Mathematics in Nature

by Adam, John A.

• ISBN13: 9780691127965
• ISBN10: 0691127964
• Format: Paperback
• Copyright: 2006-08-21
• Publisher: Princeton Univ Pr

Summary

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 Natureis 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. Professors: A supplementary Solutions Manual is available for this book. It is restricted to teachers using the text in courses. For information on how to obtain a copy, refer to: http://pup.princeton.edu/solutions.html

Author Biography

John A. Adam is Professor of Mathematics at Old Dominion University

Table of Contents

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	1	(16)
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	(14)
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	(26)
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	(23)
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	(38)
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	(21)
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	(34)
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	(21)
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	(19)
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	(18)
Phyllotaxis; the golden angle; regular pentagons and the golden ratio; some theorems on r; rational approximations to irrational numbers; continued fraction representation of τ; convergents; misconceptions about τ.
CHAPTER ELEVEN Bees, Honeycombs, Bubbles, and Mud Cracks	231	(23)
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	(41)
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	(14)
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	(27)
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	(5)
BIBLIOGRAPHY	341	(16)
INDEX	357