Table Of Content

W onders of Numbers W O R K S B Y C L I F F O R D A . P I C K O V E R The Alien IQ Test Black Holes: A Traveler's Guide Chaos and Fractals Chaos in Wonderland Computers, Pattern, Chaos, and Beauty Computers and the imagination Cryptorunes: Codes and Secret Writing Dreaming the Future Future Health: Computers and Medicine in the 21st Century Fractal Horizons: The Future Use of Fractals Frontiers of Scientific Visualization (with Stu Tewksbury) The Girl Who Gave Birth to Rabbits Keys to infinity The Loom of God Mazes for the Mind: Computers and the Unexpected The Pattern Book: Fractals, Art, and Nature The Science of Aliens Spider Legs (with Piers Anthony) Spiral Symmetry (with istvan Hargittai) Strange Brains and Genius Surfing Through Hyperspace Time: A Traveler's Guide Visions of the Future Visualizing Biological information The Zen of Magic Squares, Circles, and Stars D R . G O O G O L P R E S E N T S Wonders of Numbers Adventures in Mathematics, Mind, and Meaning Clifford A. Pickover OXPORD UNIVERSITY PRESS OXPORD UNIVERSITY PRESS Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Singapore Taipei Tokyo Toronto Copyright © 2001 by Clifford A. Pickover First published by Oxford University Press, Inc. 2001 First published as an Oxford University Press paperback, 2002 198 Madison Avenue, New York, New York 10016 www.oup.com Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval sys- tem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Pickover, Clifford A. Wonders of numbers: adventures in mathematics, mind, and meaning / by Clifford A. Pickover. p. cm. Includes bibliographical references and index. At head of title: Dr. Googol presents. ISBN 0-19-513342-0 (cloth) ISBN 0-19-515799-0 (Pbk.) 1. Mathematical recreations. 2. Number theory. I. Title. II. Title: Dr. Googol presents. QA95.P53 2000 793.7'4-dc21 99-27044 1 3 5 7 9 8 6 4 2 Printed in the United States of America on acid-free paper This book is dedicated not to a person but rather to an amusing mathematical wonder: the Apocalyptic Magic Square—a rather bizarre six-by-six magic square in which all of its entries are prime numbers (divisible only by themselves and 1), and each row, column, and diagonal sum to 666, the Number of the Beast. T H E A P O C A L Y P T I C M A G I C S Q U A R E 3 107 5 131 109 311 7 331 193 11 83 41 103 53 71 89 151 199 113 61 97 197 167 31 367 13 173 59 17 37 73 101 127 179 139 47 For additional wondrous features of this square, see Chapter 101. We are in the position of a little child entering a huge library whose walls are covered to the ceiling with books in many different tongues.The child does not understand the languages in which they are written. He notes a definite plan in the arrangement of books, a mysterious order which he does not comprehend, but only dimly suspects. —Albert Einstein Amusement is one of humankind's strongest motivating forces. Although mathematicians sometimes belittle a colleague's work by calling it "recreational" mathematics, much serious mathematics has come out of recreational problems, which test mathematical logic and reveal mathematical truths. —Ivars Peterson, Islands of Truth The mathematician's job is to transport us to new seas, while deepening the waters and lengthening horizons. —Dr. Francis 0. Googol Acknowledgments A C K N O W L E D G M E N T S F R O M C L I F F O R D A . P I C K O V E R Legendary mathematician Dr. Francis O. Googol currently resides on a small island off the coast of Sri Lanka. Because he desires privacy to continue his research, he has allowed my name to appear on this book's title page. In the past, I have frequently collaborated with Dr. Googol and edited his work. You can reach Dr. Googol by writing to me, and you can read more about the extraordi- nary life of Dr. Googol in the "Word from the Publisher" that follows this sec- tion. Dr. Googol admits to pillaging a few of my older papers, books, lectures, and patents for ideas, but he has brought them up to date with reader comments and startlingly fresh insight and presentation. A C K N O W L E D G M E N T S F R O M D R . F R A N C I S G O O G O L Martin Gardner and Ian Stewart, two scintillating stars in the universe of recreational mathematics and mathematics education, are always a source of