Mathematics in Nature by J. A. Adam is an excellent collection of
many natural phenomena that can be readily observed in nature together
with mathematical arguments and models that help to explain them. The
scope of the presented phenomena is very wide: in addition to more
common topics such as Fibonacci numbers and animal coats, Adam
discusses rainbows, snowflakes, meandering rivers, spider webs, bird
flight, clouds, ship waves, tidal bores, mud cracks. This
list could go on for quite a while! The book focuses on how these
phenomena and patterns can be explained mathematically by developing
and analyzing appropriate models. The mathematical depth varies quite
a bit from chapter to chapter and from problem to problem. In most of
the book, only algebra, geometry, trigonometry, and sometimes
elementary calculus are used; however, complex numbers appear occasionally as
do differential equations in the form of the heat and wave equations.
This book is not written as a textbook but rather as a supplement to a
textbook or simply to be read for fun and out of interest. Its
intended audience are math, science and engineering undergraduate
students and really anybody who is interested in, and not afraid of,
mathematical explanations of natural phenomena. Though not meant to be
a textbook, the author mentions in the preface of the paperback
edition that an instructor's manual with a set of problems is in
preparation, which would make it an attractive addition for
interdisciplinary modeling courses.
The book begins with an outline of what mathematical modeling
encompasses. Melting of snow balls is used as a simple example to
illustrate the modeling process and to demonstrate both strengths and
limitations of specific models. Two chapters on ballpark estimates and
problems of scale deal with questions such as how many blades of
grass there are on earth and why larger animals have lower pulse rates. Rainbows, halos, eclipses, contrail shadows and many other
optical phenomena are discussed next using geometric and physical
optics. Linear and nonlinear waves are covered in a sequence of
chapters. The topics range from sand dunes, ship waves, and pebbles
thrown into puddles to the waves generated by water striders. Clouds
and spider webs lead to a discussion of Kelvin-Helmholtz and Rayleigh
instabilities, while tidal bores motivate a discussion of nonlinear
solitary waves (which eventually leads to an interesting discourse of
how tides affect the declining length of days). Among the mathematical
concepts that are introduced in these chapters are the wave equation,
the Burgers equation, dispersion relations, phase and group velocities,
and deep and shallow water waves. Following a treatment of Fibonacci
numbers and the golden ratio, the author discusses hexagonal patterns,
foam and soap bubbles, meandering rivers, and trees (e.g. how high can
they grow, and how much light is intercepted by their leaves). Next
are various aspects of bird flight such as soaring, hovering, and the
V-flight of flocks of birds. The book ends with animal coats and the
role diffusion plays, or does not play, in their development. A short
appendix touches briefly on fractals.
I thoroughly enjoyed reading this book. It is exceedingly well written
and captures one's attention with many interesting questions and
explanations. The style in which the book is written varies quite a
bit. Some parts are written in light informal prose that involves few or no
mathematical expressions; other parts involve more lengthy mathematical
derivations that are, relatively speaking, more difficult to follow. Overall, I
found the style in which the natural phenomena and mathematical models are
explored and developed very engaging. Though readers may be familiar with some
of the covered topics such as Fibonacci numbers or dimensional analysis, the
breadth and scope of the phenomena included in this book make it a real gem. The
very comprehensive and extensive list of references to books and survey papers
is a highly valuable resource for anybody interested in exploring some of the
topics further. Given that the majority of the material requires only algebra
and trigonometry, this book also provides an excellent source of interesting applications
which could
be used in precalculus and calculus courses.
There are a few minor things which could be improved in this book. For
instance, a few additional diagrams or illustrations would often help.
In the text, references to books or articles are often made by the
author's name alone, even when that author appears with several books
and articles in the list of references. The book contains a number of
excellent color figures in the middle of the book but these are not
referred to in the main text. These are minor issues, however, and
overall I found the book very well and carefully written. I recommend
it highly!
Editor's note: John A. Adam has a new book, A Mathematical
Nature Walk, published by Princeton University Press in 2009. If you are
interested in reviewing this book, please
contact the Book Reviews editor.