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Preface to the Paperback Edition | |

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Preface: The motivation for the book; Acknowledgments; Credits | |

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Prologue: Why I Might Never Have Written This Book | |

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The Confluence of Nature and Mathematical Modeling | |

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Confluence | |

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Examples and qualitative discussion of patterns in nature | |

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Organization of the book | |

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Modeling | |

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Philosophy and methodology of modeling | |

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A mathematical model of snowball melting | |

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Estimation: The Power of Arithmetic in Solving Fermi Problems | |

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Various and sundry examples | |

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Golfballs | |

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Popcorn | |

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Soccer balls | |

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Cells | |

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Sand grains | |

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Human blood | |

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Loch Ness | |

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Dental floss | |

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Piano tuners | |

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Human hair | |

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The "dinosaur" asteroid | |

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Oil | |

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Leaves | |

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Grass | |

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Human population | |

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Surface area | |

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Volume | |

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Growth | |

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Newspaper [pi] | |

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The atmosphere | |

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Earth tunnel | |

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"Band" tectonics | |

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Mountains | |

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Cloud droplets | |

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The "Black Cloud" | |

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Shape, Size, and Similarity: The Problem of Scale | |

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what happens as things get bigger? | |

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Surface area/volume and strength/weight ratios and their implications for the living kingdom | |

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Geometric similarity | |

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Its usefulness and its limitations | |

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Falling | |

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Diving | |

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Jumping | |

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Flying | |

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Power output | |

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Running | |

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Walking | |

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Flying again | |

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Relative strength | |

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Cell viability | |

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The sphericity index | |

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Brain power | |

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Vision and hearing | |

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Dimetrodon | |

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The Buckingham [pi] theorem | |

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Various examples | |

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Models Based on Elastic Similarity | |

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Meteorological Optics I: Shadows, Crepuscular Rays, and Related Optical Phenomena | |

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Apparent size of the sun and moon | |

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Contrail shadows | |

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Tree pinhole cameras | |

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Length of the earth's shadow (and the moon's) | |

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Eclipses | |

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Reflections from a slightly rippled surface-glitter paths and liquid gold | |

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How thick is the atmosphere? | |

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Crepuscular rays and cloud distances | |

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Twilight glow | |

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The distance to the horizon | |

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How far does the moon fall each second? | |

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The apparent shape of the setting sun | |

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Why is the sky blue? | |

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Rayleigh scattering-a dimensional analysis argument | |

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A Word About Solid Angles | |

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Meteorological Optics II: A "Calculus I" Approach to Rainbows, Halos, and Glories | |

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Physical description and explanation of rainbows and supernumerary bows | |

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Derivation of Snell's law of refraction | |

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The primary bow | |

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The secondary bow | |

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A little about Airy's theory | |

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Halos-ice crystal formation and refraction by ice prisms | |

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Common halo phenomena (and some rarer forms) | |

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The circumhorizontal arc | |

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The glory | |

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Historical details | |

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Why some textbooks are wrong | |

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Snowflakes and the famous uniqueness question | |

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Mirages inferior and superior | |

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"Crocker Land" and the "Fata Morgana" | |

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The equations of ray paths | |

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Iridescence | |

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Birds | |

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Beetles and other bugs | |

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Interference of light in soap films and oil slicks | |

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Clouds, Sand Dunes, and Hurricanes | |

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Basic descriptions and basic cloud science | |

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Common cloud patterns-a descriptive account of cloud streets | |

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Billows | |

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Lee waves, and gravity waves | |

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Size and weight of a cloud | |

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Why can we see further in rain than in fog? | |

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Sand dunes | |

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Their formation and their possible relationship with cloud streets | |

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Booming dunes and squeaking sand | |

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Mayo's hurricane model | |

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More basic science and the corresponding equations | |

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Some numbers | |

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The kinetic energy of the storm | |

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(Linear) Waves of All Kinds | |

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Descriptive and introductory theoretical aspects | |

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The "wave equation" | |

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Gravity-capillarity waves | |

