
Mathematics in Nature
Modeling Patterns in the Natural World
By: John A. Adam
Paperback | 20 November 2006
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392 Pages
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From rainbows, river meanders, and shadows to spider webs, honeycombs, and the markings on animal coats, the visible world is full of patterns that can be described mathematically. Examining such readily observable phenomena, this book introduces readers to the beauty of nature as revealed by mathematics and the beauty of mathematics as revealed in nature.
Generously illustrated, written in an informal style, and replete with examples from everyday life, Mathematics in Nature is an excellent and undaunting introduction to the ideas and methods of mathematical modeling. It illustrates how mathematics can be used to formulate and solve puzzles observed in nature and to interpret the solutions. In the process, it teaches such topics as the art of estimation and the effects of scale, particularly what happens as things get bigger. Readers will develop an understanding of the symbiosis that exists between basic scientific principles and their mathematical expressions as well as a deeper appreciation for such natural phenomena as cloud formations, halos and glories, tree heights and leaf patterns, butterfly and moth wings, and even puddles and mud cracks.
Developed out of a university course, this book makes an ideal supplemental text for courses in applied mathematics and mathematical modeling. It will also appeal to mathematics educators and enthusiasts at all levels, and is designed so that it can be dipped into at leisure.
Industry Reviews
| Preface to the Paperback Edition | p. xiii |
| Preface: The motivation for the book; Acknowledgments; Credits | p. xv |
| Prologue: Why I Might Never Have Written This Book | p. xxiii |
| The Confluence of Nature and Mathematical Modeling | p. 1 |
| Confluence | |
| Examples and qualitative discussion of patterns in nature | |
| Organization of the book | |
| Modeling | |
| Philosophy and methodology of modeling | |
| A mathematical model of snowball melting | |
| Estimation: The Power of Arithmetic in Solving Fermi Problems | p. 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 | |
| Growth | |
| Newspaper [pi] | |
| The atmosphere | |
| Earth tunnel | |
| "Band" tectonics | |
| Mountains | |
| Cloud droplets | |
| The "Black Cloud" | |
| Shape, Size, and Similarity: The Problem of Scale | p. 31 |
| 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 | |
| The Buckingham [pi] theorem | |
| Various examples | |
| Models Based on Elastic Similarity | |
| Meteorological Optics I: Shadows, Crepuscular Rays, and Related Optical Phenomena | p. 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 | |
| A Word About Solid Angles | |
| Meteorological Optics II: A "Calculus I" Approach to Rainbows, Halos, and Glories | p. 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 | |
| Clouds, Sand Dunes, and Hurricanes | p. 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 | |
| (Linear) Waves of All Kinds | p. 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 | |
| More Mathematics of Ship Waves | |
| Stability | p. 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 | |
| Bores and Nonlinear Waves | p. 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 | |
| The Fibonacci Sequence and the Golden Ratio ([tau]) | p. 213 |
| Phyllotaxis | |
| The golden angle | |
| Regular pentagons and the golden ratio | |
| Some theorems on [tau] | |
| Rational approximations to irrational numbers | |
| Continued fraction representation of [tau] | |
| Convergents | |
| Misconceptions about [tau] | |
| Bees, Honeycombs, Bubbles, and Mud Cracks | p. 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 | |
| The Isoperimetric Property of the Circle | |
| River Meanders, Branching Patterns, and Trees | p. 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 | |
| The Statics and Bending of a Simple Beam | |
| Basic equations | |
| Bird Flight | p. 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. | |
| How Did the Leopard Get Its Spots? | p. 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 | |
| The Analogy with the Normal Modes of Rectangular and Circular Membranes | |
| Fractals: An Appetite Whetter | p. 336 |
| Bibliography | p. 341 |
| Index | p. 357 |
| Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9780691127965
ISBN-10: 0691127964
Published: 20th November 2006
Format: Paperback
Language: English
Number of Pages: 392
Audience: College, Tertiary and University
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.4 x 15.2 x 2.6
Weight (kg): 0.71
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