Mathematics and the Imagination

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ISBN: 9780486417035 | 0486417034
Cover: Paperback
Copyright: 3/28/2001

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Anyone who gambles, plays cards, loves puzzles, or simply seeks an intellectual challenge will love this amusing and thought-provoking book. With wit and clarity, the authors deftly progress from simple arithmetic to calculus and non-Euclidean geometry. Their subjects: assorted geometries-plane and fancy; famous puzzles that made mathematical history; tantalizing paradoxes; and more. "Charming and exciting"-Saturday Review of Literature. 169 text figures.

Introduction

New Names For Old

(24)

Easy words for hard ideas

Transcendental

Nonsimple 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

Beyond the Googol

(38)

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---Aleph0

Arithmetic for morons

Common sense hits a snag

Cardinality of the continuum

Extravagances of a mathematical madman

The tortoise unmasked

Motionless motion

Private life of a number

The house that Cantor built

π, i, e (PIE)

(47)

Chinamen and chandeliers

Twilight of common sense

π, 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 π

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 of Semantics

Imaginary numbers

The -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

Assorted Geometries---Plane and Fancy

112

(44)

The talking fish and St. Augustine

A new alphabet

High priests and numbo jumbo

Pure and applied mathematics

Euclid and Texas

Mathematical tailors

Geometry---a game

Ghosts, table-tipping, and the land of the dead

Foruth-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 tractix 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

Pastimes of Past and Present Times

156

(37)

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

Paradox Lost and Paradox Regained

193

(30)

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 do?

Chance and Chanceability

223

(42)

The clue of the billiard cue

A little chalk, a lot of talk

Watson gets his leg pulled by probable inference

Finds it all absurdly simple

Passionate oysters, waltzing ducks, and the syllogism

The twilight of probability

Interesting behavior of a modes coin

Biological necessity and a pair of dice

What is probability?

A poll of views: a meteorologist, a bootlegger, a bridge player

The subjective view---based on insufficient reason, contains an element of truth

The jackasses on Mars

The statistical view

What happens will probably happen

Experimental eurythmics; or, Pitching pennies

Relative frequencies

The adventure of the dancing men

Scheherezade and John Wilkes Booth---a challenge to statistics

The red and the black

Charles Peirce predicts the weather

How far is ``away''?

Herodotus explains

The calculus of chance

The benefits of gambling

De Mere and Pascal

Mr. Jevons omits an acknowledgment

The study of craps---the very guide of life

Dice, pennies, permutations, and combinations

Measuring probabilities

D'Alembert drops the ball

Count Buffon plays with a needle

The point

A black ball and a white ball

The binomial theorem

The calculus of probability re-examined

Found to rest on hypothesis

Laplace needs no hypothesis

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

Rubber-Sheet Geometry

265

(34)

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

Odd 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 Mobius strip

Just as black as it is painted

Forments 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

Change and Changeability

299

(58)

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

One 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 unbelieveable crisscross

Epilogue. Mathematics and the Imagination

357

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Mathematics and the Imagination

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