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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.
"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."â€”Dominic Lenton, Engineering & Technology
"A catalogue of playful inquiries and their mathematical solutions."â€”Conservation Magazine
"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."â€”Mathematics Teacher
"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."â€”David S. Mazel, MAA Reviews
"[A]dam's love of both nature and mathematics is obvious, and his chatty style and sense of humourâ€”look out for the question about spontaneously combusting haystacksâ€”enliven a book that will get readers thinking as well as itching for a pleasant stroll."â€”Physics World
"Indeed, Adam has deliberately reworked topics treated in Mathematics in Nature to make them accessible to a larger audience. Beyond insights into specific questions about nature, the general reader will find here a remarkably lucid explanation of how mathematicians create a formulaic model that mimics the key features of some natural phenomenon. Adam particularly highlights the importance in this process of solving inverse problems. Ordinary math becomes adventure."â€”Booklist
"If you are a walker, as I am, your daypack probably contains sunscreen, a poncho, a floppy hat, and a pair of binoculars. After reading this snappy guide to the mathematics of the outdoors, by John Adam, a professor of mathematics at Old Dominion University in Virginia, you might consider tossing in a programmable calculator. . . . 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."â€”Natural History
"There are now few (if any) areas of science where mathematics does not play a role and, by extension, many of the sights and sounds of nature can be studied using mathematics. This is the motivation behind A Mathematical Nature Walk by John Adam, which considers some of the natural phenomena that might be encountered on a walk in the countryside (or even just a wander around one's own garden)."â€”Sarah Shepherd, iSquared
"[S]urprising and entertaining. . . . Adam's book is lucidly written, making it suitable for people of all ages."â€”Good Book Guide
"The dedicated reader stands a lot to gain from delving into the text and thinking hard about the problems posed. As the saying goes, 'mathematics is not a spectator sport,' so if this book is read with pencil and paper at hand, to scribble along and confirm understanding of the mathematical trains of thoughtâ€”all the better."â€”Philip McIntosh, Suite101.com
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
DBrane
Posted January 2, 2010
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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Overview
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 ...