anthologies also contains Heinlein’s *And He Built a Crooked House*, which is also mentioned in the chapter on the fourth dimension. As for art, the author mentions (among others) Escher in the chapter on the fourth dimension and Dali in the chapter on the golden ratio.

Perhaps because of the informality of approach and particular choice of topics covered, I doubt this book will find much use as a text for most American college courses in geometry. Because this seems to be more of a “fun with geometry” kind of text than a systematic mathematical development of the subject (though, as I said earlier, there are theorems that are proved), its primary value, I would think, is as a source of interesting facts and supplemental reading. The overall impression I got while reading *Gems* is that the author absolutely loves geometry, knows lots of interesting facts about it, and wants to share these tidbits with the book’s audience in as vivid, entertaining and informative manner as possible. I think he has succeeded, and so will close this review on the same note that I started it: this is a lovely book. Anyone who has an interest in geometry should take a look at it.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.