Proofs Without Words II: More Exercises in Visual Thinking

edited by Roger B. Nelson

The Mystery of the Missing Words

February 2003

Most of the "proofs" in this book will be of more use and interest to math teachers -- how many people want to see 10 different diagrams showing trigonometric identities? Still, much is of great use to teachers; the standard textbooks are often redundant in their figures and thus bore students in their predictability. As well, students are in need of stretching their mathematical intuitions and understanding -- when all right triangles are shown with the legs parallel to the pages' sides, when all variables are either x or y, people think that math is a matter of grinding through standard procedures. Proofs Without Words II links subjects that are usually treated disparately: geometry is connected to combinatorics (a fancy name for counting), calculus, and linear algebra. Many of the most technical figures are accompanied by equations and words explaining the use. The back cover blurb does admit that many of the proofs aren't actually wordless, but I'm sure no one will sue them for false advertising -- these problems would be difficult to interpret otherwise.

However, the math enthusiast will be most rewarded by the figures with the least amount of words, for they provide mini-mysteries to be solved. The book starts with six different figures proving the Pythagorean Theorem -- and only one has any kind of "language" on it (a few variables and their products). These examples' author range from a 10th century Arab mathematician, a 3rd century Chinese mathematician, and Leonardo da Vinci to current contributors to the journals from the Mathematical Association of America. A few of these proofs are easier to interpret, as they involve cutting up squares into various pieces and rearranging them (but how to prove these various pieces are congruent?), one uses similar triangles, and the da Vinci one still has me in a fog. The Arabic proof was very elegant and involves a tiling pattern that looks like it would work well on a modern kitchen floor. The most elegant of all the proofs in the book is the one on the front cover: a geometric series represented by stacked equilateral triangles, fitting inside a larger triangle. The combinatorics proofs are also very elegant ways to visualize special sums and numbers.

None of these figures would be considered proofs by most people because one needs to have various parts explained; however, all crucial parts of the proofs are in the figures. As the editor writes, what makes these proofs good is that they show why a statement is true. Many mathematicians discover new theorems by playing around with figures representing already known objects -- some of these figures can show how certain relations were discovered. Unfortunately, we are usually shown a cleaned-up, perfectly deductive version of theorems in school, making it difficult for us to make the leap to new mathematical discoveries and understandings. The figures in this book make for a good course of mental calisthenics, and they provide inspiration to one to find one's own visualizations.

I recommend this book for high school and college math teachers, particularly those who teach trigonometry, calculus, and discrete math. If you know a student gifted in math, this book is appropriate for any student who is familiar with some basic geometry (Pythagorean Theorem, area of triangles and rectangles, similar triangles); they will be able to figure out a few of the counting and geometry proofs, and will grow into the other figures in time. For the intelligent child who enjoys math, this provides an extra challenge, and as they learn the math various proofs refer to, the pieces will fall into place.

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Mary Pat Campbell, last updated Feb 2003