CHAPTER ONE Fun and Ga me 2
An Introduction to Rigorous Thought
.L Silly Stories Each with a Moral 4
Conundrums that Evoke 7Tchniques of Effective Thinking
1.2 Nudges 4
Leading Questions and Hints for Resolving the Stories
' The Punch Lines 18
Solutions and Further Commentary
a4 From Play to Power 27
Discovering Strategies of Thought for Life
CHAPTER TWO Number Contemplation 3s
2. Counting 40
How the Pigeonhole Principle Leads to Precision
Through Estimation
2.2 Numerical Patterns in Nature 49
Discovering the Beauty of the Fibonacci Numbers
2.3 Prime Cuts of Numbers 64
How the Prime Numbers Are the Building Blocks ofAll
Natural Numbers
2.A Crazy Clocks and Checking Out Bars 82
Cyclical Clock Arithmetic and Bar Codes
2.5 Public Secret Codes and How to Become a Spy 95
Encrypting Information Using Modular Arithmetic and Primes
The Irrational Side of Numbers 110
Are There Numbers Beyond Fractions?
Get Real 121
The Point of Decimals and Pinpointing Numbers on the Real Line
CHAPTER THREE 1.36
Beyond Numbers 138
What Does Infinity Mean?
Comparing the Infinite 145
Pairing Up Collections via a One-to-One Correspondence
The Missing Member 162
Georg Cantor Answers: Are Some Infinities Larger Than Others?
Travels Toward the Stratosphere of Infinities 1 73
The Power Set and the Question of an Infinite Galaxy of Ifinjities
Straightening Up the Circle 190
Exploring the Infinte Within Geometrical Objects
CHAPTER FOUR 206
Pythagoras and His Hypotenuse 208
I-low a Puzzle Leads to the Proof of One of the Gems
of Mathematics
A View of an Art Gallery 218
Using Computational Geometry to Place Security Cameras
in Museums
The Sexiest Rectangle 232
Finding Aesthetics in Life,,Art, and Math Through the
Golden Rectangle
Soothing Symmetry and Spinning Pinwheels 249
Can a Floor Be Tiled Without Any Repeating Pattern?
The Platonic Solids Turn Amorous 269
Discovering the Symmetry and Interconnections Among the
Platonic Solids
The Shape of Reality? 289
How Straight Lines Can Bend in Non-Euclidean Geometries
The Fourth Dimension 307
Can You See It?
CHAPTER FIVE Conrtortios of Space 326
5. Rubber Sheet Geometry 328
Discovering the Topological Idea of Equivalence by Distortion
5.2 The Band That Wouldn't Stop Playing 346
Experimenting with the MObius Band and Klein Bottle
5 3 Feeling Edgy? 359
Exploring Relationships Among Vertices, Edges, and Faces
S 4 Knots and Links 374
Untangling Ropes and Rings
5.5 Fixed Points, Hot Loops, nd Rainy Days 389
How the Certainty of Fixed Points Implies Certain
Weather Phenomena
CHAPTER SIX Chaos and Fracta s 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 Beauty of Fractals 430
How to Create Works of infinite Intricacy Through
Repeated Processes
6,4 The Mysterious Art of Imaginary Fractals 458
Creating Julia and Mandelbrot Sets by Stepping Out n the
Complex Plane
6.5 Predetermined Chaos 482
How Repeated Simple Processes Result in Utter Chaos
6.6 Between Dimensions 503
Can the Dimensions of Fractals Fall Through the Cracks?
CHAPTER SEVEN Taming Uncertainty 514
71 Chance Surprises 516
Some Scenarios Involving Chance That Confound Our Intuition
72 Predicting the Future in an Uncertain World 523
How to Measure Uncertainty Using the Idea of Probability
Random Thoughts 541
Are Coincidences as Truly Amazing as They First Appear?
Down for the Count 554
Systematically Counting All Possible Outcomes
Collecting Data Rather than Dust 571
The Power and Pitfalls of Statistics
What the Average American Has 585
Different Means of Describing Data
Parenting Peas, Twins, and Hypotheses 610
Making Inferences from Data
CHAPTER EIGHT j 628
Applications of Rigorous Thinking
Great Expectations 630
Deciding How to Weigh the Unknown Future
Risk 645
Deciding Personal and Public Policy
Money Matters 663
Deciding Between Faring Well and Welfare
Peril at the Polls 682
Deciding Who Actually Wins an Election
Cutting Cake for Greedy People 700
Deciding How to Slice Up Scarce Resources