For the Reader xi

Preface xiii

Chapter 1: Three Examples of the Mutual Embrace 1

1.1 Unphysical Laws 1

1.2 When Math Goes Wrong 6

1.3 Math from Physics 13

Chapter 2: Measuring Gravity 18

2.1 First, a Little Theory 18

2.2 Out in the Author's Garage 21

Chapter 3: Feynman's Infinite Circuit 24

3.1 An Infinity of Resistors 24

3.2 An Infinity of Reactances, and

Recursion 27

3.3 Convergence--or Not? 32

3.4 Three More Infinite, All-Resistor

Networks 36

Chapter 4: Air Drag--A Mathematical View 44

4.1 Air Drag Treated Broadly 44

4.2 Air Drag Treated with Some Detail 51

Chapter 5: Air Drag--A Physical View 62

5.1 The Quadratic Force Law 62

5.2 Long Falls through a Real Atmosphere 70

Chapter 6: Really Long Falls 82

6.1 Falling into the Sun 82

6.2 Falling from Heaven to Hell 86

Chapter 7: The Zeta Function--and Physics 94

7.1 A Curious Double Integral 94

7.2 Fourier Series and the Zeta Function 95

7.3 The Zeta Function in Physics 100

Chapter 8: Ballistics--With No Air Drag (Yet) 107

8.1 Shooting a Cannon in a Vacuum 107

8.2 What Makes a Champion Shot-Putter? 112

8.3 Another Cannon Question 116

Chapter 9: Ballistics--With Air Drag 120

9.1 Thin Air Cannot Be Ignored! 120

9.2 Air Drag and Baseball 126

Chapter 10: Gravity and Newton 136

10.1 The Beginnings of Modern Gravity 136

10.2 Newton's Superb Theorems 140

10.3 The Moon Test and Blowing-Up Planets 148

10.4 A Surprising Gravity Calculation 152

10.5 Gravitational Contraction 157

Chapter 11: Gravity Far Above the Earth 170

11.1 Kepler's Laws of Planetary Motion 170

11.2 Weighing the Planets 175

Chapter 12: Gravity Inside the Earth 186

12.1 Newton's Experiment 186

12.2 Gravity Inside the Earth 191

12.3 Pressure at the Center of the Earth 200

12.4 Travel Inside the Earth 203

12.5 Epilogue 209

Chapter 13: Quilts & Electricity 215

13.1 Recreational Mathematics 215

13.2 Electric Quilts 220

13.3 Three Impossibility Proofs 225

Chapter 14: Random Walks 233

14.1 Ronald Ross and the Flight of Mosquitoes 233

14.2 Karl Pearson Formulates a Famous Problem 236

14.3 Gambler's Ruin 241

14.4 The Monte Carlo Method 245

Chapter 15: Two More Random Walks 261

15.1 Brownian Motion 261

15.2 Shrinking Walks 269

Chapter 16: Nearest Neighbors 285

16.1 Cannibals Can Be Fun! 285

16.2 Neighbors Beyond the Nearest 291

16.3 What Happens When We Have Lots

of Cannibals 294

16.4 Serious Physics 296

Chapter 17: One Last Random Walk 299

17.1 Resistor Mathematics 299

17.2 Electric Walks 301

17.3 Monte Carlo Circuit Simulation 305

17.4 Symmetry, Superposition, and Resistor Circuits 313

Chapter 18: The Big Noise 321

18.1 An Interesting Textbook Problem 321

18.2 The Polar Equations of the Big-Noise Flight 322

18.3 The Acceleration on a Big-Noise Flight Path 328

SOLUTIONS TO THE CHALLENGE PROBLEMS 333

SPECIAL BONUS DISCUSSION 371

Warning: Do Not Read before Reading Disscussion 17 373

Chapter 19: Electricity in the Fourth Dimension 373

19.1 The Tesseract 373

19.2 Connecting a Tesseract Resistor Cube 376

Acknowledgments 385

Index 387