Keys to Infinity

Clifford A. Pickover
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"We live on a placid island of ignorance in the midst of black seas of infinity, and it is not meant that we should voyage far." -- H. P. Lovecraft
"The heavens call to you, and circle about you, displaying to you their eternal splendors, and your eye gazes only to earth." -- Dante
"I could be bounded in a nutshell and count myself a king of infinite space." -- Hamlet

   I think all of us first become interested in the concept of infinity early in childhood. Perhaps our initial fascination starts when we hear about large numbers, or outer space, or death, or eternity, or God. For example, when I was a boy, I often visited my father's library to examine his large collection of old books. The one that stimulated my early thoughts about infinity was not a mathematics book, nor a book on philosophy, nor one on religion. It was a history book published in 1921 titled The Story of Mankind. In the book, Hendrik Willem Van Loon starts with a little parable next to a sketch of a mountain:

High up in the North in the land called Svithjod, there stands a rock. It is a hundred miles high and a hundred miles wide. Once every thousand years a little bird comes to the rock to sharpen its beak. When the rock has thus been worn away, then a single day of eternity will have gone by.
This idea of eternity -- a temporal infinity -- is enough to start any child wondering about the inexhaustible fabric of numbers, space, and time.

   A few years later, I wandered through my father's library and was rewarded with another book by Van Loon titled The Arts, published in 1937. I pulled the dusty book from the shelf and was delighted to find the following philosophical gem, a conversation between a student and a wise, old teacher:

"Master, will you not tell us what the highest purpose may be to which mortal man may aspire?"

   A strange light now came into the eyes of Lao-Kung as he lifted himself from his seat. His trembling feet carried him across the room to the spot where stood the one picture that he loved best. It was a blade of grass, for within itself it contained the spirit of every blade of grass that had ever grown since the beginning of time.

   "There," the old man said, "is my answer. I have made myself equal of the Gods, for I too have touched the hem of Eternity."

Lao-Kung, like many of the ancient philosophers and writers, considered the concept of God to be intimately intertwined with the infinite. For example, facing this page is a view of Heaven from Dante's Divine Comedy showing the numbers of angels increasing to infinity the higher one ascends. St. Augustine believed not only that God was infinite, but also that God could think infinite thoughts. According to Augustine, God "knows all numbers".

   Augustine's works, along with the simple Van Loon quotes, provided a seed in an early childhood from which my interest in infinity and large numbers grew, and in particular provided an early stimulus for Keys to Infinity.

Infinite Worlds

"The trouble with integers is that we have examined only the small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions." -- Ronald Graham
Prepare yourself for a strange journey as Keys to Infinity unlocks the doors of your imagination with thought-provoking mysteries, puzzles, and problems on topics ranging from huge numbers to life itself. Each chapter is a world of paradox and mystery.

   For example, consider my favorite chapter "Welcome to Worm World" which describes the evolution of huge worms on checkerboard worlds. Readers of all ages can study their behavior with just a pencil and paper. (Growing international interest in this topic has led to a recent short publication in Discover Magazine.) In "Welcome to Worm World" you'll be among the first to learn about the "Internet Superhighway WormWorld Tournament" where researchers around the world competed to find the longest evolving worms. How do the worms' behavior change as Worm World grows to the size of our universe?

   Consider each chapter as a launch-pad for thinking and experimenting. For example, in "Ladders to Heaven" you are asked to imagine what it would be like to climb an incredibly long ladder stretching from the earth to the moon. You can only use ropes and other mountain climbing gear. Impossible, you say? Read further and find out what scientists have to say about such a gargantuan task.

   In "The Leviathan Number", you'll learn about a monstrous number so large as to make the number of electrons, protons, and neutrons in the universe pale in comparison. (It also makes a googol -- 1 followed by 100 zeros -- look kind of small.) In this chapter, you'll learn about large numbers beyond the ability of humans to grasp or compute, apocalyptic numbers, superfactorial functions, apocalyptic powers.... What can we know about numbers too large to compute or imagine?

