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? Why can you see farther in rain than in fog? 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 the book 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.
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).
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
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"
Posamentier
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"
Will Wilson
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
Brian Sleeman
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
Neil Downie
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"
Libbrecht
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"
Hans Christian von Baeyer
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"
Peter Pesic
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"
A Mathematical Nature Walk 5 out of 5based on
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DBrane
More than 1 year ago
The author's previous book, Mathematics in Nature: Modeling Patterns in the Natural World got a great review in the June/July 2005 Notices of the AMS (a publication of the American Mathematical Society, it is one of the most prestigious math journals in the country).The last sentence of the review states: "On Growth and Form (D'Arcy Thompson) is a classic; Mathematics in Nature has the potential to become one too."
He recently published his third book A Mathematical Nature Walk and it is a gem. First I'll quote from the front flap. "How tall is that tree? How far away is that cloud and how heavy is it? Why are the droplets on that spider's web spaced apart so evenly? ." John Adam presents ninety-six questions about many natural phenomena. and then shows how to answer them using mostly basic mathematics. Can you weigh a pumpkin just by looking at it?"
This book is less technical than Mathematics in Nature, mostly pre calculus, and some very basic calculus and simple differential equations. There's more than enough information on each of the levels. When I showed the book to my calculus students they got very excited, some said that they were going to buy it as soon as possible!
At my college, for the past twenty years or so, I gave good students the opportunity to contract with me to do honors work and get honors credit for the course. This would entail doing a project, relevant to the course, not necessarily more difficult, but of interest to the student and not "run of the mill". No matter whether it was pre calculus or calculus I always had trouble finding appropriate topics. I wish I had this book years ago.
Other topics of interest.. Can the shape of an egg be modeled trigonometrically? algebraically? by calculus? by geometry? How far away is the storm? How high can a tree grow? Why do some trees have tumors? How long will it take the sun to collapse? His style is conversational. 'Why can haystacks explode if they're too big?' is quintessential John Adam!
I would say that this book will become a classic. I am beginning my forty sixth year of teaching and have taught at all levels from 8th grade pre-Algebra to graduate level mathematical physics. If I were an education administrator for high school math teachers (I taught high school math in New York City for thirteen years), I would mandate it as required reading. It should be a text for a course for budding math teachers. It would show the novice high school teacher and, of course, the veteran, how relatively easy math can have real life applications unlike those dumb word problems they teach in the traditional courses. I believe John Adam's book will ultimately be ranked on the same level as Polya's classic, How to Solve It.
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