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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 | |