Mathematics in Nature:
Modeling Patterns in the Natural World

John A. Adam

Table of Contents

Preface: Motivation for the book; acknowledgements.

Chapter 1: The confluence of nature and mathematical modeling: (i) confluence; examples and qualitative discussion of patterns in nature; organization of the book; (ii) modeling: philosophy and methodology of modeling. Appendix: a mathematical model of snowball melting.

Chapter 2: 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 and growth, newspaper pi, the atmosphere, earth tunnel, “band” tectonics, mountains, cloud droplets, the “Black Cloud”.

Chapter 3: Shape, size and similarity: the problem of scale. 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 pi theorem; various examples. Appendix: models based on elastic similarity.

Chapter 4: 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. Appendix: a word about solid angles.

Chapter 5: 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.

Chapter 6: 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.

Chapter 7: 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. Appendix: more mathematics of ship waves.

Chapter 8: Stability. The 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 9: 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.

Chapter 10: 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

Chapter 11: Bees, honeycombs, bubbles and mudcracks. 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; mudcracks and related geometric theorems. Appendix: the isoperimetric property of the circle.

Chapter 12: 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. Appendix: the statics and bending of a simple beam: basic equations.

Chapter 13: 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.

Chapter 14: 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. Appendix: the analogy with the normal modes of rectangular and circular membranes.