Mathematics and Optimal Form: Coll.Pa

As children we all asked similar questions, "Why doesn't the tree keep on growing", "Why is a whale/fish/snake shaped as it is?" "Why can't I fly?" As we grow older we stop asking the questions but at the same time do not know the answers. This book goes a long way not only on explaining why animals, plants, stars, planets and dust have certain shapes. It also is useful in explaining why giraffes don't grow to be 800 feet tall or why a tree isn't as big as a skyscraper.But that is only one part. The other discussion concerns the recurring shapes found in nature (and apparently throughout the Universe), and I am talking about more than the proverbial snow flake. Mathematics is kept to a minimum and pictures, diagrams, illustrations and drawings are plentiful.The author attempts to tell WHY certain shapes seem universal and then migrate that concept from the inantimate to the living world to explain shapes in nature. This is a book that deserves multiple, leisurely readings.

Hildebrandt and Tromba provide in Mathematics and Optimal Form an excellent survey of geometric optimization problems and their relation to physical and biological science, in the spirit of D'arcy Thompson's On Growth And Form and written for the lay reader. Mathematics and Optimal Form sets out to explain some of the recurring and on their face complex structures seen in nature.It is rare to see among mass-audience science books illustrations liberally included unless they are there for a "gee whiz" factor. (Case in point: Hawking's Universe in a Nutshell. ) In Mathematics and Optimal Form the illustrations, including graphs, diagrams, and photographs of microorganisms, soap films, and buildings are present on nearly every page and serve to clarify and truly illuminate the text.The section on minimal surfaces is especially strong. The authors manage to include a basic introduction to the topology of manifolds and give an enlightening history of Plateau's Problem and related work including applications to architecture. (Additionally, this chapter includes some spectacular photographs!)History is the focus of two early chapters. The approach to the Greek geometers, a few pre-Enlightenment mathematicians, and development of the calculus is in places lacking relevance (and is better treated elsewhere.) There are a few high points of this section, Heron's theorem on the shortest path from a point to a line to another point being one of them; this could have been included in a near-absent section on geodesics, and the brachistochrone (and related) problems--the chronological approach did not work here and ought to have been redone or avoided.The absence of formal or semi-formal mathematics, or even equations, is frustrating. While the book is written for a lay audience it is written for the mathematically literate. Basic literacy including familiarity with the differential calculus should have been assumed. The authors are capable and clear writers; inclusion of formulae and a few derivations would only have served to enhance the book. At the very least, it is ridiculous to mention things like "differential equation" without supplying one--anyone who knows what a differential equation is must must have some familiarity with them--or at least explaining what it and other mentioned but unused concepts are. The authors do provide notes and references for those seeking more mathematical detail.Overall, this was a very worthwhile read. Unlike most humans it could benefit from putting on a little weight and an inflammed appendix.