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The Zen of Magic Squares, Circles, and Stars:
An Exhibition of Surprising Structures across Dimensions
Clifford A. Pickover

Paperback | 2004 | $38.95 | £32.95 | ISBN: 9780691115979
432 pp. | 6 x 9 1/4 | 191 line illus.
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Humanity's love affair with mathematics and mysticism reached a critical juncture, legend has it, on the back of a turtle in ancient China. As Clifford Pickover briefly recounts in this enthralling book, the most comprehensive in decades on magic squares, Emperor Yu was supposedly strolling along the Yellow River one day around 2200 B.C. when he spotted the creature: its shell had a series of dots within squares. To Yu's amazement, each row of squares contained fifteen dots, as did the columns and diagonals. When he added any two cells opposite along a line through the center square, like 2 and 8, he always arrived at 10. The turtle, unwitting inspirer of the ''Yu'' square, went on to a life of courtly comfort and fame.

Pickover explains why Chinese emperors, Babylonian astrologer-priests, prehistoric cave people in France, and ancient Mayans of the Yucatan were convinced that magic squares--arrays filled with numbers or letters in certain arrangements--held the secret of the universe. Since the dawn of civilization, he writes, humans have invoked such patterns to ward off evil and bring good fortune. Yet who would have guessed that in the twenty-first century, mathematicians would be studying magic squares so immense and in so many dimensions that the objects defy ordinary human contemplation and visualization?

Readers are treated to a colorful history of magic squares and similar structures, their construction, and classification along with a remarkable variety of newly discovered objects ranging from ornate inlaid magic cubes to hypercubes. Illustrated examples occur throughout, with some patterns from the author's own experiments. The tesseracts, circles, spheres, and stars that he presents perfectly convey the age-old devotion of the math-minded to this Zenlike quest. Number lovers, puzzle aficionados, and math enthusiasts will treasure this rich and lively encyclopedia of one of the few areas of mathematics where the contributions of even nonspecialists count.


"A perpetual idea machine, Clifford Pickover is one of the most creative, original thinkers in the world today."--Journal of Recreational Mathematics

"Pickover just seems to exist in more dimensions than the rest of us."--Ian Stewart, Scientific American

"Clifford Pickover is many things--scientist, scholar, author, editor, and visionary."--Games

"It is a safe bet to conjecture that this is the best recreational mathematics book that will be published in this year. . . . Pickover writes with his usual style and straightforward simplicity in this book. The material is presented well and can be understood by anyone with a basic middle school mathematics background. This is a cool book!"--Charles Ashbacker, Journal of Recreational Mathematics

"Through accessible and readable prose and through detailed, highquality line illustrations, Pickover ably transports the general reader from culturally embedded traditional topics to a new and surprising frontier."--Harold Don Allen, Mathematics Teacher

"Pickover writes about his subject with contagious enthusiasm and comprehensive erudition."--Choice

"A splendid recreational book. . . . An extremely alluring page-turner."--Andrew Bremner, Notices of the American Mathematical Society


"A refreshing new look at a timeless topic, brimming over with ideas, littered with surprising twists. Anyone who loves numbers, anyone who enjoys puzzles, will find The Zen of Magic Squares, Circles, and Stars compulsive (and compulsory!) reading."--Ian Stewart, University of Warwick, author of Flatterland and Does God Play Dice?

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Table of Contents:

Preface xi
Acknowledgments xix
Introduction 1
CHAPTER ONE: Magic Construction 37
CHAPTER TWO: Classification 65
CHAPTER THREE: Gallery I : Squares, Cubes, and Tesseracts 147
CHAPTER FOUR: Gallery 2: Circles and Spheres 297
CHAPTER FIVE: Gallery 3: Stars, Hexagons, and Other Beauties 325
Some Final Thoughts 369
Notes 375
For Further Reading 395
Index 397
About the Author 403


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