Reseñas proporcionadas por Syndetics
CHOICE Review
So vast a legacy have Euclid's Elements spawned that sometimes, aside from a manifest common ancestry, little mutual resemblance remains detectable between present day descendents. The millennia have seen geometry globalized, projectivized, dimensionalized, algebracized, arithmetized, complexified, discretized, and computerized. A lifetime would hardly suffice now to pin down the precise range and scope of the term "geometry" in modern mathematics. Still, one can list themes whose presence and confluence push mathematical disciplines toward the geometrical: incidence, separation, symmetry, dimension, distance, measure, angle. The spirit of Martin's Geometric Constructions, among the volumes under review, lies closest to the Euclidean source. Though the title might suggest just a compendium of specific, ever more baroque, ruler and compass constructions (of the sort familiar from midcentury college geometry texts), the author has a different purpose. Rather, he invites the reader to explore and characterize the comparative constructive strengths of various sets of construction tools: ruler and compass; ruler alone; compass alone; ruler and rusty compass; marked ruler, "tomahawk," etc. The theorems forming the book's backbone characterize algebraically the coordinates of all the points that one may construct with a given set of tools. This reduces questions about the solvability of classic construction problems to mere algebra exercises (but says little about how to find elegant or practical constructions). One can find some of these theorems in older sources (this reviewer first encountered this stream of ideas in H. Dorrie's wonderful classic 100 Great Problems of Elementary Mathematics, CH, Feb'66); this book's advantage is its unified viewpoint. The author had the use of software (The Geometer's Sketchpad) to write the book, but he does not make the tie-in very explicit for the student. Henle's Modern Geometries develops the basics of Euclidean and non-Euclidean geometry, but delays consideration of the traditional axiomatic method nearly to the end in favor of analytic tools. Many recent works have a similar slant, but this book has certain features to recommend it: an explicit treatment of Klein's Erlanger Programme; three chapters on solid geometry, including quaternions; a nice treatment of wallpaper symmetry; Bachmann's axioms (which make transformations rather than points and lines the geometric primitives); and a healthy dose of asides concerning the significance of geometry and its history to physics, philosophy, and art. Rich in detail and skillfully set forth, this book should appeal to a wide range of undergraduates. Cromwell writes that he intends Polyhedra to fill a gap in the literature, between merely recreational treatments and instructions for making models on the one side, and advanced treatments that emphasize general theorems in all dimensions on the other. Thus he offers a rigorous mathematical of polyhedra in three dimensions. Though not a history, the book has a wealth of historical detail and a mostly historical organization. So much has he to say concerning the mathematics of the ancients that Euler's formula does not appear for nearly 200 pages, but then things start moving quickly. Among the book's high points, there is a chapter on rigidity and flexibility of polyhedra, which discusses a classic theorem of Cauchy and a spectacular modern counterexample due to Connelly, and another on stellations of the icosahedron. Beautifully illustrated and immaculately produced, the book should bring pleasure and enlightenment to all readers. Flavors of Geometry consists of four expository geometrical essays by eminent authorities. They treat convexity (K. Ball); hyperbolic geometry (J. Cannon, W. Floyd, R. Kenyon, W. Parry); Henon mapping dynamics and related matters (J. Smillie and G. Buzzard); and randomized algorithms for estimating volume (B. Bollobas). The first two essays, especially, should attract undergraduate attention. Ball organizes his treatment of convexity around a broad slogan, "all convex bodies behave a bit like Euclidean balls." He produces both positive and negative evidence for this, culminating in a treatment of Dvoretzky's famous and deep theorem that high-dimensional convex bodies always have high-dimensional, nearly elliptical cross-sections. Hyperbolic geometry and non-Euclidean geometry mean the same, except that the former suggests a research tool in topology and group theory, while the latter designates an undergraduate topic. The essay here concisely and usefully bridges the gap. For upper-division undergraduate and graduate students. Mathematicians maintain a distinction between the "elementary" and the "simple." Indeed, a paucity of tools or a decision to avoid tools renders many an "elementary" subject fiendishly complex. In Design Theory, Lindner and Rodger carefully and judiciously choose their tools to reveal their subject's underlying simplicity without sacrificing accessibility. Viewed as a geometrical subject, design theory concentrates exclusively on the themes of incidence and symmetry, but this treatment ignores the role of symmetry; the index does not even contain the word "group." As the authors say themselves, "the best feature of this book is its pictures." More than the illustrations in any other book in the literature, these will help readers to absorb easily a D. V. Feldman University of New Hampshire
