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Is it possible to make mathematical drawings that help to understand mathematical ideas, proofs and arguments? The authors of this book are convinced that the answer is yes and the objective of this book is to show how some visualization techniques may be employed to produce pictures that have both mathematical and pedagogical interest.
Mathematical drawings related to proofs have been produced since antiquity in China, Arabia, Greece and India but only in the last thirty years has there been a growing interest in so-called "proofs without words". Hundreds of these have been publised in Mathematics Magazine and The College Mathematics Journal, as well as in other journals, books, and on the Internet.
Often times, a person encountering a "proof without words" may have the feeling that the pictures involved are the result of a serendipitous discovery or the consequence of an exceptional ingenuity on the part of the picture's creator. In this book the authors show that behind most of the pictures "proving" mathematical relations are some well-understood methods. As the reader shall see, a given mathematical idea or relation may have many different images that justify it, so that depending on the teaching level or the objectives for producing the pictures, one can choose the best alternative.
Content: Visualizing mathematics by creating pictures
Representing numbers by graphical elements
Representing numbers by lengths of segments
Representing numbers by areas of plane figures
Representing numbers by volumes of objects
Identifying key elements
Employing isometry
Employing similarity
Area-preserving transformations
Escaping from the plane
Overlaying tiles
Playing with several copies
Sequential frames
Geometric dissections
Moving frames
Iterative procedures
Introducing colors
Visualization by inclusion
Ingenuity in 3 D
Using 3D models
Combining techniques
Visualization in the classroom
Hints and solutions to the challenges