Browse Books

Reviewer: Harvey Cohn

Rarely can a single work serve as both a guide to a serious innovative course and as an attractive coffee-table book. This book has that distinction, and is either way irresistible to a wide range of scientists. It embraces many concepts: a one-dimensional theory of linkages, a two-dimensional (2D) folding theory, and a three-dimensional (3D) theory of unfolding polyhedra. In each case, it is surprising to see the book show how many figures of art, science, and history have been speaking prose. A well-known linkage is the pantograph, used by Thomas Jefferson to copy (and enlarge) simultaneous signatures. Indeed, Alfred B. Kempe (of four-color fame (1877)) found a bad proof for the preemptive result that any signature (indeed, curve) can be manufactured by linkages. For mechanical purposes, notably, James Watt's linkage, in 1784, converted circular motion (almost) into linear motion in a discovery that was as laudable as the steam engine. Later on, in 1864, Charles N. Peaucellier found an exact linkage. Current interest may more likely reside in robotry. Paper folding is the core of the art, and it is hard to match the significance of the fact that a square of paper can be folded three times and a five-pointed star can be cut out in one straight cut. This was demonstrated by Betsy Ross to General Washington, who then agreed to put the stars in the American flag. A spectacular result in pure mathematics is that of H. Abe, who showed that angle trisection is possible by creasing and folding (1980)-normally, compasses and rulers construct quadratic radicals. Karl F. Gauss (1796), and later Pierre Wantzel (1837), demonstrated that trisection requires a cubic radical. Abe produced it by folding: for example, two points and two lines can be folded so each point finds a line. The section on unfolding polyhedra is the deepest. Albrecht Duehrer (1525) was the pioneer, unfolding an Archimedian polyhedron of 38 faces onto a plane without overlapping; this suggests an unsolved problem of what can be done to an arbitrary convex polyhedron. Mainstream geometric techniques and theories of rigidity, Gaussian curvature and geodesics, come into play in ways too numerous to summarize. For a computer scientist, the thrill will surely be enhanced by the presence of nondeterministic polynomial time (NP) and NP-complete problems. Any mathematician should value the book for its challenges of geometric imagination. Online Computing Reviews Service