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 Khayyam, 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 Moebius 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