Mathematics and the Imagination

Introduction I. NEW NAMES FOR OLD Easy words for hard ideas Transcendental Non-simple curve Simple curve Simple group Bolsheviks and giraffes Turbines Turns and slides Circles and cycles Patho-circles Clocks Hexagons and parhexagons "Radicals, hyperradicals, and ultraradicals (nonpolitical)" New numbers for the nursery Googol and googolplex Miracle of the rising book The mathescope II. BEYOND THE GOOGOL Counting?the language of number "Counting, matching, and "Going to Jerusalem" Cardinal numbers Cosmic chess and googols The sand reckoner Mathematical induction The infinite and its progeny Zeno Puzzles and quarrels Bolzano Galileo's puzzle Cantor Measuring the measuring rod The whole is no greater than some of its parts The first transfinite?Alepho Arithmetic for morons Common sense hits a snag Cardinality madman The tortoise unmasked Motionless motion Private life of a number The house that Cantor built III. "Pie, i, e (PIE)" Chinamen and chandeliers Twilight of common sense "Pie, i, e" Squaring the circle and its cousins Mathematical impossibility "Silk purse, sow's ear, ruler and compass" Rigor mortis Algebraic equations and transcendental numbers Galois and Greek epidemics Cube duplicators and angle trisectors Biography of pie "Infancy: Archimedes, the Bible, the Egyptians" "Adolescence: Vieta, Van Ceulen" "Maturity: Wallis, Newton, Leibniz" "Old Age: Dase, Richter, Shanks" Victim of schizophrenia Boon to insurance companies (e) Logarithms or tricks of the trade Mr. Briggs is surprised Mr. Napier explains "Biography of e; or e, the banker's boon" Pituitary gland of mathematics: the exponential function (i) "Humpty Dumpty, Doctor or Semantics" Imaginary numbers "The v-1, or "Where am I?" " "Biography of i, the self-made amphibian" "Omar Khayyám, Cardan, Bombelli, and Gauss" i and Soviet Russia Program music of mathematics "Breakfast in bed; or, How to become a great mathematician" Analytic geometry Geometric representation of i Complex plane "A famous formula, faith, and humility" IV. ASSORTED GEOMETRIES?PLANE AND FANCY The talking fish and St. Augustine A new alphabet High priests and mumbo jumbo Pure and applied mathematics Euclid and Texas Mathematical tailors Geometry?a game "Ghosts, table-tipping, and the land of the dead" Fourth-dimension flounders Henry More to the rescue Fourth-dimension?a new gusher A cure for arthritis Syntax suffers a setback The physicist's delight Dimensions and manifolds Distance formulae Scaling blank walls Four-dimensional geometry defined Moles and tesseracts A four-dimensional fancy Romance of flatland Three-dimensional cats and two-dimensional kings Gallant Gulliver and the gloves Beguiling voices and strange footprints Non-Euclidean geometry Space credos and millinery Private and public space Rewriting our textbooks The prince and the Boethians The flexible fifth The mathematicians unite?nothing to lose but their chains Lobachevsky breaks a link Riemann breaks another Checks and double checks in mathematics The tractrix and the pseudosphere Great circles and bears The skeptic persists?and is stepped on Geodesics Seventh Day Adventists Curvature Lobachevskian Eiffel Towers and Riemannian Holland Tunnels V. PASTIMES OF PAST AND PRESENT TIMES Puzzle acorns and mathematical oaks Charlemagne and crossword puzzles "Mark Twain and the "farmer's daughter" The syntax of puzzles Carolyn Flaubert and the cabin boy "A wolf, a goat, and a head of cabbage" Brides and cuckolds I'll be switched "Poisson, the misfit" "High finance; or, The international beer wolf" Lions and poker players The decimal system Casting out nines "Buddha, God, and the binary scale" "The march of culture; or, Russia, the home of the binary system" The Chinese rings The tower of Hanoi "The ritual of Benares: or, Charley horse in the Orient" "Nim, Sissa Ben Dahir, and Josephus" Bismarck plays the boss The 15 puzzle plague The spider and the fly A nightmare of relatives The magic square Take a number from 1 to 10 Fermat's last theorem Mathematics' lost legacy VI. PARADOX LOST AND PARADOX REGAINED Great paradoxes and distant relatives Three species of paradox Paradoxes strange but true Wheels that move faster on top than on bottom The cycloid family "The curse of transportation; or, How locomotives can't make up their minds" Reformation of geometry Ensuing troubles Point sets?the Arabian Nights of mathematics Hausdorff spins a tall tale Messrs. Banach and Tarski rub the magic lamp Baron Munchhausen is stymied by a pea Mathematical fallacies "Trouble from a bubble; or, Dividing by zero" The infinite?troublemaker par excellence Geometrical fallacies Logical paradoxes?the folk tales of mathematics Deluding dialectics of the poacher and the prince; of the introspective barber; of the number 111777; of this book and Confucius; of the Hon. Bertrand Russell "Scylla and Charybdis; or, What shall poor mathematics d "Twits Napoleon, who does" The Marquis de Condorcet has high hopes M. le Marquis omits a factor and loses his head Fourier of the Old Guard Dr. Darwin of the New The syllogism scraps a standby Mr. Socrates may not die "Ring out the old logic, ring in the new" VIII. RUBBER-SHEET GEOMETRY Seven bridges over a stein of beer Euler shivers Is warmed by news from home Invents topology Dissolves the dilemma of Sunday strollers Babies' cribs and Pythagoreans Talismen and queer figures Position is everything in topology Da Vinci and Dali Invariants Transformations The immutable derby "Competition for the caliph's cup; or, Sifting out the suitors by science" Mr. Jordan's theorem Only seems idiotic Deformed circles Old facts concerning Times Square and a balloonist's head Eccentric deportment of several distinguished gentlemen at Princeton Their passion for pretzels Their delving in doughnuts Enforced modesty of readers and authors The ring Lachrymose recital around a Paris pissoir "Who staggered how many times around the walls of what?" In and out the doughnut Gastric surgery?from doughnut to sausage in a single cut N-dimensional pretzels The Möbius strip Just as black as it is painted Foments industrial discontent Never takes sides Bane of painter and paintpot alike The iron rings "Mathematical cotillion; or, How on earth do I get rid of my partner?" "Topology?the pinnacle of perversity; or, Removing your vest without your coat" Down to earth?map coloring Four-color problem Euler's theorem The simplest universal law Brouwer's puzzle The search for invariants IX. CHANGE AND CHANGEABILITY The calculus and cement Meaning of change and rate of change Zeno and the movies "Flying Arrow" local?stops at all points" Geometry and genetics The arithmetic men dig pits Lamentable analogue of the boomerang History of the calculus Kepler Fermat Story of the great rectangle Newton and Leibniz Archimedes and the limit "Shrinking and swelling; or, "Will the circle go the limit?" Brief dictionary of mathematics and physics "Military idyll; or, The speed of the falling bomb" The calculus at work The derivative Higher derivatives and radius of curvature Laudable scholarship of automobile engineers The third derivative as a shock absorber The derivative finds its mate Integration Kepler and the bungholes "Measuring lengths; or, The yawning regress" Methods of approximation Measuring areas under curves Method of rectangular strips The definite Indefinite On the inverse of the other "The outline of history and the descent of man: or, y=ex" Sickly curves and orchidaceous ones The snowflake Infinite perimeters and postage stamps Anti-snowflake Super-colossal pathological specimen?the curve that fills space The unbelievable crisscross EPILOGUE. MATHEMATICS AND THE IMAGINATION