   In "Fractal Milkshakes and Infinite Archery" you'll learn about a bubbly froth lurking in the fabric of our number system. The foam is comprised of an infinite regression of circles known as Ford circles. (A graphical representation of the froth is placed below the leading quotation in this book to whet your appetite.)

   In "Slides in Hell", you'll be asked to descend immense porous slides with zany mathematical properties. Want to gamble from which hole in the slide you'll fall? Have you ever dreamed of playing God, simulating life or preventing cancer? Then the chapter "Creating Life Using The Cancer Game" is for you. Want to fly through immense grids of dots -- as big as the universe -- with startling properties? Then take a look at "Grid of the Gods". Hop aboard a flying saucer stealing humans from earth, and compute the sex of the one-billionth abductee in "Alien Abduction Algebra."

   Keys to Infinity is for anyone who has pondered the immensity of numbers, dreamed of daring challenges, and wondered about the infinitely small. I hope that Keys to Infinity will stimulate creative thinking, enhance computer programming skills, and suggest the usefulness of simple mathematics for solving curious, practical, or mind-shattering problems. BASIC and C source programs are included for those of you who own computers. Some of the larger programs are gathered together in an Appendix at the end of the book.

My Keys

"I looked round the trees. The thin net of reality. These trees, this sun. I was infinitely far from home. The profoundest distances are never geographical." -- John Fowles
To help you on your journey, I offer various keys:
  1. Essays on all of the previously mentioned topics and more, everything from vampire numbers to the loom of creation.
  2. Puzzles, such as the fiendishly difficult "Cyclotron Puzzle", with hints to remind you there're often more ways of looking at the world than are immediately obvious.
  3. Quotations from novelists, philosophers, and famous scientists.
  4. Program Codes, so you can experiment further using personal computers as an aid to your pencil and paper explorations.
  5. Fractal and other images of infinity to stimulate your imagination. (Fractals are intricately shaped objects that reveal infinite detail as they are continually magnified.)
Some of the topics in the book may appear to be curiosities, with little practical application or purpose. However, I have found all of these experiments to be useful and educational, as have the many students, educators, and scientists who have written to me during the last few years. It is also important to keep in mind that throughout history, experiments, ideas and conclusions originating in the play of the mind have found striking and unexpected practical applications. I urge you to explore all of the topics in this book with this principle in mind.

   As in all my previous books, you are encouraged to pick and choose from the smorgasbord of topics. Many of the articles are brief and give you just a flavor of an application or method. Often, additional information can be found in the referenced publications. In order to encourage your involvement, computational hints and recipes for producing some of the computer-drawn figures are provided. For many of you, seeing pseudocode will clarify concepts in ways mere words cannot.

   I have created all of the compter graphics images in Keys to Infinity and have provided a brief description of the color plates towards the end. The book chapters are arranged somewhat randomly to retain the playful spirit of the book, and to give you unexpected pleasures. Some of the more technical chapters are placed at the end. Throughout the book, there are suggested exercises for future experiments and thought, and directed reading lists. Some information is repeated so that each chapter contains sufficient background information, and you may therefore skip chapters. The basic philosophy of this book is that creative thinking is learned by experimenting.

   Perhaps I should say what this book is not about. It does not contain the standard number-crunching problems found in scientific texts -- most often these do not stimulate creativity, nor do they have artistic appeal. Also, the problems and topics in this book are not of a "linear" variety, where variables are fed into an equation and a succinct answer is returned. In fact many of the exercises are of the "stop-and-think" variety, and can be explored without using a computer.

   The book is not intended for mathematicians looking for a formal mathematical treatise. Various books in the past have given fascinating accounts of infinity in mathematics, culture and art. For example, Eli Maors' To Infinity and Beyond and Rudy Rucker's Infinity and the Mind describe the history of number theory and various ideas connected to the concept of infinity. Their topics include: number series, prime and irrational numbers, Cantor sets, and non-Euclidean geometries. They also discuss infinity in the Kabbalist and Christian concepts of God, and also astronomers' evolving concepts of the size and structure of the universe. Two other useful books are Stan Gibilisco's Reaching for Infinity and Ray Hemmings' and Dick Tahta's Images of Infinity (see General Reading).

