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256 pages, 8 plates with 17 colour photos; 22 b/w photos, 97 b/w illustrations

How heavy is that cloud? Why can you see farther in rain than in fog? Why are the droplets on that spider web spaced apart so evenly? If you have ever asked questions like these while outdoors, and wondered how you might figure out the answers, this is a book for you.

An entertaining and informative collection of fascinating puzzles from the natural world around us, *A Mathematical Nature Walk* will delight anyone who loves nature or math or both.

John Adam presents ninety-six questions about many common natural phenomena – and a few uncommon ones – and then shows how to answer them using mostly basic mathematics. Can you weigh a pumpkin just by carefully looking at it? What causes the variations in the colors of butterfly wings, bird feathers, and oil slicks? And why are large haystacks prone to spontaneous combustion? These are just a few of the questions you'll find inside.

Many of the problems are illustrated with photos and drawings, and *A Mathematical Nature Walk* also has answers, a glossary of terms, and a list of some of the patterns found in nature. About a quarter of the questions can be answered with arithmetic, and many of the rest require only precalculus. But regardless of math background, readers will learn from the informal descriptions of the problems and gain a new appreciation of the beauty of nature and the mathematics that lies behind it.

"[A] snappy guide to the mathematics of the outdoors [...] A sharp eye and an ingenious mind are at work on every page [...] Read this book with pencil and paper in hand. Then go forth, enjoy the view, and impress your friends."

- Laurence A. Marschall, *Natural History*

"Mathematics professor John Adam has come up with a novel combination. This book will provide anyone with a solid grounding in mathematics with enough conversation starters to keep fellow walkers' brains working as hard as their legs."

- Dominic Lenton, *Engineering & Technology*

"A catalogue of playful inquiries and their mathematical solutions."

- *Conservation Magazine*

"Adam has written a terrific book that takes his earlier work a step further [...] [T]his is a well written guide not only to seeing our world with simplified and useful models and mathematics, but to asking good questions of what we see and then answering those questions on our own. I found the book delightful, engaging, and interesting. It's written for anyone with a calculus background, and that's all one needs. If you're looking for a fun book with a touch of complexity, this is a good one."

- David S. Mazel, *MAA Reviews*

"For teachers who are interested in seeing how what they teach might be used or for students or parents who might be interested in seeing how mathematics might be used, this is an intriguing book."

- *Mathematics Teacher*

"[A]dam's love of both nature and mathematics is obvious, and his chatty style and sense of humour – look out for the question about spontaneously combusting haystacks – enliven a book that will get readers thinking as well as itching for a pleasant stroll."

- *Physics World*

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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**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*.

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