But First, a Word from our Sponsors ix Welcome! xi Surfing the Book xv Chapter One: Fun and Games: An Introduction to Rigorous Thought 2 1.1 Silly Stories Each with a Moral 4 Conundrums that Evoke Techniques of Effective Thinking 1.2 Nudges 4 Leading Questions and Hints for Resolving the Stories 1.3 The Punch Lines 18 Solutions and Further Commentary 1.4 From Play to Power 27 Discovering Strategies of Thought for Life Chapter Two: Number Contemplation 38 2.1 Counting 40 How the Pigeonhole Principle Leads to Precision Through Estimation 2.2 Numberical Patterns in Nature 49 Discovering the Beauty of the Fibonacci Numbers 2.3 Prime Cuts of Numbers 64 How the Primes Numbers are the Building Blocks of All Natural Numbers 2.4 Crazy Clocks and Checking Out Bars 82 Cyclical Clock Arithmetic and Bar Codes 2.5 Public Secret Codes adn How to Become a Spy 95 Encrypting Information Using Modula Arithmetic and Primes 2.6 The Irrational Side of Numbers 110 Are There Numbers Beyond Fractions? 2.7 Get Real 121 The Point of Decimals and Pinpointing Numbers on the Real Line Chapter Three: Infinity 136 3.1 Beyond Numbers 138 What Does Infinity Mean? 3.2 Comparing the Infinite 145 Pairing Up Collections via a One-to-One Corerspondence 3.3 The Missing Member 162 Georg Cantor Answers: Are Some Infinities Larger Than Others? Chapter Four: Geometric Gems 206 4.1 Pythagoras and His Hypotenuse 208 How a Puzzle Leads to the Proof of one of the Gems of Mathematics 4.2 A View of an Art Gallery 218 Using Computational Geometry to Place Security Cameras in Musuems 4.3 The Sexiest Rectangle 232 Finding Aesthetics in Life, Art, and Math Through the Golden Rectangle 4.4 Soothing Symmetry and Spinning Pinwheels 249 Can a Floor be Tiled without any Repeating Pattern? 4.5 The Platonic Solids Turn Amorous 269 Discovering the Symmetry and Interconnections Among the Platonic Solds 4.6 The Shape of Reality? 289 How Straight Lines Can Bend in Non-Euclidean Geomet
But First, a Word from our Sponsors ix Welcome! xi Surfing the Book xv Chapter One: Fun and Games: An Introduction to Rigorous Thought 2 1.1 Silly Stories Each with a Moral 4 Conundrums that Evoke Techniques of Effective Thinking 1.2 Nudges 4 Leading Questions and Hints for Resolving the Stories 1.3 The Punch Lines 18 Solutions and Further Commentary 1.4 From Play to Power 27 Discovering Strategies of Thought for Life Chapter Two: Number Contemplation 38 2.1 Counting 40 How the Pigeonhole Principle Leads to Precision Through Estimation 2.2 Numberical Patterns in Nature 49 Discovering the Beauty of the Fibonacci Numbers 2.3 Prime Cuts of Numbers 64 How the Primes Numbers are the Building Blocks of All Natural Numbers 2.4 Crazy Clocks and Checking Out Bars 82 Cyclical Clock Arithmetic and Bar Codes 2.5 Public Secret Codes adn How to Become a Spy 95 Encrypting Information Using Modula Arithmetic and Primes 2.6 The Irrational Side of Numbers 110 Are There Numbers Beyond Fractions? 2.7 Get Real 121 The Point of Decimals and Pinpointing Numbers on the Real Line Chapter Three: Infinity 136 3.1 Beyond Numbers 138 What Does Infinity Mean? 3.2 Comparing the Infinite 145 Pairing Up Collections via a One-to-One Corerspondence 3.3 The Missing Member 162 Georg Cantor Answers: Are Some Infinities Larger Than Others? Chapter Four: Geometric Gems 206 4.1 Pythagoras and His Hypotenuse 208 How a Puzzle Leads to the Proof of one of the Gems of Mathematics 4.2 A View of an Art Gallery 218 Using Computational Geometry to Place Security Cameras in Musuems 4.3 The Sexiest Rectangle 232 Finding Aesthetics in Life, Art, and Math Through the Golden Rectangle 4.4 Soothing Symmetry and Spinning Pinwheels 249 Can a Floor be Tiled without any Repeating Pattern? 