Geometrical Kaleidoscope

Back in my formative years as an undergraduate, my favorite mathematics professor did me a considerable favor by introducing me to the world of Dover books. He told me at the time that Dover did the mathematics community a valuable service by rescuing high-quality books from extinction, and also making them available to people at a price that even students could afford. The (many) years since then have born out the wisdom of his advice on numerous occasions. Lately, however, Dover Publications has broadened its scope and, in addition to publishing new editions of classic texts, now publishes a fair number of original texts. The book now under review is one of these, and it’s a nice little book indeed.

The subject matter of Geometrical Kaleidoscopes may reasonably be described as “advanced Euclidean geometry”, in the sense that the book assumes the reader is already comfortable with the content of good high-school courses in geometry and trigonometry. Without spending much time time on a review of this material, the text starts where a typical high school course leaves off, discussing, for example, triangle centers (centroid, circumcenter, orthocenter, etc.), the nine-point circle, advanced problems in geometric constructions (including constructions with inaccessible points and with tools other than compass and straightedge), and the use of geometric transformations (especially rotations) to solve geometric problems. In addition, some famous results the reader will encounter here include Ceva’s theorem, Morley’s theorem, Heron’s formula for the area of a triangle (and Brahmagupta’s formula for the area of a cyclic quadrilateral), and the Steiner-Lehmus theorem.

This text does not deal with foundations or rigorous axiomatics; statements are proved, but are done so making cheerful use of reasonable inferences from diagrams. Non-Euclidean geometry is not discussed; neither is solid geometry. All of these choices strike me as entirely reasonable and appropriate for a book at this level.

Another good pedagogical choice made by the author was to spend an entire chapter giving multiple solutions to an interesting construction problem: you are given an angle whose vertex O is inaccessible, and therefore can’t be used in the construction. You are also given a point M in the interior of the angle. The problem is to construct the line OM, using as given two other points, one on either side of the angle. The chapter begins with an explanation of why it is desirable to have more than one solution to a problem (something undergraduates should know, but often don’t) and then offers no less than eight different solutions to this problem, illustrating a range of ideas and techniques.

The chapters of the text combine theoretical development of the material with solved problems illustrating the underlying geometric ideas. (For example, the first problem in the book asks the reader to explain, given three non-collinear points A, B and G, how to construct a triangle with A and B as two of its vertices and G as its centroid.) In addition, most of the chapters end with a handful of problems for the reader, solutions to which are available at the back of the book.

There were a few topics in plane Euclidean geometry whose omission from the text I thought was unfortunate. For example, given the fact that there are many different proofs of the Pythagorean theorem, covering a variety of different approaches and ideas, a chapter on that theorem would have been nice. I would have also liked to have seen a more extended discussion of impossibility results in geometric constructions. In addition, it seemed odd that the author would talk about Ceva’s theorem but not discuss the theorem of Menelaus, with which Ceva’s theorem is usually paired.

No book can cover everything, however, and the omission of these topics certainly falls within the author’s discretion. There is another omission, though, that I find much more serious, in fact indefensible: there is no index for this book. This seems to be a growing trend, one that I wish would come to an immediate halt.

This book reminded me in some ways of Geometry for College Students by Martin Isaacs, which has nontrivial overlap with this text and which also uses a mix of theoretical development and worked-out problems to illustrate the basic ideas. Isaacs’ book is more of a traditional text, though, and includes a lengthy (and, I find in my upper-level geometry course, necessary) review of the basic facts of Euclidean geometry: congruence, similarity, parallelograms, etc. The Isaacs text also goes more deeply into the theorems of Menelaus and Ceva, discusses some of the more sophisticated results about geometrical constructions, and has a chapter on vector geometry.

On the other hand, the Pritsker book covers some topics not found in Isaacs: there is, for example, a very nice chapter on the use of rotations in geometric theorem-proving; an instructor covering geometric transformations in class should find this very valuable. In addition, the shortness of this text, combined with the fact that it sells for less than one-third of the price of Geometry for College Students, makes it perhaps a very attractive candidate as supplemental reading for a geometry course.

Summary: the lack of an index is a seriously annoying feature, but, aside from that, this is a very pleasant book. The exposition is clear enough that a college student, or even a good high school student, could read it independently, and it is a very useful reference for any faculty member teaching a course in junior/senior undergraduate level geometry, especially a course with a population of aspiring secondary school teachers. I highly recommend it.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.