Preface to the Paperback Edition ............................. xiii
Preface. The motivation for the book; Acknowledgments;
Credits .............................................. xv
Prologue. Why I Might Never Have Written This Book .......... xxiii
CHAPTER ONE
The Confluence of Nature and Mathematical Modeling .............. l
Confluence: examples and qualitative discussion of patterns
in nature; organization of the book. Modeling: philosophy
and methodology of modeling, appendix: A mathematical model
of snowball melting.
CHAPTER TWO
Estimation: The Power of Arithmetic in Solving Fermi
Problems ....................................................... 17
Various and sundry examples: golfballs, popcorn, soccer
balls, cells, sand grains, human blood, Loch Ness, dental
floss, piano tuners, human hair, the "dinosaur" asteroid,
oil, leaves, grass, human population, surface area, volume,
and growth, newspaper π, the atmosphere, earth tunnel,
"band" tectonics, mountains, cloud droplets, the "Black
Cloud."
CHAPTER THREE
Shape, Size, and Similarity: The Problem of Scale .............. 31
Dimensional analysis I — what happens as things get
bigger? Surface area/volume and strength/weight ratios
and their implications for the living kingdom; geometric
similarity, its usefulness and its limitations; falling,
diving, jumping, flying, power output, running, walking,
flying again, relative strength, cell viability. The
sphericity index, brain power, vision and hearing.
Dimetrodon. Dimensional analysis II — the Buckingham π
theorem; various examples. APPENDIX: models based on
elastic similarity.
CHAPTER FOUR
Meteorological Optics I: Shadows, Crepuscular Rays,
and Related Optical Phenomena .................................. 57
Apparent size of the sun and moon; contrail shadows; tree
pinhole cameras; length of the earth's shadow (and the
moon's); eclipses; reflections from a slightly rippled
surface—glitter paths and liquid gold; how thick is the
atmosphere? Crepuscular rays and cloud distances; twilight
glow; the distance to the horizon; how far does the moon
fall each second? The apparent shape of the setting sun. Why
is the sky blue? Rayleigh scattering—a dimensional analysis
argument. APPENDIX: a word about solid angles.
CHAPTER FIVE
Meteorological Optics II: A "Calculus I" Approach to
Rainbows, Halos, and Glories ................................... 80
Physical description and explanation of rainbows and
supernumerary bows. Derivation of Snell's law of
refraction. The primary bow; the secondary bow; a little
about Airy's theory. Halos—ice crystal formation and
refraction by ice prisms; common halo phenomena (and some
rarer forms); the circumhorizontal arc; the glory;
historical details; why some textbooks are wrong;
snowflakes and the famous uniqueness question; mirages,
inferior and superior; "Crocker Land" and the "Fata
Morgana"; the equations of ray paths; iridescence: birds,
beetles and other bugs; interference of light in soap films
and oil slicks.
CHAPTER SIX
Clouds, Sand Dunes, and Hurricanes ............................ 118
Basic descriptions and basic cloud science; common cloud
patterns— a descriptive account of cloud streets, billows,
lee waves, and gravity waves; size and weight of a cloud;
why can we see further in rain than in fog? Sand dunes,
their formation and their possible relationship with cloud
streets; booming dunes and squeaking sand; Mayo's hurricane
model; more basic science and the corresponding equations;
some numbers; the kinetic energy of the storm.
CHAPTER SEVEN
(Linear) Waves of All Kinds ................................... 139
Descriptive and introductory theoretical aspects; the
"wave equation"; gravity-capillarity waves; deep water
waves; shallow water waves; plane wave solutions and
dispersion relations; acoustic-gravity waves; the
influence of wind; planetary waves (Rossby waves); wave
speed and group speed; an interesting observation about
puddles; applications to water striders; edge waves and
cusps, ship waves and wakes in deep and shallow water.
APPENDIX: more mathematics of ship waves.
CHAPTER EIGHT
Stability ..................................................... 173
Kelvin-Helmholtz (shear) instability; internal gravity
waves and wave energy; billow clouds again; convection
and its clouds; effects of the earth's rotation; the
Taylor problem; spider webs and the stability of thin
cylindrical films.
CHAPTER NINE
Bores and Nonlinear Waves ..................................... 194
Examples; basic mechanisms; mathematics of bores;
hydraulic jumps; nonlinear wave equations: Burger's
equation; Korteweg-de Vries equation; basic wavelike
solutions; solitary waves; Scott Russell's "great wave
of translation"; tides: differential gravitational
forces; the power of "tide": the slowing power of tidal
friction; tides, eclipses and the sun/moon density ratio.
CHAPTER TEN
The Fibonacci Sequence and the Golden Ratio (τ) ............... 213
Phyllotaxis; the golden angle; regular pentagons and the
golden ratio; some theorems on τ; rational approximations
to irrational numbers; continued fraction representation
of τ; convergents; misconceptions about τ.
CHAPTER ELEVEN
Bees, Honeycombs, Bubbles, and Mud Cracks ..................... 231
The honeycomb cell and its geometry; derivation of its
surface area and consequent minimization; collecting
nectar: optimizing visits to flowers. Soap bubbles and
minimal surfaces. Plateau's rules; the average geometric
properties of foam; the isoperimetric property of the
circle and the same-area theorem; Princess Dido and her
isoperimetric problem; mud cracks and related geometric
theorems. appendix: the isoperimetric property of the circle.
CHAPTER TWELVE
River Meanders, Branching Patterns, and Trees ................. 254
Basic description; a Bessel function model; analogy of
meanders with stresses in elastic wires; brief account
of branching systems in rivers and trees; river drainage
patterns and the Fibonacci sequence again. Trees;
biomimetics; the geometric proportions of trees and
buckling; shaking of trees; geometric-, elastic-, and
static stress similarity models; how high can trees
grow? — a Bessel function model; the interception of
light by leaves; Aeolian tones; the whispers of the
forest, appendix: the statics and bending of a simple
beam: basic equations.
CHAPTER THIRTEEN
Bird Flight ................................................... 295
Wing loading; flapping flight; soaring flight; formation
flight; drag and lift; sinking and gliding speeds;
hovering; helicopters and hummingbirds. Lift and
Bernoulli — some misconceptions about lift; Reynolds'
number again. The shape of water from a tap.
CHAPTER FOURTEEN
How Did the Leopard Get Its Spots? ............................ 309
Random walks and diffusion; a simple derivation of the
diffusion equation; animal and insect markings;
morphogenesis: the development of patterns; pattern
formation by activator and inhibitor mechanisms;
seashells; mechanisms of activation and inhibition;
reaction-diffusion equations—a linear model; butterfly
wing spots: a simplistic but informative mathematical
model. Other applications of diffusion models: the
size of plankton blooms; earth(l)y applications of
historical interest: the diurnal and annual temperature
variations below the surface; the "age" of the earth.
APPENDIX: the analogy with the normal modes of rectangular
and circular membranes.
APPENDIX
Fractals: An Appetite Whetter ................................. 336
BIBLIOGRAPHY .................................................. 341
INDEX ......................................................... 357
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