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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.
—Martin GardnerAbout a year after the publication of Proofs Without Words: Exercises in Visual Thinking by the Mathematical Association of America in 1993, William Dunham, in his delightful book The Mathematical Universe, An Alphabetical Journey through the Great Proofs, Problems, and Personalities (John Wiley & Sons, New York, 1994), wrote
Mathematicians admire proofs that are ingenious. But mathematicians especially admire proofs that are ingenious and economical—lean, spare arguments that cut directly to the heart of the matter and achieve their objectives with a striking immediacy. Such proofs are said to be elegant.
Mathematical elegance is not unlike that of other creative enterprises. It has much in common with the artistic elegance of a Monet canvas that depicts a French landscape with a few deft brushstrokes or a haiku poem that says more than its words. Elegance is ultimately an aesthetic, not a mathematical property.
… an ultimate elegance is achieved by what mathematicians call a “proof without words,” in which a brilliantly conceived diagram conveys a proof instantly, without need even for explanation. It is hard to get more elegant than that.
Since the books mentioned above were published, a second collection Proofs Without Words II: More Exercises in Visual Thinking was published by the MAA in 2000, and this book constitutes the third such collection of proofs without words (PWWs). I should note that this collection, like the first two, is necessarily incomplete. It does not include all PWWs that have appeared in print since the second collection appeared, or all of those that I overlooked in compiling the first two books. As readers of the Association's journals are well aware, new PWWs appear in print rather frequently, and they also appear now on the World Wide Web in formats superior to print, involving motion and viewer interaction.
I hope that the readers of this collection will find enjoyment in discovering or redis-covering some elegant visual demonstrations of certain mathematical ideas, that teachers will share them with their students, and that all will find stimulation and encouragement to create new proofs without words.
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