La Perfezione Visible: Matematica e Arte by Michele Emmer, and: L’occhio di Horus: Itinerari Nell’Immaginario Matematico ed. by Michele Emmer (review)

ric space, but others can be used to model complex natural shapes, such as coastlines and mountains. Chaos and fractal geometry go handin -hand. Both fields deal with intricately shaped objects, and chaotic processes often produce fractal patterns. This fascinating book should be of interest to scientists, mathematicians, programmers and artists interested in the field of chaos and fractal geometry . This would also be a good textbook for students and teachers. For those readers new to these topics, the book starts gradually by introducing concepts of self-similarity, fractal snowflakes and fractal dimensions. Other topics discussed later in the book include pink noise, Brownian motion, Cantor sets, multifractals, iteration, bifurcation maps, the Mandelbrot set, Fibonacci numbers, percolation and cellular automata. Ample diagrams indicate the graphical results of the formulas and mathematical theories. My favorite sections discuss fractals in number theory-an area that may be new to readers familiar only with the standardJulia and Mandelbrot sets so richly illustrated on computer-art posters and T-shirts. The best section of all discusses the enigmatic Morse-Thue number sequence and its amazing fractal properties . This book has something for beginners and for advanced students of fractals and chaos. HUYGENS AND BARROW, NEWTON AND HOOKE byV. 1.Arnol'd. Birkhauser, Boston, MA, U.S.A., 1990. 118 pp., illus. Paper, $19.95. ISBN: 0-8176-2383-3. Reviewed fly Robert S. Lansdon, 3830 Annapolis Ct., So. San Francisco, CA 94080, U.S.A. Subtitled Pioneers in Mathematical Analysisand Catastrophe TheoryJrom Eooloents to Quasicrystals, this volume was translated from the Russian by Eric]. F. Primrose. Consisting of an extension and update of nonspecialist talks given by Vladimir Arnol'd during 1988 and 1989, this small book is a delight of mathematical exposition and history, with high-quality color plates of computer-generated patterns from dynamical systems and of computed views of quasicrystals, including the Fourier transforms (essentially diffraction patterns) of the quasicrystals . Figure 41 shows a seven-sided tiling that shows up as a 'stochastic web' in simulations of particle diffusion in plasma physics [1]. This work can be seen as very much in the spirit of Kepler, combining the classification of solid shapes and questions about real crystals with the theory of motion of celestial bodies. (Also related are fluid dynamic models of stars.) Arnol'd uses Plato's beloved icosahedron and pentagons to explain the five-fold symmetric tiling of the plane. (Such almost-periodic tilings are often called quasicrystalline, by analogy with the classical symmetry groups of crystal classification.) Arnol'd alludes to six-dimensional space-the natural habitat of the five-fold quasicrystal. Figure 38 shows the tiling of Roger Penrose that uses three types of tile to aperiodically tile the plane [2]. Figure 40 illustrates a quasiperiodic Penrose tiling from a Markov partition of the torus. Markov partitions of tori are used to study the chaotic behavior of orbits on tori by means of geometrical transformations and sequences of symbols. Besides weaving together a number of mathematical themes, Arnol'd concentrates on the history of a rich period of mathematical and scientific development. The textual organization is motivated as much by theme as by chronology, in keeping with Arnol'd's encyclopedic mathematical knowledge , deep understanding of analogies and connections, mastery of discourse and sense of intellectual fun. The book is as much an esthetic treat as a monograph. The author dips into three generations of history, starting with Newton's teacher Isaac Barrow (one of the fathers of calculus), bringing to light connections and providing insights that would no doubt have amazed the natural philosophers and geometers of the book's title. In several places Arnol'd shows that important results were known many years before they were cast into the forms in which we know them. 'Newtonian' gravitation and the wave nature of light provide several examples . Arnol'd seems to enjoy showing that results attributed to mathematicians of the late nineteenth and the twentieth centuries were known to Newton (although Newton might not have been aware of how much he knew). The pendulum, soap films and the Platonic solids provide historical and mathematical material that the author integrates with anecdotes that illustrate the society, personalities...