4.5 The Platonic Solids Turn Amorous 269 Discovering the Symmetry and Interconnections Among the Platonic Solds 4.6 The Shape of Reality? 289 How Straight Lines Can Bend in Non-Euclidean Geometries 4.7 The Fouther Dimension 307 Can You See It? Chapter Five: Contortions of Space 326 5.1 Rubger Sheet Geometry 328 Discovering the Topological Idea of Equivalence by Distortion 5.2 The Band That Wouldn't Stop Playing 46 Experimenting with the Mobius Band and Klein Bottle 5.3 Feeling Edgy? 359 Exploring Relationships Among Vertices, Edges, and Faces 5.4 Knots and Links 374 Untangling Ropes and Rings 5.5 Fixed Points, Hot Loops, and Rainy Days 389 How the Certainty of Fixed Points Implies Certain Weather Phenomena Chapter Six: Chaos and Fractals 402 6.1 Images 404 Viewing a Gallery of Fractals 6.2 The Dynamics of Change 412 Can Change be Modeled by Repeated Applications of Simple Processes? 6.3 The Infinitely Detailed Beuty of Fractals 430 How to Create Workds of Ifinite Intricacy THrough Repeated PRocesses 6.4 The Mysterious Art of Imaginary Fractals 458 Creating Juila and Mandelbrot Sets by Stepping Out in the Complex Plane 6.5 Predetermined Chase 482 How Repeated SImple Processes Result in Utter Chaos 6.6 Between Dimnsions 503 Can the Dimensions of Fractals Fall Through the Cracks? Chapter Seven: Taming Uncertainty 514 7.1 Chance Surprises 516 Some Scenarioos Involving Chance that Confound our Intuition 7.2#
But First, a Word from our Sponsors ix Welcome! xi Surfing the Book xv Chapter One: Fun and Games: An Introduction to Rigorous Thought 2 1.1 Silly Stories Each with a Moral 4 Conundrums that Evoke Techniques of Effective Thinking 1.2 Nudges 4 Leading Questions and Hints for Resolving the Stories 1.3 The Punch Lines 18 Solutions and Further Commentary 1.4 From Play to Power 27 Discovering Strategies of Thought for Life Chapter Two: Number Contemplation 38 2.1 Counting 40 How the Pigeonhole Principle Leads to Precision Through Estimation 2.2 Numberical Patterns in Nature 49 Discovering the Beauty of the Fibonacci Numbers 2.3 Prime Cuts of Numbers 64 How the Primes Numbers are the Building Blocks of All Natural Numbers 2.4 Crazy Clocks and Checking Out Bars 82 Cyclical Clock Arithmetic and Bar Codes 2.5 Public Secret Codes adn How to Become a Spy 95 Encrypting Information Using Modula Arithmetic and Primes 2.6 The Irrational Side of Numbers 110 Are There Numbers Beyond Fractions? 2.7 Get Real 121 The Point of Decimals and Pinpointing Numbers on the Real Line Chapter Three: Infinity 136 3.1 Beyond Numbers 138 What Does Infinity Mean? 3.2 Comparing the Infinite 145 Pairing Up Collections via a One-to-One Corerspondence 3.3 The Missing Member 162 Georg Cantor Answers: Are Some Infinities Larger Than Others? Chapter Four: Geometric Gems 206 4.1 Pythagoras and His Hypotenuse 208 How a Puzzle Leads to the Proof of one of the Gems of Mathematics 4.2 A View of an Art Gallery 218 Using Computational Geometry to Place Security Cameras in Musuems 4.3 The Sexiest Rectangle 232 Finding Aesthetics in Life, Art, and Math Through the Golden Rectangle 4.4 Soothing Symmetry and Spinning Pinwheels 249 Can a Floor be Tiled without any Repeating Pattern? 4.5 The Platonic Solids Turn Amorous 269 Discovering the Symmetry and Interconnections Among the Platonic Solds 4.6 The Shape of Reality? 289 How Straight Lines Can Bend in Non-Euclidean Geomet