inspira- tion. Martin Gardner, a mathematician, journalist, humorist, rationalist, and prolific author, has long stunned the world by giving countless people an incen- tive to study and become fascinated by mathematics. Many other individuals have provided intellectual stimulation over the years: Arthur C. Clarke, J. Clint Sprott, Ivars Peterson, Paul Hoffman, Theoni Pappas, Douglas Hofstader, Charles Ashbacher, Dorian Devins, Rudy Rucker, John Conway, Jack Cohen, and Isaac and Janet Asimov. Dr. Googol thanks Brian Mansfield for his creative advice and encouragement. Aside from drawing the various number mazes, Brian also created all of the car- toon representations of Dr. Googol from rare photographs in Googol's private archives. Dr. Googol also thanks Kevin Brown, Olivier Gerard, Dennis Gordon, Robert E. Stong, and Carl Speare for further advice and encouragement. He also owes a special debt of gratitude to Dr. John J. O'Connor and Pro- fessor Edmund F. Robertson (School of Mathematics and Statistics, University of St. Andrews, Scotland) for their wonderful "MacTutor History of Mathe- matics Archive," http://www-history.mcs.st-andrews.ac.uk/history/index.html. This web page allows users to access biographical data of more than 1300 mathematicians, and Dr. Googol used this wonderful archive extensively for back- ground information for Chapters 29, 33, and 38. A Word from the Publisher about Dr. Googol Francis Googol's date of birth is unknown. According to court records, he was born in London, England, and has held various "jobs" including mathematician, world explorer, and inventor. A prolific author of over 300 publications, Googol achieved his greatest fame with his book Number Madness, in which he argued that Neanderthals invented a primitive form of calculus. He also conducted pioneering studies of parabolas and statistics and was knighted in 1998. Dr. Googol is a prac- tical scientist, always testing his theories using apparatuses of his own design. Today, Dr. Googol has an obsessive predilection for quantifying anything that he views—from the curves of women's bodies to the number of brush strokes used to paint his portrait. It is rumored that he even published anonymously a paper in Nature on the length of rope necessary for breaking a criminal's neck without decapitation. In short, Googol is obsessed with the idea that anything can be counted, correlated, and understood as some sort of pattern. Clements Markham (former president of the Geographical Society) once remarked, "His mind is mathematical and statistical with little or no imagination." When asked his advice on life, Googol responded: "Travel and do mathematics." Francis Googol, great-great-great-grandson of Charles Darwin, was born to a family of bankers and gunsmiths of the Quaker faith. His family life was happy. Googol's mother, Violetta, lived to 91, and most of her children lived to their 90s or late 80s. Perhaps the longevity of his ancestors accounts for Googol's very long life. When Francis Googol was born, 13-year-old sister Elizabeth asked to be his primary caretaker. She placed Googol's cot in her room and began teaching him numbers, which he could point to and recognize before he could speak. He would cry if the numbers were removed from sight. As an adult, Googol became bored by life in England and felt the urge to explore the world. "I craved travel," he said, "as I did all adventure." For the next A Word from the Publisher about Dr. Googol © ix decade, he embarked on a shattering odyssey of self-discovery; in fact, his biog- raphy reads more like Pirsig's Zen and the Art of Motorcycle Maintenance or Simon's Jupiter's Travels than like the life story of a mathematical genius. Googol suddenly moved like a roller coaster over some of the world's most mysterious physical and psychological terrain: studies of the female monkeys at Kathmandu, camel rides through Egyptian desserts, death-defying escapes in the jungles of Tanzania. . . . Anyone who hears about Googol's journeys is enthralled by Googol's descriptions of the exotic places and people, by his ability to adjust to adversity, by his humor and incisiveness, but above all by the realization that to understand his world, he had to make himself vulnerable to it so that it could change him.