PROOFS WITHOUT WORDS:

EXERCISES IN VISUAL THINKING

ROGER B. NELSEN

Introduction

see (se) v., saw, seen, seeing. -v.t.

5. to perceive (things) mentally; discern;
understand: to see the point of an argument.

-THE RANDOM HOUSE DICTIONARY
OF THE ENGLISH LANGUAGE (2nd ED.)
UNABRIDGED.

"Proofs without words" (PWWS) have become regular features in the journals published by the Mathematical Association of America notably Mathematics Magazine and The College Mathematics Journal. PWWs began to appear in Mathematics Magazine about 1975, and, in an editors' note in the January 1976 issue of the Magazine, J. Arthur Seebach and Lynn Arthur Steen encouraged further contributions of PWWs to the Magazine. Although originally solicited for "use as end-of-article fillers," the editors went on to ask "What could be better for this purpose than a pleasing illustration that made an important mathematical point?"

A few years earlier Martin Gardner, in his popular "Mathematical Games" column in the October 1973 issue of the Scientific American, discussed PWWs as "look-see" diagrams. Gardner points out that "in many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance." This dramatically illustrates the dictionary quote above: in English "to see" is often "to understand."

In the same vein, the editorial policy of The College Mathematics journal throughout most of the 1980s stated that, in addition to expository articles, "The journal also invites other types of contributions, most notably: proofs without words, mathematical poetry, quotes, ..." (their italics). But PWWs are not recent innovations - they have a long history. Indeed, in this volume you will find modern renditions of proofs without words from ancient China, classical Greece, and India of the twelfth century.

Of course, "proofs without words" are not really proofs. As Theodore Eisenberg and Tommy Dreyfus note in their paper "'On the Reluctance to Visualize in Mathematics" [in Visualization in Teaching and Learning Mathematics, MAA Notes Number 191, some consider such visual arguments to be of little value, and "that there is one and only one way to communicate mathematics, and 'proofs without words' are not acceptable." But to counter this viewpoint, Eisenberg and Dreyfus go on to give us some quotes on the subject:

[Paul] Halmos, speaking of Solomon Lefschetz (editor of the Annals), stated: "He saw mathematics not as logic but as pictures." Speaking of what it takes to be a mathematician, he stated: "To be a scholar of mathematics you must be born with ... the ability to visualize" and most teachers try to develop this ability in their students. [George] P61ya's "Draw a figure ..." is classic pedagogic advice, and Einstein and Poincar6's views that we should use our visual intuitions are well known.

So, if "proofs without words" are not proofs, what are they? As you will see from this collection, this question does not have a simple, concise answer. But generally, PWWs are pictures or diagrams that help the observer see why a particular statement may be true, and also to see how one might begin to go about proving it true. In some an equation or two may appear in order to guide the observer in this process. But the emphasis is clearly on providing visual clues to the observer to stimulate mathematical thought.

I should note that this collection is not intended to be complete. It does not include all PWWs which have appeared in print, but is rather a sample representative of the genre. In addition, as readers of the Association's journals are well aware, new PWWs appear in print rather frequently, and I anticipate that this will continue. Perhaps some day a second volume of PWWs will appear!

I hope that the readers of this collection will find enjoyment in discovering or rediscovering some elegant visual demonstrations of certain mathematical ideas; that teachers will want to share many of them with their students; and that all will find stimulation and encouragement to try to create new "proofs without words."

Roger B. Nelsen
Lewis and Clark College
Portland, Oregon

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