Keys to Infinity

357 Pages·1995·5.89 MB·English

by  Clifford A. Pickover

KEYS TO INFINITY
KEYS TO INFINITY
Clifford A. Pickover
John Wiley & Sons, Inc.
New York • Chichester • Brisbane • Toronto • Singapore
This text is printed on acid-free paper.
Copyright © 1995 by Clifford A. Pickover
Published by John Wiley & Sons, Inc.
Design, composition, editorial services, and production management provided by
Professional Book Center, Denver, Colorado.
Unless noted, all illustrations have been created by the author or are reproduced from copy-
right-free sources. The two-dimensional color computer graphic renditions in the insert were
produced using custom C-language programs running on an IBM RISC System/6000 or on
an IBM POWER Visualization System. IBM's software Visualization Data Explorer and the
Piatt Galsoft renderer were used to aid in the production of the three-dimensional computer
graphics.
Reproduction or translation of any part of this work beyond that permitted by Section 107 or
108 of the 1976 United States Copyright Act without the permission of the copyright owner
is unlawful. Requests for permission or further information should be addressed to the Per-
missions Department, John Wiley & Sons, Inc.
This publication is designed to provide accurate and authoritative information in regard to
the subject matter covered. It is sold with the understanding that the publisher is not engaged
in rendering professional services. If legal, accounting, medical, psychological, or any other
expert assistance is required, the services of a competent professional person should be
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Library of Congress Cataloging-in-Publication Data
Pickover, Clifford A.
Keys to infinity / Clifford A. Pickover.
p. cm.
ISBN 0-471-11857-5 (acid-free paper)
1. Infinite. I. Title.
QA9.P515 1996
793.7'4—dc20 94-45541
Printed in the United States of America
10 9 8 7 6 5 4 3 21
This book is dedicated to all those
who do not have a book
dedicated to them.
Acknowledgments
I thank Martin Gardner, J. Clint Sprott, Robert Stong, Arlin Anderson, Clay Fried,
and Manfred Schroeder for useful comments. I thank Don Webb for permission to
use our collaboration, "Valley of the Horses," and J. Clint Sprott for his permission
to use our collaboration, "Escape from Fractalia," in this book.
The angelic engravings interspersed throughout this book come from G. Dore,
The Dore Illustrations for Dante's Divine Comedy (Dover 1976). Born in Strasbourg
in 1832, Gustave Dore was perhaps the most successful illustrator of the nineteenth
century. Dore began work on these Divine Comedy illustrations in 1857 and usually
drew his designs directly onto wood blocks for printing.
The quotes from George Zebrowski come from an article titled "Is Science Ra-
tional?" that appeared in the June 1994 issue of OMNI magazine (p. 45). Permission
to reprint the lyrics from "The Song That Doesn't End" was granted by Norman L.
Martin, songwriter.
I have published some of the ideas in this book in the following scholarly and
professional journals: "The Loom of Creation" (Computers and Graphics), "Fractal
Milkshakes" {Leonardo), "Automated Computer Art" (Computers and Graphics),
"Recursive Worlds" (Dr. Dobb's Journal), "Random Number Generators" (The Vis-
ual Computer), "Fractal Batrachions" (Computers and Graphics), "Factorions"
(Mathematical Spectrum), "Undulation of Monks" (Mathematical Spectrum), "Vam-
pire Numbers" (Theta), "Fractal Curlicues" (The Visual Computer), "Carotid-Kun-
dalini Functions" (Fractal Report Newsletter), "Logit Terrain" (IEEE Computer
Graphics and Applications), and "Slides in Hell" (Skeptical Inquirer).
vi
Contents
Preface xiii
Chapter 1 Too Many Threes 3