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Deep water waves | |

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Shallow water waves | |

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Plane wave solutions and dispersion relations | |

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Acoustic-gravity waves | |

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The influence of wind | |

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Planetary waves (Rossby waves) | |

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Wave speed and group speed | |

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An interesting observation about puddles | |

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Applications to water striders | |

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Edge waves and cusps | |

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Ship waves and wakes in deep and shallow water | |

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More Mathematics of Ship Waves | |

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Stability | |

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Kelvin-Helmholtz (shear) instability | |

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Internal gravity waves and wave energy | |

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Billow clouds again | |

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Convection and its clouds | |

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Effects of the earth's rotation | |

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The Taylor problem | |

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Spider webs and the stability of thin cylindrical films | |

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Bores and Nonlinear Waves | |

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Examples | |

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Basic mechanisms | |

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Mathematics of bores | |

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Hydraulic jumps | |

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Nonlinear wave equations | |

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Burger's equation | |

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Korteweg-de Vries equation | |

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Basic wavelike solutions | |

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Solitary waves | |

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Scott Russell's "great wave of translation" | |

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Tides | |

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Differential gravitational forces | |

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The power of "tide" | |

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The slowing power of tidal friction | |

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Tides | |

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Eclipses and the sun/moon density ratio | |

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The Fibonacci Sequence and the Golden Ratio ([tau]) | |

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Phyllotaxis | |

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The golden angle | |

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Regular pentagons and the golden ratio | |

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Some theorems on [tau] | |

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Rational approximations to irrational numbers | |

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Continued fraction representation of [tau] | |

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Convergents | |

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Misconceptions about [tau] | |

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Bees, Honeycombs, Bubbles, and Mud Cracks | |

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The honeycomb cell and its geometry | |

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Derivation of its surface area and consequent minimization | |

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Collecting nectar | |

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Optimizing visits to flowers | |

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Soap bubbles and minimal surfaces | |

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Plateau's rules | |

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The average geometric properties of foam | |

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The isoperimetric property of the circle and the same-area theorem | |

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Princess Dido and her isoperimetric problem | |

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Mud cracks and related geometric theorems | |

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The Isoperimetric Property of the Circle | |

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River Meanders, Branching Patterns, and Trees | |

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Basic description | |

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A Bessel function model | |

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Analogy of meanders with stresses in elastic wires | |

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Brief account of branching systems in rivers and trees | |

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River drainage patterns and the Fibonacci sequence again | |

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Trees | |

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Biomimetics | |

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The geometric proportions of trees and buckling | |

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Shaking of trees | |

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Geometric-, elastic-, and static stress similarity models | |

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How high can trees grow?-a Bessel function model | |

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The interception of light by leaves | |

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Aeolian tones | |

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The whispers of the forest | |

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The Statics and Bending of a Simple Beam | |

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Basic equations | |

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Bird Flight | |

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Wing loading | |

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Flapping flight | |

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Soaring flight | |

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Formation flight | |

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Drag and lift | |

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Sinking and gliding speeds | |

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Hovering | |

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Helicopters and hummingbirds | |

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Lift and Bernoulli-some misconceptions about lift | |

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Reynolds' number again | |

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The shape of water from a tap. | |

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How Did the Leopard Get Its Spots? | |

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Random walks and diffusion | |

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A simple derivation of the diffusion equation | |

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Animal and insect markings | |

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Morphogenesis | |

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The development of patterns | |

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Pattern formation by activator and inhibitor mechanisms | |

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Seashells | |

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Mechanisms of activation and inhibition | |

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Reaction-diffusion equations-a linear model | |

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Butterfly wing spots | |

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A simplistic but informative mathematical model | |

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Other applications of diffusion models | |

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The size of plankton blooms | |

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Earth(l)y applications of historical interest | |

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The diurnal and annual temperature variations below the surface | |

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The "age" of the earth | |

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The Analogy with the Normal Modes of Rectangular and Circular Membranes | |

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Fractals: An Appetite Whetter | |

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Bibliography | |

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Index | |