A Mathematical Nature Walk

Preface xv

Acknowledgments xix

Introduction 1

AT THE BEGINNING. 11

(General questions to challenge our powers of observation, estimation, and physical intuition)

Q.1-Q.6: Rainbows 11

Q.7: Shadows 11

Q.8-9: Clouds and cloud droplets 12

Q.10: Light 12

Q.11: Sound 12

Q.12-13: The rotation of the Earth 12

Q.14: The horizon 12

Q.15: The appearance of distant hills 12

IN THE "PLAYGROUND" 13

(just to get our feet wet.)

Q.16: Loch Ness--how long to empty it? 13

Q.17: The Grand Canyon--how long to fill it with sand? 14

Q.18: Just how large an area is a million acres? 15

Q.19: Twenty-five billion hamburgers--how many have you eaten? 16

Q.20: How many head of cattle would be required to satisfy the (1978) daily demand for meat in the United States? 16

Q.21: Why could King Kong never exist? 17

Q.22: Why do small bugs dislike taking showers? 18

Q.23: How fast is that raindrop falling? 18

Q.24: Why can haystacks explode if they're too big? 20

In the garden 24

Q.25: Why can I see the "whole universe" in my garden globe? 24

Q.26: How long is that bee going to collect nectar? 25

Q.27: Why are those drops on the spider's web so evenly spaced? 27

Q.28: What is the Fibonacci sequence? 31

Q.29: So what is the "golden angle"? 35

Q.30: Why are the angles between leaves "just so"? 36

IN THE NEIGHBORHOOD 43

Q.31: Can you infer fencepost (or bridge) "shapes"

just by walking past them? 43

Q.32: Can you weigh a pumpkin just by carefully looking at it? 48

Q.33: Can you determine the paths of low-flying ducks? 53

IN THE SHADOWS 58

Q.34: How high is that tree? (An estimate using elliptical light patches) 58

Q.35: Does my shadow accelerate? 59

Q.36: How long is the Earth's shadow? 61

Q.37: And Jupiter's? And Neptune's? 63

Q.38: How wide is the Moon's shadow? 63

IN THE SKY 64

Q.39: How far away is the horizon (neglecting refraction)? 64

Q.40: How far away is that cloud? 66

Q.41: How well is starlight reflected from a calm body of water? 67

Q.42: How heavy is that cloud? 71

Q.43: Why can we see farther in rain than in fog? 72

Q.44: How far away does that "road puddle" mirage appear to be? 73

Q.45: Why is the sky blue? 77

Q.46: So how much more is violet light scattered than red? 79

Q.47: What causes variation in colors of butterfly wings, bird plumage, and oil slicks? 80

Q.48: What causes the metallic colors in that cloud? 84

Q.49: How do rainbows form? And what are those fringes underneath the primary bow? 85

Q.50: What about the secondary rainbow? 92

Q.51: Are there higher-order rainbows? 93

Q.52: So what is that triple rainbow? 95

Q.53: Is there a "zeroth"-order rainbow? 98

Q.54: Can bubbles produce "rainbows"? 99

Q.55: What would "diamondbows" look like? 100

Q.56: What causes that ring around the Sun? 101

Q.57: What is that shaft of light above the setting Sun? 109

Q.58: What is that colored splotch of light beside the Sun? 111

Q.59: What's that "smiley face" in the sky? 113

Q.60: What are those colored rings around the shadow of my plane? 116

Q.61: Why does geometrical optics imply infinite intensity at the rainbow angle? 118

IN THE NEST 122

Q.62: How can you model the shape of birds' eggs? 122

Q.63: What is the sphericity index? 123

Q.64: Can the shape of an egg be modeled trigonometrically? 124

Q.65: Can the shape of an egg be modeled algebraically? 127

Q.66: Can the shape of an egg be modeled using calculus? 130

Q.67: Can the shape of an egg be modeled geometrically? 134

IN (OR ON) THE WATER 137

Q.68: What causes a glitter path? 137

Q.69: What is the path of wave intersections? 140

Q.70: How fast do waves move on the surface of water? 141

Q.71: How do moving ships produce that wave pattern? 148

Q.72: How do rocks in a flowing stream produce different patterns? 152

Q.73: Can waves be stopped by opposing streams? 154

Q.74: How far away is the storm? 157

Q.75: How fast is the calm region of that "puddle wave" expanding? 158

Q.76: How much energy do ocean waves have? 160

Q.77: Does a wave raise the average depth of the water? 162

Q.78: How can ship wakes prove the Earth is "round"? 164

In the forest 168

Q.79: How high can trees grow? 168

Q.80: How much shade does a layer of leaves provide for the layer below? 172

Q.81: What is the "murmur of the forest"? 174

Q.82: How opaque is a wood or forest? 176

Q.83: Why do some trees have "tumors"? 179

IN THE NATIONAL PARK 183

Q.84: What shapes are river meanders? 183

Q.85: Why are mountain shadows triangular? 189

Q.86: Why does Zion Arch appear circular? 191

IN THE NIGHT SKY 194

Q.87: How are star magnitudes measured? 194

Q.88: How can I stargaze with a flashlight? 196

Q.89: How can you model a star? 197

Q.90: How long would it take the Sun to collapse? 205

Q.91: What are those small rings around the Moon? 207

Q.92: How can you model an eclipse of the Sun? 210

AT THE END. 217

Q.93: How can you model walking? 217

Q.94: How "long" is that tree? 221

Q.95: What are those "rays" I sometimes see at or after sunset? 224

Q.96: How can twilight help determine the height of the atmosphere? 228

Appendix 1: A very short glossary of mathematical terms and functions 231

Appendix 2: Answers to questions 1-15 234

Appendix 3: Newton's law of cooling 238

Appendix 4: More mathematical patterns in nature 240

References 243

Index 247