Calculating Queen Dido

Mathematicians are not natural communicators, so the effort that Stefan Hildebrandt and Anthony Tromba have made to popularise key ideas in mathematics is innately laudable. In their book, they explain the crucial role of geometry in natural phenomena, how shapes and forms in nature arise because of the "principle of economy of means", otherwise known as the calculus of variations.

Their thesis is that of Leibniz, in his Essays on Theodicy, where, he says "our world is organised to be the best of all possible worlds ... (which) is created in such a way that a preestablished harmony exists between freedom and necessity". All imperfections, in the authors' view, are attributable to the fact that "this world, being merely the best selection out of what is possible, may be much worse than we might hope for".

This philosophical premise is used to motivate the existence of various shapes and forms in the natural world such as soap bubbles, liquid drops, cracks and even living organisms. We are told how simple mathematical rules, such as the minimisation of distance between two points or the maximisation of area within a perimeter, can respectively explain phenomena as diverse as the paths of light rays, or the circular shapes of medieval walled cities. This last has a particularly fascinating history, to do with the so-called problem of Queen Dido: when Dido arrived as an exile in Carthage, she was sold land by the erstwhile king on condition that she could buy only as much land as she could enclose by the skin of an ox. By realising that the maximal area in an isoperimetric problem was that of a circle, she was able, from a rather unpromising beginning, to acquire about 60 acres of land. The authors use the minimisation of distance between several points to discuss the well-known travelling salesman problem where it is the job of the salesman to find the shortest route between a set of cities, a problem that gets unpleasantly complicated as the number of cities gets large.

They also use the brachysto-chrone problem (that of finding the trajectory of quickest descent between two points in a vertical plane) to introduce the geometry of cycloids, and then relate this to the trajectory of a perfect pendulum. Having so far dealt with problems for which the ideal is also the real solution, the authors then use soap films to illustrate the situation when complexity gets in the way of the ideal being the real, or the only solution; nevertheless they demonstrate that these "imperfect" solutions are still the "best selection of what is possible", in the spirit of Leibniz. En route, readers are given personalised and entertaining accounts of the history of science, ranging from academic rivalries to actual wars, with illuminating snippets concerning a host of subjects from the music of the spheres to the burning mirrors of Archimedes that defeated the Romans. Interspersed with all of this are the principles of the mathematical proof, the necessity of proving existence and uniqueness of mathematical solutions, and (occasionally rather abstruse) examples of the methods of mathematical logic.

When contrasted with this panoply of mathematical minutiae, the world of the physical sciences is rather poorly represented. Newtonian and relativistic dynamics are crammed together in the epilogue, the section on crack propagation is almost entirely pictorial, little mention is made of physical theories of foams and the section on crystal growth is limited to an obscure account of the Wulff construction. This is especially evident in the chapter on soap films, when 40 pages are devoted to the mathematical details of many possible (and some improbable) geometries that can arise in such films, with scarcely a page devoted to surface tension, the physical cause of the phenomenon. Another major drawback is that the figures referred to in the text are rarely where they are supposed to be, a particular impediment when intricate geometries are being discussed. Another important issue about this otherwise worthy book concerns its readership; far too technical in parts to hold the lay reader's undivided attention, it is much too elementary to interest the specialist. But this is only to be expected of a book that seeks to popularise the most abstract of all the sciences: the ability to make sections of it entertaining is perhaps more important than the inability to make all of it absorbing.

Anita Mehta is an EPSRC visiting fellow, Clarendon Laboratory, Oxford.