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A Mathematical Nature Walk

John A. Adam
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
David S. Mazel
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John Adam is starting to make quite a splash with his books on mathematics in nature. A few years ago, he penned the informative and lively Mathematics in Nature: Modeling Patterns in the Natural World which explored nature in great detail and with engaging prose and analyses.

Now Adam has written a terrific book that takes his earlier work a step further. This is, again, a mathematical treatment of nature but his approach is fun and enticing. This book asks the reader to journey with the author on his many walks. The author walks outside, looks at what is around him, and asks questions — 96 to be precise. The questions are sometimes simple, sometimes not so simple. But with the questions we find answers based a model (sometimes simplified but still useful) with diagrams, equations, and explanations that give the reader insight to the answers. What’s more, the derivations are well presented so that readers will want to explore the topics on their own.

Let’s look at some examples.

Question 17: The Grand Canyon — How long to fill it with sand? There are other “simplistic” questions like this one and we find here the idea of using our intuition with some (gross) approximations to derive an educated guess. It is reminiscent of Archimedes’s estimate of the number of poppy seeds to fill the universe. (See Archimedes: What Did He Do Besides Cry Eureka?)

Question 31: Can you infer fence post (or bridge) “shapes” just by walking past them? This question looks at an inverse problem. Adam begins his answer with a model of a fence made of either flat posts or round posts. He develops the model to show how light would appear through both sorts of fences as a function of the viewing angle. The treatment is methodical and explained in equations, text, diagrams, and graphs. (The answer is that you can tell the shapes by passing by the fence.)

There are some obvious questions with answers that are not always obvious, such as:

Question 45: Why is the sky blue? and others about rainbows, mirages, and bubbles with rainbows. Some of this material is a repeat of parts of his earlier book, but I welcomed their inclusion in the context of this book. And I was happy to relearn the material.

Sunrises and sunsets play a large role in the questions, as you would expect, but where I met a surprise is the role of ice in the sky. Here we find upper and lower sun pillars, shafts of light above the setting sun caused by the ice crystals in the atmosphere. The author then asks and answers how we can represent the shape of a bird’s egg, both algebraically and with calculus. Later, the author looks to the sky to discuss eclipses and a simple model for a star. For an ending, Adam models walking itself; while the model is crude, it shows how a little thought mixed with algebra and some simplifying assumptions allows us to study what is in front of us everyday.

Overall, this 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 received his Ph. D. from Georgia Tech in electrical engineering and is a practicing engineer in Washington, DC. His research interests are in the dynamics of billiards, signal processing, and cellular automata.


Preface xv
Acknowledgments xix
Introduction 1

(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

(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

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

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

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

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

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

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

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