"Pickover contemplates realms beyond our known reality."
-- The New York Times
"Pickover inspires a new generation of da Vincis to build unknown flying machines and create new Mona Lisas."
-- Christian Science Monitor
"Pickover has published nearly a book a year in which he stretches the limits of computers, art, and thought."
-- Los Angeles Times
"Pickover just seems to exist in more dimensions than the rest of us."
-- Ian Stewart, Scientific American
"Add two doses of Isaac Asimov, and one dose each of Martin Gardner and Carl Sagan, and you get Clifford Pickover, one of the most entertaining and thought provoking writers of our time." -- Michael Shermer, Editor-in-Chief of Skeptic
"Clifford A. Pickover is the heir apparent to Carl Sagan..." -- Robert J. Sawyer, author of Calculating God
"In recent years, Pickover has taken up the helm once worn by Isaac Asimov as the most compelling popular explainer of cutting-edge scientific ideas." -- In Pittsburgh
"Jews in Hyperspace: It's Mel Brooks meets Michio Kaku!" -- Mac Tonnies, author of After the Martian Apocalypse
"A Jewish Da Vinci Code" -- Mariana Tomas, entrepreneur
The Zen of Magic Squares, Circles, and Stars:
An Exhibition of Surprising Structures Across Dimensions
Clifford A. Pickover
| "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
Order from Amazon.com.
- "At first glance magic squares may seem frivolous (Ben Franklin's opinion, even as he spent countless hours studying them!), but I think that is wrong. The great nineteenth-century German mathematician Leopold Kronecker said 'God Himself made the whole numbers--everything else is the work of men,' and Cliff Pickover's stimulating book hints strongly at the possibility that God may have done more with the integers than just create them. I don't believe in magic in the physical world, but magic squares come as close as we will probably ever see to being mathematical magic."
- Paul J. Nahin, University of New Hampshire, author of Duelling Idiots and Other Probability Puzzlers
- "Clifford Pickover has compiled such a wonderfully voluminous collection of magic squares and related configurations that the physical book itself threatens to take the shape of a magic cube. Whether or not you achieve arithmetical satori contemplating these engagingly intricate patterns, you will surely come to appreciate their history, beauty, and richness."
- John Allen Paulos, Temple University, author of Innumeracy and A Mathematician Reads the Newspaper
- "Who would have thought that the simple numbers we used when learning to count as children could be arranged into so many geometric patterns with interesting properties? If playing with numbers is your thing, this latest work by Cliff Pickover will provide you will countless hours of mystical entertainment and mental challenges."
- Julien Clinton Sprott, University of Wisconsin-Madison
- "Pickover carries the mystique of magic squares and their relatives into the twenty-first century with his new book, The Zen of Magic Squares, Circles, and Stars. Whether you're seeking an introduction to magic squares, an in-depth study, some historical information, or just some enjoyable magic figure problems, tricks, properties, or novelties-this book is for you. Pickover does not leave a magic square unturned, and tantalizes us just enough to want to explore further. From the very famous to the less known, Pickover brings them all together in this amazing collection as he points out their roles in science, our lives and the universe."
- Theoni Pappas author of The Joy of Mathematics and Math-A-Day
- "Every generation seems to demand its own updated book dedicated to magic squares. Pickover's work meets the needs of the present generation. It is an enthusiastic, whimsical, and idiosyncratic compendium of magic squares and their variants, some of which are centuries old, some as fresh as corn picked this morning."
- Sherman Stein, University of California, Davis
- "Well documented and illustrated, this book is essentially a huge catalogue of these arithmetical beasts called magic squares. No book that I know of presents such a large range of 'mysterious, odd, and fun' patterns, as the author puts it. This includes the numerous geometrical diagrams that reveal some hidden symmetries not obvious to the eye contemplating the number arrays. Throughout, Pickover convincingly substantiates his claim that the field of magic square study is still wide open."
- Arturo Sangalli, author of The Importance of Being Fuzzy
- "Pickover inspires a new generation of da Vincis to build unknown flying machines and create new Mona Lisas."
- Christian Science Monitor
- "Pickover has published nearly a book a year in which he stretches the limits of computers, art, and thought."
- Los Angeles Times "Bucky Fuller thought big, Arthur C. Clarke thinks big, but Cliff Pickover outdoes them both."
- "Pickover just seems to exist in more dimensions than the rest of us."
- Ian Stewart, author of Does God Play Dice?
- "Strange and beuatiful, stunningly realistic."
- Scientific American
- "Pickover is van Leeuwenhoek's 20th century equivalent."
- "Clifford Pickover is many things -- scientist, scholar, author, editor, and visionary..." - GAMES
Jacket Flap: 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. (Continued below)
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.
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Table of Contents
Introduction and History
3 Gallery 1: Squares, Cubes, and Tesseracts
4 Gallery 2: Circles, and Spheres
5 Gallery 3: Stars and Other Beauties
Some Final Thoughts
For Further Reading
Clifford A. Pickover is the author of over twenty highly-acclaimed books, such as Time: A Traveler's Guide and Computers, Pattern, Chaos, and Beauty, and has written on diverse topics ranging from computers and creativity, art, mathematics, and astronomy-to human behavior and intelligence, time travel, alien life, and science fiction. Dr. Pickover received his Ph.D. from Yale University and is a prolific inventor. His web site pickover.com has received over a half-million visits.
A note on pentamagic squares. As described in my book, a magic square is p-MULTIMAGIC if the square formed by replacing each element by its kth power (for k=1, 2, ?, p) is also magic. (If this definition seems unclear, the book give numerous examples for you to study.) Bimagic squares (2-multimagic) are presented in the book, and references for trimagic squares (3-multimagic) are given. The new multimagic record is a pentamagic square (5-multimagic) built in 2001 by André Viricel and Christian Boyer. Look at Pentamagic, where you can study and download this pentamagic square. This website is in English, French and German.
Return to Cliff Pickover's home page which includes computer art, educational puzzles, higher dimensions, fractals, virtual caverns, JAVA/VRML, alien creatures, black hole artwork, and animations.