   Since there have been so many excellent books on the subject of infinity, my current book, Keys to Infinity, attempts to provide unusual views on the way that the human mind makes sense of the world through the use of computer tools, games, puzzles, numbers, and mathematical relations. Many chapters directly touch on the concept of infinity, while others are meant to stimulate readers' minds in a more general sense regarding the unlimited extent of time, space, or quantity. I leave more direct discussions of infinity in number theory and culture to my predecessors.


"The universe is not only stranger than we imagine, it's stranger than we can imagine." - Arthur C. Clarke
Keys to Infinity emphasizes creativity, fun, and expansion of the mind. For most chapters, no specialized knowledge is required. As I just mentioned, even though there are many chapters with mathematical ideas and computer programming hints, almost all problems are of the "stop-and-think" variety that do not require programming or sophisticated mathematics to allow you to explore and imagine.

   Many of the questions I pose in the book are unanswered. Some may be unanswerable. As Stanford psychologist Roger Shepard recently noted at a Sante Fe Institute workshop on the limits of scientific knowledge, even if our computers and mathematical tools continue to improve, we may not understand the world any better. He says, "We may be headed toward a situation where knowledge is too complicated to understand." As Princeton astrophysicist Piet Hut has pointed out, the structure of the physical universe may represent the ultimate limit on human knowledge. John Horgan (Scientific American) believes that particle physicists may never be able to test theories that unify gravity and the other forces of nature because the predicted effects become apparent beyond the range of any conceivable experiment.

   Finally, Ralph Gomory, the former director of research at IBM, and who is now president of the Alfred P. Sloan foundation in New York City, believes that our educational system does not place enough emphasis on what is unknown or even unknowable. To solve this problem, the Sloan Foundation may start a program on the limits of knowledge. Hopefully Keys to Infinity will stimulate students in thinking about both the unknown and unknowable.

The Electric Smorgasbord

"There was from the very beginning no need for a struggle between the finite and infinite. The peace we are so eagerly seeking has been there all the time." -- D. T. Suzuki
In many chapters of Keys to Infinity I quote colleagues from around the world who have responded to my questions, and I thank them for permission to reproduce excerpts from their comments. My questions were sent through electronic mail and often posted to electronic bulletin boards. Two common sources for such information exchange were "rec.puzzles" and "sci.math" -- electronic bulletin boards (or "newsgroups") that are part of a large, worldwide network of interconnected computers called Usenet. (The computers exchange news articles with each other on a voluntary basis.)

   Some define "Usenet" as the set of people (not computers) who exchange puzzles, tips, and news articles tagged with one or more universally-recognized labels signifying a particular "newsgroup". There are thousands of newgroups on topics ranging from bicycles, to physics, to music. Usenet started out at Duke University around 1980 as a small network of UNIX machines. Today there is no UNIX limitation, and there are versions of the news-exchange programs that run on computers ranging from DOS PCs to mainframes. Most Usenet sites are at universities, research labs, or other academic and commercial institutions. The largest concentrations of Usenet sites outside the U.S. seem to be in Canada, Europe, Australia and Japan.

General Reading

"Mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won't ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite" - Paul Erd&oe.s
   1. Maor, E. (1991) To Infinity and Beyond. Princeton Univ. Press: New Jersey.

   2. Rucker, R. (1983) Infinity and the Mind. Bantam: New York.

   3. Hemmings, R. and Tahta, D. (1984) Images of Infinity. Leapfrogs Insight Series: Vermont.

   4. Gibilisco, S. (1990) Reaching for Infinity. TAB Books: Pennsylvania.

   5. Gamow, G. (1988) One, Two, Three... Infinity. Dover: New York

   6. Horgan, J. (1994) Anti-omniscience. Scientific American. August, 271(2): 20-22.

   7. Kasner, E., and Newman, J. (1989) Mathematics and the Imagination. Tempus: Redmond, Washington.

   8. Schroeder, M. (1991) Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. Freeman: New York. Amazon.Com in separate window.

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