A Mathematical Nature Walk

John A. Adam is professor of mathematics at Old Dominion University. He is the coauthor of "Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin" and the author of "Mathematics in Nature" (both Princeton).

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