Mathematics in Nature is a simple title for a complex book. It suggests an excursion into the Fibonacci numbers, or perhaps a predator-prey model. But this is so much more. Extensively researched, this is 360 pages of mathematical explanations for readily observed events in nature.
John Adam has combined his interests in the great outdoors and applied mathematics to compile one surprising example after another of how mathematics can be used to explain natural phenomena. And what examples! Why do spotted cats have striped tails, but no striped cats have spotted tails? Why do clouds often form in straight lines? When chased by a bear on a mountain, should you run uphill or downhill? Why is the sky blue, or, more interestingly, why isn't it violet?
Before diving into all of these and many more interesting questions, Adam takes a couple of chapters to address more general issues concerning the importance of estimation, and the value and limitations of mathematical models. Adam's genesis for this text was developing a class in mathematical modeling, and for those that are new to the genre, this is a great introduction. Using the example of a melting snowball, the author walks through the sorts of issues faced by a mathematical modeller, including the operating assumptions made, the first set of questions asked, the selection of appropriate equations, and then the process of refinement as limitations of the model become apparent and the reasonableness of the solutions is considered. One could do worse than to use this example in the first meeting of an introductory mathematical modeling course.
After setting the table, Adam brings out course after course of models of natural phenomena. Some are familiar to any veteran of a liberal arts math course, although explained in greater depth. Problems of scale are discussed, such as why King King couldn't exist, and why a large animal has a slower heartbeat than a smaller one. The Fibonacci numbers are here, including explanations of why they occur so frequently, proofs concerning ratios of terms in the Fibonacci sequence, and extensive citations. There's also an involved examination of whether or not a honeycomb is, as common knowledge holds, formed of hexagons because they're the optimal way to enclose volume with a given perimeter. But many more topics are much less familiar. In addition to those mentioned above, there are discussions of the width of the glitter path of the setting sun on a lake, the proper angles for double and triple rainbows, the meandering behavior of rivers, tidal bores, and the calm spot in the center of the ring of concentric circles formed by throwing a pebble into a pond.
Adam suggests in his preface that his intended audience consists of "undergraduate students in mathematics, science, and engineering (and their professors)," "high school teachers and their students who may have the opportunity for 'mathematical enrichment'," and "anyone interested in the beauty of nature, regardless of their mathematical background." There are numerous non-mathematical discussions of the phenomena and the results of the models, but this book will be far more valuable to someone who can follow at least some of the mathematics. Even though many of the chapters push some of the involved mathematics to an appendix, there's still plenty of mathematics worked through the text to frustrate a novice. The level varies from differential calculus and introductory differential equations to elliptical integrals and complex (as in complex numbers) series. That's not to say it's a thorough mathematical treatment, either. There are lots of phrases like, "It can be shown that" or "It turns out that the value of the constant is ..." In trying to be all things to all people, the text is somewhat uneven in tone, allowing conversational discussions to obscure the mathematics at some points, and losing the main ideas in a forest of mathematical details at others.
This isn't necessarily a bad thing, as the constant switching between conversation, calculus level mathematics, and specific topics from applied mathematics seems to mirror Adam's own method of approaching these problems and conveys his enthusiasm for this subject. It also illustrates the potential for an instructor's guide. The uneven level of presentation and requisite mathematical experience presents opportunities for these examples to be used in several different courses, but the text is organized by topic, not mathematical level. There is an excellent discussion of how tides are slowing down the earth's rotation, using not much more than introductory calculus topics, but if you're teaching calculus and searching for an interesting example to use in class, you wouldn't know to look for it. An initially reasonable but incorrect model for the movement of sap up through a tree is included, but an instructor on the hunt for such an example would have to stumble on it (or read this review). An instructor's guide would make this book a top notch classroom resource.
Other minor quibbles include the fact that while there are color plates, there are no references to them in the relevant sections of the text, and that there are instances when a labeled diagram could really help out those not familiar with the standard physical constants, or with unfamiliar phenomena, such as tidal bores. And although a Dimetrodon provides an interesting application of how mathematics can help paleontologists determine the use of the sail on its back, it is not a dinosaur, as my first grade son has repeatedly told me.
But these certainly aren't worth ruminating over. John Adam has done a great deal of reading and exposition, indulging his passions to create this compilation of mathematical models of natural phenomena, and the sheer number of examples he manages to cram into this book is testament to his efforts. There are other texts on the market which explore the connection between mathematics and nature, such as Nature's Numbers by Ian Stewart, but none this wide-ranging. And besides, Adam has probably already cited them. For anyone searching for a real mathematical examination of the connection between mathematics and nature, this is a great place to start.
Appendix: There's an unexpected message that the numerous examples in this book make manifest. Trigonometry is important. In my experience, students arrive for their first year of college mathematics less and less well-versed in trigonometry as the years go by. I don't know if they're spending more time with probability and statistics, or simply spending more time on polynomials, but I find myself spending more and more time reviewing basic trigonometric facts that I used to take for granted. But from problems of scale to river meanders, so many of the models involve trigonometry one could make the argument that you can't understand the world around you without being familiar with it. For an instructor to put any of these examples to use in a classroom, the students have to be more familiar with sines and cosines than simply knowing their definitions. If anyone wanted to make a case for a return to an emphasis on trigonometric functions, this book provides it.
Steve Morics (Steven_Morics@redlands.edu} is Associate Professor of Mathematics at the University of Redlands in southern California. His interests include combinatorics, mathematics and politics, music, and getting outdoors as much as possible. He'll be a bit more observant as a result of reading this book.