# A Mathematical Nature Walk

by John A. Adam

"Finally a book that shows the general reader how mathematics can explain the natural phenomena that we continuously encounter but rarely understand. John Adam answers questions about nature's secrets—many of which we haven't even thought to ask. This is a delightful book."—**Alfred S. Posamentier, coauthor of The Fabulous Fibonacci Numbers**

## Overview

"Finally a book that shows the general reader how mathematics can explain the natural phenomena that we continuously encounter but rarely understand. John Adam answers questions about nature's secrets—many of which we haven't even thought to ask. This is a delightful book."—**Alfred S. Posamentier, coauthor of The Fabulous Fibonacci Numbers**

"John Adam's *A Mathematical Nature Walk* is a true gem of popular scientific writing. He adroitly does what all good science writers should do: he inspires readers first to observe and then to analyze the world outside their windows."—**Raymond Lee, author of The Rainbow Bridge**

"With a mathematician's eye and a playful wit, John Adam takes a walk through the woods and returns with stories aplenty! His narratives are about nature and how things work, about looking analytically at the world around us, and about the art of creating mathematical models. For anyone with a mathematical bent who has ever asked 'what is that?,' this book will provide an interesting read and a valuable resource."—**Kenneth G. Libbrecht, author of The Snowflake: Winter's Secret Beauty**

"Do not miss this memorable walk with John Adam, filled with delightful surprises that bring together nature, mathematics, and the infectious pleasure of thought, culminating in a special kind of wonder."—**Peter Pesic, author of Sky in a Bottle**

"For generations, field guides to plants and animals have sharpened the pleasure of seeing by opening our minds to understanding. Now John Adam has filled a gap in that venerable genre with his painstaking but simple mathematical descriptions of familiar, mundane physical phenomena. This is nothing less than a mathematical field guide to inanimate nature."—**Hans Christian von Baeyer, author of Information: The New Language of Science**

"When you see a spider's web bedecked with morning dew like strings of pearls or the lazy bends in a distant river valley, you are seeing mathematics as well as beauty. You will find equations in A Mathematical Nature Walk for the evanescent colors of the sky—as well as for why you can't fly over a rainbow. John Adam can help you see a world of algebra in a drop of water, and a Fibonacci sequence in a wild flower."—**Neil Downie, author of Vacuum Bazookas, Electric Rainbow Jelly, and 27 Other Saturday Science Projects**

"John Adam presents a wonderful set of mathematical inquiries into a broad range of natural phenomena. This rich book will be interesting to mathematically minded readers who are inspired by nature."—**Will Wilson, Duke University**

"In *A Mathematical Nature Walk*, John Adam encourages readers to explore everyday observations of the natural world from a mathematical point of view. The problems are presented in an engaging style and most of the mathematics is well within the grasp of beginning college students."—**Brian Sleeman, University of Leeds**

## Product Details

- ISBN-13:
- 9780691128955
- Publisher:
- Princeton University Press
- Publication date:
- 04/20/2009
- Pages:
- 280
- Product dimensions:
- 6.20(w) x 9.40(h) x 1.10(d)

## 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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