Table of Contents
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
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
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
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.