THE ZEN of MAGIC SQUARES, CIRCLES, and STARS Also by Clifford A

THE ZEN OF MAGIC SQUARES, CIRCLES, AND STARS Also by Clifford A. Pickover The Alien IQ Test Black Holes: A Traveler’s Guide Chaos and Fractals Chaos in Wonderland Computers and the Imagination Computers, Pattern, Chaos, and Beauty Cryptorunes Dreaming the Future Fractal Horizons: The Future Use of Fractals Frontiers of Scientific Visualization (with Stuart Tewksbury) Future Health: Computers and Medicine in the 21st Century The Girl Who Gave Birth t o Rabbits Keys t o Infinity The Loom of God Mazes for the Mind: Computers and the Unexpected The Paradox of God and the Science of Omniscience The Pattern Book: Fractals, Art, and Nature The Science of Aliens Spider Legs (with Piers Anthony) Spiral Symmetry (with Istvan Hargittai) The Stars of Heaven Strange Brains and Genius Surfing Through Hyperspace Time: A Traveler’s Guide Visions of the Future Visualizing Biological Information Wonders of Numbers THE ZEN OF MAGIC SQUARES, CIRCLES, AND STARS An Exhibition of Surprising Structures across Dimensions Clifford A. Pickover Princeton University Press Princeton and Oxford Copyright © 2002 by Clifford A. Pickover Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Cataloging-in-Publication Data Pickover, Clifford A. The zen of magic squares, circles, and stars : an exhibition of surprising structures across dimensions / Clifford A. Pickover. p. cm Includes bibliographical references and index. ISBN 0-691-07041-5 (acid-free paper) 1. Magic squares. 2. Mathematical recreations. I. Title. QA165.P53 2002 511'.64—dc21 2001027848 British Library Cataloging-in-Publication Data is available This book has been composed in Baskerville BE and Gill Sans. Printed on acid-free paper ∞ www.pup.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 The peculiar interest of magic squares lies in the fact that they possess the charm of mystery. They appear to betray some hidden intelligence which by a preconceived plan produces the impression of intentional design, a phenomenon which finds its close analogue in nature. —Paul Carus, in W. S. Andrews’s Magic Squares and Cubes The mathematical phenomenon always develops out of simple arithmetic, so useful in everyday life, out of numbers, those weapons of the gods: the gods are there, behind the wall, at play with numbers. —Le Corbusier, The Modulor To study magic squares is to study the self. To study the self is to forget the self. To forget the self is to be enlightened. —Abhinavagupta Isvarapratyabhijna1 The magic square is the hammer that shatters the ice of our unconscious. —Qingfu Chuzhen2 Translation: the Zen of magic squares. The phrase “magic squares” is written literally as “square puzzles.” On its own, this Chinese word for puzzle refers to the location of soldiers and weapons on a battlefield as described in Sun Zi’s Art of War. However, together with the Chinese word for “square,” the phrase denotes “magic squares.” The small square at the bottom encloses the name of the calligrapher, Siu-Leung Lee. This book is dedicated not to a person but to a meditative aid, the Durga Yantra, to which numbers can be applied in magic ways. Contents Preface xi Acknowledgments xix Introduction 1 C H A P T E R O N E Magic Construction 37 C H A P T E R T W O Classification 65 C H A P T E R T H R E E Gallery 1: Squares, Cubes, and Tesseracts 147 C H A P T E R F O U R Gallery 2: Circles and Spheres 297 C H A P T E R F I V E Gallery 3: Stars, Hexagons, and Other Beauties 325 Some Final Thoughts 369 Notes 375 For Further Reading 395 Index 397 About the Author 403 ix Preface Art and science will eventually be seen to be as closely connected as arms to the body. Both are vital elements of order and its discovery. The word “art” derives from the Indo-European base “ar,” meaning to join or fit together. In this sense, science, in the attempt to learn how and why things fit, becomes art. And when art is seen as the ability to do, make, apply or portray in a way that withstands the test of time, its connection with science becomes more clear. —Sven Carlson, Science News If we wish to understand the nature of the Universe we have an inner hidden advantage: we are ourselves little portions of the universe and so carry the answer within us. —Jacques Boivin, The Single Heart Field Theory Lepidoptera Overdrive Sometimes you are a butterfly collector, a person with a net. You don’t always know what the meadows will yield, but you know when the breeze is right, where the meadows are fertile, and what type of net to use. Often the specific catch is a surprise, and this is the enjoyment of the quest. There are no