Mathematics in Nature Modeling Patterns in the Natural World

ISBN-10: 0691127964
ISBN-13: 9780691127965
Edition: 2004
Authors: John A. Adam
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Description: 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  More...

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Book details

Copyright year: 2004
Publisher: Princeton University Press
Publication date: 9/10/2006
Binding: Paperback
Pages: 392
Size: 6.50" wide x 9.50" long x 1.00" tall
Weight: 1.584
Language: English

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

John A. Adam is professor of mathematics at Old Dominion University. He is the author of "A Mathematical Nature Walk" and "Mathematics in Nature", and coauthor of "Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin" (all Princeton).

Preface to the Paperback Edition
Preface: The motivation for the book; Acknowledgments; Credits
Prologue: Why I Might Never Have Written This Book
The Confluence of Nature and Mathematical Modeling
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
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
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
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
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
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
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
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
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])
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
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
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
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?
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
Bibliography
Index

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