TABLE OF CONTENTS: 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
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