guarantees. There are often unexpected pleasures. Follow me as we search through unknown forests. Each structure we uncover contains magnificent patterns as beautiful as any swallowtail’s wing. Hopefully you will enjoy looking at the catches or magnifying them further to learn more about their internal structures. Sometimes you may feel as if you are walking through an exhibit of butterflies in a gallery. At other times you may feel as if you are standing on the edge of a gigantic crystal, surrounded by thousands of butterfly wings beating in unison, producing a hum that reminds you of the chanting of monks. xi xii Preface Beyond Magic Squares Magic squares have fascinated humans since the dawn of civiliza- tion. Even in ancient Babylonian times, people considered these squares to have magical powers, and in the eighth century A.D., some squares were considered useful for turning ordinary metal into gold. The patterns have also been used as religious symbols, protective charms, and tools for divination. When the squares lost their mystical meanings, laypeople continued to use them as fascinating puzzles, while seasoned mathematicians studied them as problems in number theory. Albrecht Dürer, the fourteenth- century painter and printmaker, used them in his artworks, and today magic squares continue to intrigue us with their elegant, beautiful, and strange symmetries. A magic square is a square array of integers in which the rows, columns, and diagonals have the same sums. However, in this book we will go far beyond ordinary magic squares and consider many unusual variations, some in higher dimensions, all with mind-boggling patterns. Many of the squares possess an intrinsic Preface xiii beauty and complexity hard to describe or understand without careful study. For example, some squares remind me of fractals, geometric patterns with similar structures contained within larger structures. The numerical diagrams tantalize us with hidden patterns as if some mathematician-God played a role in their design. In this book, you will find squares within squares; cubes within cubes; four-dimensional objects; patterns based on ancient, sacred geometry; and structures that defy classification. Mathematician Benoit Mandlebrot likens the creation of nested structures to Zen philosophy: They are repeated without end. Infinite regression is something fully described in Zen Buddhist books, and often appears in the works of Leibniz and Kant.1 While thinking about these perplexing patterns, you may be at first confused, but then you will find that your mind “clicks” as you suddenly appreciate the structure. It is not an exaggeration to think of this moment of understanding as a miniature epiph- any, a revelation, a piece of satori in which each magic structure is a mandala for the mind. Due to space limitations, I sometimes exhibit just a few hidden patterns to plant a seed, allowing you to consult the references or explore with a computer to make addi- tional discoveries. Revelation In my mind, we don’t invent magic squares, but rather we discover them. Magic squares are out there in the realm of eternal ideas. They have an independent existence from us. These ideas are con- troversial, and there are certainly other points of view.2 However, to me, magic squares transcend us and our physical reality. The statement “A magic square has particular symmetries and proper- ties” is either true or false. As you look through this book, you can see that it turns out to be true. Was the statement true before the invention and discovery of magic squares? I believe it was. Magic squares exist whether humans know about them or not. xiv Preface I think mathematics is a process of discovery. Mathematicians are like archeologists. Physicist Roger Penrose felt the same way about fractal geometry. In his book The Emperor’s New Mind, he says that fractals (for example, intricate patterns such as the Julia set or Mandelbrot set) are out there waiting to be found: It would seem that the Mandelbrot set is not just part of our minds, but it has a reality of its own . The computer is being used in essentially the same way that an experimental physicist uses a piece of experimental apparatus to explore the structure of the physical world. The Mandelbrot set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot set is just there.3 I think we are uncovering truths and ideas independent of the computer or mathematical tools we have invented. Penrose went a step further about fractals: When one sees a mathematical truth, one’s consciousness breaks through into this world of ideas.

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