Briefly Noted

November 1, 2003

Short Book Reviews

Origami Design Secrets: Mathematical Methods for an Ancient Art. By Robert J. Lang, A.K. Peters Ltd., Natick, Massachusetts, 2003, 585 pages, $48.00.

Any adult who has folded a newspaper into a hat, or any child who has tossed a paper airplane across a classroom, is an origamist. I doubt that aficionados of origami are as numerous as passionate orchid growers, who turn up in the thousands at their national meetings, but dedicated origamists are certainly as enthusiastic. They have their clubs, their specialties, associations, Web sites, publications, meetings, and exhibitions.

Beyond that, it turns out that origami is a branch of applied mathematics. Proof: Origami has its definitions, fundamentals, theorems, algorithms, and even applications to the non-fun world (e.g., to the design of large space telescopes). Q.E.D. To read a brief survey of these aspects of paper folding, readers should take a look at Barry A. Cipra's splendid survey article "In the Fold: Origami Meets Mathematics" (SIAM News, Vol. 34, No. 8, October 2001).

Robert J. Lang, author of the present work, is a PhD in applied physics from Caltech and a world-class origamist. He is a pioneer in computational origami and in the development of formal design algorithms for folding. He is currently a full-time artist and a consultant on origami and its applications to engineering problems. Though stressing the algorithmic through detailed instructions and step-by-step diagrams for several dozen origamized objects, this magisterial work, splendidly produced, covers all aspects of the art and science. If you want to make a paper tree-frog, an orchid blossom, or a Black Forest cuckoo clock to wow your children, if you need evidence to convince a philosopher of mathematics that origami is a subfield of geometry, Lang's book is where you should go.


Four Colors Suffice: How the Map Problem Was Solved. By Robin Wilson, Princeton University Press, Princeton, New Jersey, 2002, 262 pages (including lots of pictures, both of people and of maps), $24.95.

I suppose that most readers of SIAM News know what the notorious four-color problem is and that it was "solved" only fairly recently. For those in need of a reminder: Prove that every map drawn on the plane can be colored with at most four colors so that neighboring countries are colored differently.

Robin Wilson, Fellow of Keble College, Oxford, has put together an entertaining and instructive history of the problem. Beginning with Euler, and proceeding through all the mathematical worthies who've had a finger in the problem, the book goes beyond Kenneth Appel, Wolfgang Haken, and John Koch, who, in 1977, published their "proof" of the problem. This proof required a substantial computer assist to deal with an "unavoidable set of 1936 reducible configurations" (lowered in 1994 by Robertson, Sanders, Seymour, and Thomas to 633 reducible configurations).

Over the past decades work on this problem has been attended by a bit of personal rancor. Mathematicians have questioned the published details. Philosophers have questioned the methodology because of the computer usage. Those of us who batten on scandal and who are less interested in the details of reducible configurations than in what a proof "is" or "should be" would wish that Wilson had written a bit more on these topics.

---Philip J. Davis

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