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

Description

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.

User’s Reviews

Editorial Reviews: Review “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 Review “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?”Pickcover 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”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”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 From the Back Cover “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?”Pickcover 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”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”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 About the Author Clifford Pickover is the author of over twenty books on a broad range of topics in science and art, a columnist for Odyssey, and an inventor. His books include Surfing Through Hyperspace: Understanding Higher Universes in Six Easy Lessons, Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning, and The Loom of God: Mathematical Tapestries at the Edge of Time. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The enthusiasm of the author for the subject comes through very clearly. He covers the history of magic squares and along with his obvious mathematical genius, he covers with great reverence the mystical and religious aspects of the subject. There is even a chapter that gives a number of methods for one to learn to make “instant” magic squares any time or place. The book is chock full of about every magic square, star, cube (or any other geometric construction) one could imagine. There are even tessaracts. Just looking at these magical constructs and making an effort to understand something about them is worth the price of admission. The book is so comprehensive on the subject that one could spend a very long time studying it-or one could benefit greatly by a good serious reading of it.William Swyter

⭐Awesome history about numbers and the beauty of magic squares!

⭐Let me be brief and to the point.I have been looking for a book on the subject of “Magic Squares” for sometime. When I first spotted this title< I was hooked by its appeal Thus, my expectations were very high. My expecations were exceeded significantly. Clifford Pickover is a brillian writer. John Tyler Gibson, Livonia, Michigan. ⭐this book brings answers that are not found in hundreds of other books dealing with the subject, it is a wonderful and reliable work! ⭐excellent! ⭐Thanks. ⭐This is a book about mathematical artifacts, but it has practically no mathematical content of its own. A casual reader who wants to gaze at these beautiful objects and come away impressed but with little understanding will find this a marvellous book. However, a mathematically inclined reader is not satisfied with someone declaring that an object has such-and-such a property, he wants to know WHY.Chapter 1 of this book gives dozens of fascinating constructions, but for most of them not a shred of proof is offered that the arrays produced are the magic squares Pickover claims. It leaves me wondering whether or not Pickover could produce such proofs himself, even for the more simple constructions in the book.Pickover describes some interesting computer experiments at the end of the chapter but seems completely stymied as to why they work. The demonstration is a lovely, but simple, piece of matrix theory that I would expect my first or second year Linear Algebra students to be able to perform.He shows two "brute-force" proofs for the order 3 case, one by Hendricks and "another" by Johnson (at least here is an attempt at including a proof), but annoyingly seems unaware that the second is just a minor variation on the first. I wonder if Pickover actually tried to follow these proofs himself or if he just copied them for his book.Mathematics is not a collection of statements that the hearer must accept on "authority", it is a systematic development of theory in which every statement can be, at least in principle, demonstrated by a logical argument. The mathematics is in understanding "why", not in the acceptance of fact. Without demonstration of the claims, all that is left is the shell with no life. Beautiful, like other shells we find along the shore, but not the genuine article itself.I am reminded somewhat of Stephen Hawking's popularizations of physics in which the reader is deeply impressed with the beauty of the subject, but comes away knowing practically no actual physics to speak of, for the author carefully seals the machinery of physics from his reader and presents only the glamorous face. In the case of Hawking, however, the author's authority is unquestionable; I'm sure he could, if pressed, demonstrate every claim in his books from first principles. I suspect that Pickover could not.Aside from a few excusable errors of fact, the book shares a serious omission with almost every book on magic squares that I have seen, in that it does not present what is surely the most elementary construction known for magic squares of any odd order, as the sum of a circulant and a back-circulant matrix. Even Pickover would be able to prove that this construction works, since the reason it works is extremely obvious. Given the connection of this construction to the very important subject of orthogonal Latin Squares, you would think a serious writer would devote some space to it.Aside from all of the above, the material in the book is comprehensive and fascinating, drawing on a number of sources, displaying many artifacts that have titillated dabblers for millennia. As a museum piece I'd have to give the book an "A", but as a piece of mathematics, only a "D". ⭐Magic squares have fascinated us for many centuries. Even in ancient Babylonian times, people considered these squares to have magical powers. Albrecht Dürer, the painter and printmaker, used them in his artworks.Most of the ideas in this book can be explored with just a pencil and paper! You can even discover new patterns in old magic squares that no one has ever found before. Even the famous eighteenth-century American Benjamin Franklin loved magic squares although he once considered them a waste of time.Pickover presents interesting people and their magic squares. From Benjamin Franklin's magic squares to four-dimensional magic tesseracts, the patterns fascinate us with their elegance. The book is a treasure and has gotten some rave reviews in the press. I enjoyed the magic spheres best of all, but I think each reader will find something new and interesting as they browse. A lot of magic squares deal with the chess board. Some focus on DNA sequences! A few were made by prisioners in jail. The author has certainly searched far and wide to assemble this massive collection.

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