An original and exciting exploration of how utterly weird, and utterly beautiful, the infinite can be.-Ian Stewart, author of Does God Play Dice? What can we know about numbers too large to compute or even imagine? Do the tiny bubbles in the froth of a milkshake actually form an infinite fractal pattern? What are apocalyptic numbers and recursive worlds? These and dozens of equally beguiling mathematical mysteries, problems, and paradoxes fill this mind-bending new book. In each chapter, acclaimed author Clifford Pickover poses a delightful brain-teasing challenge that reveals the scope and splendor of the world of infinity. Try scaling the ladders to heaven, playing a game of infinite chess, or escaping from the land of Fractalia. Along the way you will encounter a myriad of intriguing topics from vampire numbers, to abduction algebra, to the infinity worms of Callisto. Every problem and puzzle is presented in a remarkably accessible style requiring no specialized mathematical knowledge. Over one hundred illustrations enhance the text and help to explain the mathematical concepts, and stunning color images created by the author reveal the breathtaking beauty of the patterns of infinity. A variety of computer programs offer additional ways to penetrate the enigma of infinity. For anyone who has ever wondered just how big infinity really is, or just how small, this book will provide an endless source of insight, creativity, and fun. Advance praise for KEYS TO INFINITY "In this the latest of Dr. Pickover's marvelous books, he breaks all finite chains to soar into the transcendental, mind-boggling regions of mathematical infinity. Written in the author's informal, clear style, it is a treasure trove of recreational problems, many published here for the first time, with special emphasis on computer programs and riveting graphics. As you soar, fasten your seat belt."-Martin Gardner, author of The Magic Numbers of Dr. Matrix "Inventive, quirky, fun! Pickover presents an engaging, inspiring romp in the realm of number and mathematical thought."-Ivars Peterson, author of The Mathematical Tourist "Join Pickover on his wonderful merry-go-round of ideas, and reach for the infinite. Keys to Infinity is an engaging book. . .a must for those wishing to explore the infinite in all its manifestations."-Theoni Pappas, author of The Joy of Mathematics "Keys to Infinity contains a near infinity of absorbing themes: from stepladders to the moon and spiral earths, to worm worlds, random chords, and self-similar curlicues. Fascinating!"-Manfred Schroeder, author of Fractals, Chaos, Power Laws "What could be more appropriate to the subject of infinity than a book like this one, so dense with wonderful puzzles, anecdotes, images, and computer programs that you could pore over it forever? In Keys to Infinity, Pickover has once again assembled a mathematical feast."-Carl Zimmer, Senior Editor Discover "Cliff Pickover has produced yet another book of mathematical puzzles, weird facts, computer art, and simple programs to challenge our minds and enthrall us with the beauty of the infinite mathematical world in which we live."-Dr. Julien C. Sprott, author of Strange Attractors