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The Heart of Mathematics: An Invitation to Effective Thinking

Table of Contents

Chapter One: Fun and Games: An Introduction to Rigorous Thought

1.1 Silly Stories each with a Moral: Conundrums That Evoke Techniques of Effective Thinking

1.2 Nudges: Leading Questions and Hints for Resolving the Stories

1.3 The Punch Lines: Solutions and Further Commentary

1.4 From Play to Power: Discovering Strategies of Thought for Life

Chapter Two: Number Contemplation

2.1 Counting: How the Pigeonhole Principle Leads to Precision Through Estimation

2.2 Numerical Patterns in Nature: Discovering the Beauty of the Fibonacci Numbers

2.3 Prime Cuts of Numbers: How the Prime Numbers Are the Building Blocks of All Natural Numbers

2.4 Crazy Clocks and Checking Out Bars: Cyclical Clock Arithmetic and Bar Codes

2.5 Public Secret Codes and How to Become a Spy: Encrypting Information Using Modular Arithmetic and Primes

2.6 The Irrational Side of Numbers: Are There Numbers Beyond Fractions?

2.7 Get Real: The Point of Decimals and Pinpointing Numbers on the Real Line

Chapter Three: Infinity

3.1 Beyond Numbers: What Does Infinity Mean?

3.2 Comparing the Infinite: Pairing Up Collections via a One-to-One Correspondence

3.3 The Missing Member: Georg Cantor Answers: Are Some Infinities Larger Than Others?

3.4 Travels Toward the Stratosphere of Infinities: The Power Set and the Question of an Infinite Galaxy of Infinities

3.5 Straightening Up the Circle: Exploring the Infinite Within Geometrical Objects

Chapter Four: Geometric Gems 

4.1 Pythagoras and His Hypotenuse: How a Puzzle Leads to the Proof of One of the Gems of Mathematics

4.2 A View of an Art Gallery: Using Computational Geometry to Place Security Cameras in Museums

4.3 The Sexiest Rectangle: Finding Aesthetics in Life, Art, and Math Through the Golden Rectangle

4.4 Smoothing Symmetry and Spinning Pinwheels: Can a Floor Be Tiled Without Any Repeating Pattern?

4.5 The Platonic Solids Turn Amorous: Discovering the Symmetry and Interconnections Among the Platonic Solids

4.6 The Shape of Reality? How Straight Lines Can Bend in Non-Euclidean Geometries

4.7 The Fourth Dimension: Can You See It?

Chapter Five: Contortions of Space

5.1 Rubber Sheet Geometry: Discovering the Topological Idea of Equivalence by Distortion

5.2 The Band That Wouldn’t Stop Playing: Experimenting with the Möbius Band and Klein Bottle

5.3 Feeling Edgy? Exploring Relationships Among Vertices, Edges, and Faces

5.4 Knots and Links: Untangling Ropes and Rings

5.5 Fixed Points, Hot Loops, and Rainy Days: How the Certainty of Fixed Points Implies Certain Weather Phenomena

Chapter Six: Chaos and Fractals

6.1 Images: Viewing a Gallery of Fractals

6.2 The Dynamics of Change: Can Change Be Modeled by Repeated Applications of Simple Processes?

6.3 The Infinitely Detailed Beauty of Fractals: How to Create Works of Infinite Intricacy Through Repeated Processes

6.4 The Mysterious Art of Imaginary Fractals: Creating Julia and Mandelbrot Sets by Stepping Out in the Complex Plane

6.5 Predetermined Chaos: How Repeated Simple Processes Result in Utter Chaos

6.6 Between Dimensions: Can the Dimensions of Fractals Fall Through the Cracks?

Chapter Seven: Taming Uncertainty

7.1 Chance Surprises: Some Scenarios Involving Chance That Confound Our Intuition

7.2 Predicting the Future in an Uncertain World: How to Measure Uncertainty Using the Idea of Probability

7.3 Random Thoughts: Are Coincidences as Truly Amazing as They First Appear?

7.4 Down for the Count: Systematically Counting All Possible Outcomes

7.5 Collecting Data Rather Than Dust: The Power and Pitfalls of Statistics

7.6 What the Average American Has: Different Means of Describing Data

7.7 Parenting Peas, Twins, and Hypotheses: Making Inferences from Data

Chapter Eight: Deciding Wisely: Applications of Rigorous Thought

8.1 Great Expectations: Deciding How to Weigh the Unknown Future

8.2 Risk: Deciding Personal and Public Policy

8.3 Money Matters: Deciding Between Faring Well and Welfare

8.4 Peril at the Polls: Deciding Who Actually Wins an Election

8.5 Cutting Cake for Greedy People: Deciding How to Slice Up Scarce Resources

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