Table of Contents
But First, a Word from our Sponsors ix Welcome! xi Surfing the Book xv Chapter One: Fun and Games: An Introduction to Rigorous Thought 2 1.1 Silly Stories Each with a Moral 4 Conundrums that Evoke Techniques of Effective Thinking 1.2 Nudges 4 Leading Questions and Hints for Resolving the Stories 1.3 The Punch Lines 18 Solutions and Further Commentary 1.4 From Play to Power 27 Discovering Strategies of Thought for Life Chapter Two: Number Contemplation 38 2.1 Counting 40 How the Pigeonhole Principle Leads to Precision Through Estimation 2.2 Numberical Patterns in Nature 49 Discovering the Beauty of the Fibonacci Numbers 2.3 Prime Cuts of Numbers 64 How the Primes Numbers are the Building Blocks of All Natural Numbers 2.4 Crazy Clocks and Checking Out Bars 82 Cyclical Clock Arithmetic and Bar Codes 2.5 Public Secret Codes adn How to Become a Spy 95 Encrypting Information Using Modula Arithmetic and Primes 2.6 The Irrational Side of Numbers 110 Are There Numbers Beyond Fractions? 2.7 Get Real 121 The Point of Decimals and Pinpointing Numbers on the Real Line Chapter Three: Infinity 136 3.1 Beyond Numbers 138 What Does Infinity Mean? 3.2 Comparing the Infinite 145 Pairing Up Collections via a One-to-One Corerspondence 3.3 The Missing Member 162 Georg Cantor Answers: Are Some Infinities Larger Than Others? Chapter Four: Geometric Gems 206 4.1 Pythagoras and His Hypotenuse 208 How a Puzzle Leads to the Proof of one of the Gems of Mathematics 4.2 A View of an Art Gallery 218 Using Computational Geometry to Place Security Cameras in Musuems 4.3 The Sexiest Rectangle 232 Finding Aesthetics in Life, Art, and Math Through the Golden Rectangle 4.4 Soothing Symmetry and Spinning Pinwheels 249 Can a Floor be Tiled without any Repeating Pattern? 4.5 The Platonic Solids Turn Amorous 269 Discovering the Symmetry and Interconnections Among the Platonic Solds 4.6 The Shape of Reality? 289 How Straight Lines Can Bend in Non-Euclidean Geometries 4.7 The Fouther Dimension 307 Can You See It? Chapter Five: Contortions of Space 326 5.1 Rubger Sheet Geometry 328 Discovering the Topological Idea of Equivalence by Distortion 5.2 The Band That Wouldn't Stop Playing 46 Experimenting with the Mobius Band and Klein Bottle 5.3 Feeling Edgy? 359 Exploring Relationships Among Vertices, Edges, and Faces 5.4 Knots and Links 374 Untangling Ropes and Rings 5.5 Fixed Points, Hot Loops, and Rainy Days 389 How the Certainty of Fixed Points Implies Certain Weather Phenomena Chapter Six: Chaos and Fractals 402 6.1 Images 404 Viewing a Gallery of Fractals 6.2 The Dynamics of Change 412 Can Change be Modeled by Repeated Applications of Simple Processes? 6.3 The Infinitely Detailed Beuty of Fractals 430 How to Create Workds of Ifinite Intricacy THrough Repeated PRocesses 6.4 The Mysterious Art of Imaginary Fractals 458 Creating Juila and Mandelbrot Sets by Stepping Out in the Complex Plane 6.5 Predetermined Chase 482 How Repeated SImple Processes Result in Utter Chaos 6.6 Between Dimnsions 503 Can the Dimensions of Fractals Fall Through the Cracks? Chapter Seven: Taming Uncertainty 514 7.1 Chance Surprises 516 Some Scenarioos Involving Chance that Confound our Intuition 7.2#