Try It Yourself • The Search for Super-3 Numbers
Chapter 2 Ladders to Heaven 9
Life on Ladder World • Insight from Computer Programs • The Infinite
Ladder • Digressions
Chapter 3 Infinity Machines 17
Infinity Antilock Brakes • Infinity Keyboard • Infinity Program • Some
Closing History of Infinity • Paradoxes Past the Speed of Light
Chapter 4 Infinity World 25
Fifth Avenue • Ancient Maps • Worlds without End • Vertical Infinity
Earths • Sociopolitical Impact • Physical Constraints to Infinity World •
Alaska-Asia Land Bridge • Peninsular Anisotropy • One Supercontinent
• No Greece or Italy • Reader Questions • Horizontal Repetition on
Infinity World • Hemicentrism • Artificial Earths • Spiral Earths •
Mecca Maps and Heart-Shaped Maps
Chapter 5 Grid of the Gods 41
Digressions • Strahlkorper
Chapter 6 To the Valley of the Sea Horses 47
Chapter 7 The Million-Dollar, Trillion-Digit, Pi Sequencing Initiative 59
Ants, God, and Pi • The Grand Pi-Sequencing Project • A Rupture in
Geometry • Survey Results • Randomness • Microbes, Hermann
Schubert, and Pi • Try It Yourself
vii
viii KEYS TO INFINITY
Chapter 8 Infinite Chess 73
Digressions
Chapter 9 The Loom of Creation 79
Chapter 10 Slides in Hell 85
Simulation • Digressions
Chapter 11 Alien Abduction Algebra 91
Digressions
Chapter 12 The Leviathan Number 97
Eternity • Superfactorials, Super-Leviathans, and Incomputability
Chapter 13 Welcome to Worm World 105
The Internet Worm World Tournament • Abkhasian Areas • The Czech
Logical Labyrinth • Hyperworms in Abkhasia
Chapter 14 Fractal Milkshakes and Infinite Archery 117
Archery in the Great Beyond • Further Exploration • A Musical Tribute
• Postscript
Chapter 15 Creating Life Using the Cancer Game 127
Creation • The Game • Genetics, Programs, and Evolution • A 40-Day
Flood
Chapter 16 No Zeros Allowed 135
Try It Yourself
Chapter 17 Infinite Star Chambers 139
Chapter 18 Infinitely Exploding Circles 147
Not for the Meek
Chapter 19 The Infinity Worms of Callisto 153
Digression
CONTENTS ix
Chapter 20 The Undulation of the Monks 159
Other Undulants • Binary Undulants beyond Imagination
Chapter 21 The Fractal Golden Curlicue Is Cool 163
Curlicues and the Infinite • Postscript
Chapter 22 The Loneliness of the Factorions 169
Digression: Narcissistic Numbers • Proof
Chapter 23 Escape from Fractalia 173
Simple Equations Make Interesting Patterns • Computer Art Critic •
Sample Artwork • Further Suggestions
Chapter 24 Are Infinite Carotid-Kundalini Functions Fractal? 179
The Magical Fractal Gaps • A Programming Tip for Some BASIC
Users • Details
Chapter 25 The Crying of Fractal Batrachion 1,489 183
$10,000 Cash Award • Visualizing Infinity • Other Batrachions •
The Crying of Fractal Batrachions
Chapter 26 Ramanujan, Infinity, and the
Majesty of the Quattuordecillion 193
Numerical Sumo Wrestlers • Absolute Reality
Chapter 27 Recursive Worlds 197
Recursion • Recursive Lattices • Beauty and the Bits • Higher
Symmetry • Digressions
Chapter 28 Chaos in Ontario 217
Digressions and Contest • The Portuguese Maneuver
Chapter 29 Cyclotron Puzzles 221
Cyclotrons to Infinity • Solutions
KEYS TO INFINITY
Chapter 30 Vampire Numbers 227
Anne Rice • True Vampires versus Pseudovampires • Vampires and the
Infinite • Digressions
Chapter 31 Computers, Randomness, Mind, and Infinity 233
Humans Cannot Produce Random Numbers • The Lure of Random
Numbers • Randomness • The Cliff RNG • Noise-Spheres • Explore
Extraterrestrial Lands • The Bizarre Logistic Equation • The Logit
Transformation • Pretty Good Random Numbers • Flipping a Quantum
Mechanical Coin • Randomness and Infinity
Appendix 1 Program Code 249
Appendix 2 For Further Exploration 278
Notes 317
Further Reading 322
Index 325