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Proofs without Words: Exercises in Visual Thinking

Roger B. Nelsen
Mathematical Association of America
Publication Date: 
Number of Pages: 
Classroom Resource Materials 1
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Tom Schulte
, on

This slim collection of varied visual “proofs” (a term, it can be argued, loosely applied here) is entertaining and enlightening. I personally find such representations engaging and stimulating aids to that “aha!” moment when symbolic argument seems not to clarify. Since such pictures can be found sprinkled in the pages of many mathematics periodicals and even in the occasional textbook, others obviously feel the same way. A collection of a gross of them in a single volume is really a delight to peruse.

The proofs are arranged by topic into six main chapters: Geometry & Algebra, Trigonometry, Calculus & Analytic Geometry, Inequalities, Integer Sums, Sequences & Series, and a final chapter of Miscellaneous. Geometry & Algebra leads off with the classic subject for this approach: the Pythagorean Theorem. This includes one of my favorites, the one by President James Garfield. Having several to compare can be particularly illustrative (literally) of how proof-like reasoning can be done in an image. I would wager that any student that can apply the Pythagorean to find a missing side of a right triangle can also, with little or no help, figure out how most of these Pythagorean proofs “work”. Of course, this particular theorem lends itself very well to such an approach, which makes it very apt initial material

The Inequalities chapter presents various approaches to the five basic means and slightly more arcane inequalities, such as the Cauchy-Schwarz Inequality and those of Napier, Bernoulli, and Aristarchus. Up to this point an ambitious high school student or college student in first semester calculus would make fine progress.

This leads me to my only complaint here. This book is part of the MAA’s Classroom Resource Materials and as a teacher I find it an excellent resource to enliven the odd lecture. However, I also like to think of the independent mathematics enthusiast with this book in hand and in that situation I feel an opportunity was missed to add a few words of explanation here and there. For example, one could point out the geometric connection to topics such as inner product spaces and norms, or include a sentence or two on combinatorial notation, or describe the simple choreography that results in a cycloid.

Of course, very few pages here could really benefit from such added verbiage and a diligent reader will still enjoy the book even if a few of the proofs require further investigation. The collection does have a good explanation for the confection-inspired optical illusion of The Problem of Calissons. Along with non-attacking queens and characteristic polynomials, this is in the final miscellany section. Whether one is seeking to ornament undergraduate lectures or simply to exercise one’s mathematically-inclined mind, this book is a worthwhile purchase.

Tom Schulte surprises the occasional drowsy undergrad with visual proofs at Oakland Community College in Michigan.

The table of contents is not available.