Tracking the Automatic Ant

  • David Gale
Springer: 1998. pp.241$30, £22.50

The Mathematical Tourist

  • Ivars Peterson
W. H. Freeman: 1998. 256pp.$14$14.95 (pbk)

The ant is a fascinating object of study, and anyone who has hefted the compendious, weighty, (and Pulitzer prize winning) tome The Ants by Bert Hölldobler and Edward O. Wilson, surely has tremendous regard for these creatures. Mathematicians, too, have now taken the ant to their hearts, and it provides one of the common links between the two books under review, which are compilations of material that has previously appeared in Science News and The Mathematical Intelligencer respectively. The former book is an updated version of Peterson's very successful first edition, published almost 10 years ago; the second collects the “Mathematical Entertainment” columns that appeared in the Intelligencer between 1991 and 1996, while Gale was editor.

The books differ in their emphasis. Peterson sets out to explain to the general reader (one assumes that the general reader of Science News has a modest mathematical background) some of the startling discoveries and recent advances that have caused excitement in mathematics. This is a difficult task, because it requires the author to have the ability not only to grasp thoroughly the mathematical concepts involved, but also to translate these concepts into entertaining prose for the non-specialist reader.

Mathematicians are really monks, but rare indeed is the illuminating manuscript. It is thus a pleasure to find a journalist like Peterson who can write so lucidly, on rare occasions perhaps a trifle earnestly, about knots and DNA, about crystallography and Penrose tilings, about discoveries shedding light on the relationship between four-dimensional geometry and the physical theories of the nature of time, and about cryptography and prime factorization.

Occasionally, the author breathlessly takes on Baedeker: “beauty spots”, “unnamed wonders”, and “innumerable strange sights to view” can be found if the tourist in the Mandelbrot Set explores the area with real part between 0.26 and 0.27 and imaginary part between 0 and 0.01i (“or why not try -0.76 to -0.74 and 0.01i to 0.03i?”). The Mathematical Tourist conveys vividly the excitement, the usefulness, and the sheer beauty of the subject.

Gale, in contrast, writes for the pleasure-seeking mathematician, professional and amateur. These columns have a similar form to those of Martin Gardner and Ian Stewart in Scientific American, with an emphasis on recreational mathematics, though as we know, this is often underpinned by eminently serious stuff. Extracting tidbits for mention here is difficult to do fairly; the book contains as much to savour on each page as a compendium of short stories by Saki.

Try for instance the following problem. Each of two boxes contains an integer printed on a card; the two integers are known to be distinct. You open one of the boxes at random, and then have to guess whether the other contains a larger or a smaller integer. Is there anything you can do to give yourself a better than even chance of guessing correctly? The surprising answer is yes. Or what about the (simple) theorem that a sufficiently high power of two will have leading digits that encode the works of Shakespeare? “To be ⃛” for example appears at the start of 29965483 = 20150205⃛. A five-line algorithm is provided for computing the first such power to yield a chosen sequence, which raises interesting possibilities: perhaps a unique personalized present for that in-law who has everything (provided of course your PC can cope).

What becomes apparent from reading these books is the role now played by computers in mathematics, suggesting a shift in nature of the subject from deductive science to empirical science. For the technophobe this may appear discouraging, perhaps even alarming, with the discovery that machines have now been used to prove theorems. But the ability to amass vast amounts of experimental data certainly has enormous ramifications; without it much of the territory explored in these books would remain uncharted.