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.